From pas at lists.imstat.org  Sun Jan  4 01:26:14 2009
From: pas at lists.imstat.org (Probability Abstract Service)
Date: Sun, 04 Jan 2009 08:26:14 +0100
Subject: [PAS] Probability Abstracts 107
Message-ID: <E07CFFE9-74D6-49DF-B2E5-CCA7FD04FA2D@unimi.it>

Probability Abstracts 107
This document contains abstracts 7696-7953
from November-1-2008 to December-31-2008.
They have been mailed on Jan 4, 2009.


This letter can be also found on line at
http://pas.imstat.org/Letters/letter_107.shtml

Wishing you all a great 2009!
stefano


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7696. LARGE GAPS BETWEEN RANDOM EIGENVALUES

Benedek Valk\'o and  B\'alint Vir\'ag

We show that in the point process limit of the bulk eigenvalues of
$\beta$-ensembles of random matrices, the probability of having no  
eigenvalue
in a fixed interval of size $\lambda$ is given by $$
(\kappa_\beta+o(1))\lambda^{\gamma_\beta} \exp(-
\frac{\beta}{64}\lambda^2+(\frac{\beta}{4}-\frac18)\lambda) $$ as
$\lambda\to\infty$, where $$ \gamma_\beta={1/4}(\frac\beta{2}+\frac{2} 
{\beta}-
3). $$ This is a slightly corrected version of a prediction by Dyson.  
Our proof
uses the new Brownian carousel representation of the limit process, as  
well as
the Cameron-Martin-Girsanov transformation in stochastic calculus.


  http://arxiv.org/abs/0811.0007

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7697. A CRITERION FOR THE VIABILITY OF STOCHASTIC SEMILINEAR CONTROL  
SYSTEMS  VIA THE QUASI-TANGENCY CONDITION

Dan Goreac

In this paper we study a criterion for the viability of stochastic  
semilinear
control systems on a real, separable Hilbert space. The necessary and
sufficient condition is given using the notion of stochastic quasi- 
tangency. As
a consequence, we prove that approximate viability and the viability  
property
coincide for stochastic linear control systems. The paper generalizes  
recent
results from the deterministic framework.


  http://arxiv.org/abs/0811.0098

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7698. COMPETITIVE OR WEAK COOPERATIVE STOCHASTIC LOTKA-VOLTERRA  
SYSTEMS  CONDITIONED TO NON-EXTINCTION

Patrick Cattiaux (IMT) and  Sylvie M\'el\'eard (CMAP)

We are interested in the long time behavior of a two-type density- 
dependent
biological population conditioned to non-extinction, in both cases of
competition or weak cooperation between the two species. This  
population is
described by a stochastic Lotka-Volterra system, obtained as limit of
renormalized interacting birth and death processes. The weak cooperation
assumption allows the system not to blow up. We study the existence and
uniqueness of a quasi-stationary distribution, that is convergence to
equilibrium conditioned to non extinction. To this aim we generalize in
two-dimensions spectral tools developed for one-dimensional  
generalized Feller
diffusion processes. The existence proof of a quasi-stationary  
distribution is
reduced to the one for a $d$-dimensional Kolmogorov diffusion process  
under a
symmetry assumption. The symmetry we need is satisfied under a local  
balance
condition relying the ecological rates. A novelty is the outlined  
relation
between the uniqueness of the quasi-stationary distribution and the
ultracontractivity of the killed semi-group. By a comparison between the
killing rates for the populations of each type and the one of the global
population, we show that the quasi-stationary distribution can be either
supported by individuals of one (the strongest one) type or supported by
individuals of the two types. We thus highlight two different long time
behaviors depending on the parameters of the model: either the model  
exhibits
an intermediary time scale for which only one type (the dominant  
trait) is
surviving, or there is a positive probability to have coexistence of  
the two
species.


  http://arxiv.org/abs/0811.0240

---------------------------------------------------------------

7699. ASYMPTOTICS FOR THE SURVIVAL PROBABILITY IN A SUPERCRITICAL  
BRANCHING  RANDOM WALK

Nina Gantert and  Yueyun Hu and Zhan Shi

Consider a discrete-time one-dimensional supercritical branching  
random walk.
We study the probability that there exists an infinite ray in the  
branching
random walk that always lies above the line of slope $\gamma-\epsilon 
$, where
$\gamma$ denotes the asymptotic speed of the right-most position in the
branching random walk. Under mild general assumptions upon the  
distribution of
the branching random walk, we prove that when $\epsilon\to 0$, the  
probability
in question decays like $\exp\{- {\beta + o(1)\over \epsilon^{1/2}}\} 
$, where
$\beta$ is a positive constant depending on the distribution of the  
branching
random walk. In the special case of i.i.d. Bernoulli$(p)$ random  
variables
(with $0<p<{1\over 2}$) assigned on a rooted binary tree, this answers  
an open
question of Robin Pemantle.


  http://arxiv.org/abs/0811.0262

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7700. DISPERSION OF VOLUME UNDER THE ACTION OF ISOTROPIC BROWNIAN FLOWS

Georgi Dimitroff and Michael Scheutzow

We study transport properties of isotropic Brownian flows. Under a  
transience
condition for the two-point motion, we show asymptotic normality of  
the image
of a finite measure under the flow and -- under slightly stronger  
assumptions
-- asymptotic normality of the distribution of the volume of the image  
of a set
under the flow. Finally, we show that for a class of isotropic flows,  
the
volume of the image of a nonempty open set (which is a martingale)  
converges to
a random variable which is almost surely strictly positive.


  http://arxiv.org/abs/0811.0276

---------------------------------------------------------------

7701. GREEDY POLYOMINOES AND FIRST-PASSAGE TIMES ON RANDOM VORONOI  
TILINGS

Leandro P. R. Pimentel and Raphael Rossignol

Let N be distributed as a Poisson random set on R^d with intensity  
comparable
to the Lebesgue measure. Consider the Voronoi tiling of R^d, (C_v)_{v 
\in N},
where C_v is composed by points x in R^d that are closer to v than to  
any other
v' in N. A polyomino P of size n is a connected union (in the usual R^d
topological sense) of n tiles, and we denote by Pi_n the collection of  
all
polyominos P of size n containing the origin. Assume that the weight  
of a
Voronoi tile C_v is given by F(C_v), where F is a nonnegative  
functional on
Voronoi tiles. In this paper we investigate the tail behavior of the  
maximal
weight among polyominoes in Pi_n for some functionals F, mainly when  
F(C_v) is
the number of faces of C_v. Next we apply our results to study self- 
avoiding
paths, first-passage percolation models and the stabbing number on the  
dual
graph, named the Delaunay triangulation. As the main application we  
show that
first passage percolation has at most linear variance.


  http://arxiv.org/abs/0811.0308

---------------------------------------------------------------

7702. QUENCHED INVARIANCE PRINCIPLE FOR THE KNUDSEN STOCHASTIC  
BILLIARD IN A  RANDOM TUBE

Francis Comets and  Serguei Popov and  Gunter M. Sch\"utz and  Marina  
Vachkovskaia

We consider a stochastic billiard in a random tube which stretches to
infinity in the direction of the first coordinate. This random tube is
stationary and ergodic, and also it is supposed to be in some sense
well-behaved. The stochastic billiard can be described as follows: when
strictly inside the tube, the particle moves straight with constant  
speed. Upon
hitting the boundary, it is reflected randomly, according to the  
cosine law:
the density of the outgoing direction is proportional to the cosine of  
the
angle between this direction and the normal vector. We also consider the
discrete-time random walk formed by the particle's positions at the  
moments of
hitting the boundary. Under the condition of existence of the second  
moment of
the projected jump length with respect to the stationary measure for the
environment seen from the particle, we prove the quenched invariance  
principles
for the projected trajectories of the random walk and the stochastic  
billiard.


  http://arxiv.org/abs/0811.0366

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7703. BIASED TUG-OF-WAR, THE BIASED INFINITY LAPLACIAN, AND COMPARISON  
WITH  EXPONENTIAL CONES

Yuval Peres and  G\'abor Pete and  Stephanie Somersille

We prove that if X\subset \R^n is compact in the path metric, Y\subset  
X is
closed, and F is a Lipschitz function on Y, then for each \beta \in \R  
there
exists a unique viscosity solution to the \beta-biased infinity  
Laplacian
equation
   \beta |\nabla u| + \Delta_\infty u=0 on X\setminus Y, where \Delta_ 
\infty u=
|\nabla u|^{-2} \sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j} . In the proof,  
we extend
the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and  
define the
\beta-biased \eps-game as follows. The starting position is x_0 \in X 
\backslash
Y. At the k-th step the two players toss a suitably biased coin (in  
our key
example, player I wins with odds of \exp(\beta\eps) to 1), and the  
winner
chooses x_k with d(x_k,x_{k-1}) < \eps. The game ends when x_k \in Y,  
and
player II pays the amount F(x_k) to player I. We prove that the value
u^{\eps}(x_0) of this game exists, and that \|u^\eps - u\|_\infty \to  
0 as \eps
\to 0, where u is the unique extension of F to X that satisfies  
comparison with
\beta-exponential cones. Comparison with exponential cones is a notion  
that we
introduce here, and generalizing a theorem of Crandall, Evans and  
Gariepy
regarding comparison with linear cones, we show that a continuous  
function
satisfies comparison with \beta-exponential cones if and only if it is a
viscosity solution to the \beta-biased infinity Laplacian equation.


  http://arxiv.org/abs/0811.0208

---------------------------------------------------------------

7704. SCALE-INVARIANT GROUPS

Volodymyr Nekrashevych and G\'abor Pete

Motivated by the renormalization method in statistical physics, Itai
Benjamini defined a finitely generated infinite group G to be scale- 
invariant
if there is a nested sequence of finite index subgroups G_n that are all
isomorphic to G and whose intersection is a finite group. He  
conjectured that
every scale-invariant group has polynomial growth, hence is virtually
nilpotent. We disprove his conjecture by showing that the following
self-similar groups of finite automata are also scale-invariant: the
lamplighter groups \F \wr \Z, where \F is any finite Abelian group; the
solvable Baumslag-Solitar groups BS(1,m); the affine groups GL(\Z,d)  
\ltimes
\Z^d. However, the conjecture remains open with some natural stronger  
notions
of scale-invariance for groups and transitive graphs. We construct
scale-invariant tilings of certain Cayley graphs of the discrete  
Heisenberg
group, whose existence is not immediate just from the scale-invariance  
of the
group.


  http://arxiv.org/abs/0811.0220

---------------------------------------------------------------

7705. SHARP TRANSITIONS IN MAKING SQUARES

Ernie Croot and  Andrew Granville and  Robin Pemantle and Prasad Tetali

In many integer factoring algorithms, one produces a sequence of  
integers
(created in a pseudo-random way), and wishes to rapidly determine a  
subsequence
whose product is a square (which we call a square product). In his  
lecture at
the 1994 International Congress of Mathematicians, Pomerance observed  
that the
following problem encapsulates all of the key issues: Select integers  
a_1, a_2,
 >... at random from the interval [1,x], until some (non-empty)  
subsequence has
product equal to a square. Find good estimate for the expected  
stopping time of
this process. A good solution to this problem should help one to  
determine the
optimal choice of parameters for one's factoring algorithm, and  
therefore this
is a central question.
   Pomerance (1994), using an idea of Schroeppel (1985), showed that  
with
probability 1-o(1) the first subsequence whose product equals a square  
occurs
after at least J_0^{1-o(1)} integers have been selected, but no more  
than J_0,
for an appropriate (explicitly determined) J_0=J_0(x). Herein we  
determine this
expected stopping time up to a constant factor, tightening Pomerance's  
interval
to $$[ (\pi/4)(e^{-\gamma} - o(1))J_0, (e^{-\gamma} + o(1)) J_0],$$  
where
$\gamma = 0.577...$ is the Euler-Mascheroni constant. We will also  
confirm the
well established belief that, typically, none of the integers in the  
square
product have large prime factors. We believe the upper of the two  
bounds to be
asymptotically sharp.


  http://arxiv.org/abs/0811.0372

---------------------------------------------------------------

7706. DISCRETE MULTIVARIATE DISTRIBUTIONS

Oleg Yu. Vorobyev and Lavrentiy S. Golovkov

This article brings in two new discrete distributions: multidimensional
Binomial distribution and multidimensional Poisson distribution. Those
distributions were created in eventology as more correct  
generalizations of
Binomial and Poisson distributions. Accordingly to eventology new laws  
take
into account full distribution of events. Also, in article its  
characteristics
and properties are described


  http://arxiv.org/abs/0811.0406

---------------------------------------------------------------

7707. EVENTOLOGICAL THEORY OF DECISION MAKING FOR STOCK MARKETS

Oleg Yu. Vorobyev and  Joe J. Goldblatt and Rebecca Finkel

The eventological theory of decision-making, the theory of eventfull
decision-making is a theory of decision-making based on eventological
principles and using results of mathematical eventology; a theoretical  
basis of
the practical eventology. The beginnings of this theory which have  
arisen from
eventfull representation of the reasonable subject and his decisions  
in the
form of eventological distributions (E-distributions) of sets of  
events and
which are based on the eventological H-theorem are offered. The  
illustrative
example of the eventological decision-making by the reasonable subject  
on his
own eventfull behaviour in the financial or share market is considered.


  http://arxiv.org/abs/0811.0420

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7708. LARGE DEVIATIONS FOR RANDOM WALKS IN RANDOM ENVIRONMENT ON A   
GALTON-WATSON TREE

Elie Aidekon (PMA)

Consider a random walk in random environment on a supercritical
Galton--Watson tree, and let $\tau_n$ be the hitting time of  
generation $n$.
The paper presents a large deviation principle for $\tau_n/n$, both in  
quenched
and annealed cases. Then we investigate the subexponential situation,  
revealing
a polynomial regime similar to the one encountered in one dimension.  
The paper
heavily relies on estimates on the tail distribution of the first  
regeneration
time.


  http://arxiv.org/abs/0811.0438

---------------------------------------------------------------

7709. ON THE MOMENTS AND DISTRIBUTION OF DISCRETE CHOQUET INTEGRALS  
FROM  CONTINUOUS DISTRIBUTIONS

Ivan Kojadinovic and  Jean-Luc Marichal

We study the moments and the distribution of the discrete Choquet  
integral
when regarded as a real function of a random sample drawn from a  
continuous
distribution. Since the discrete Choquet integral includes weighted  
arithmetic
means, ordered weighted averaging functions, and lattice polynomial  
functions
as particular cases, our results encompass the corresponding results  
for these
aggregation functions. After detailing the results obtained in [1] in  
the
uniform case, we present results for the standard exponential case,  
show how
approximations of the moments can be obtained for other continuous
distributions such as the standard normal, and elaborate on the  
asymptotic
distribution of the Choquet integral. The results presented in this  
work can be
used to improve the interpretation of discrete Choquet integrals when  
employed
as aggregation functions.


  http://arxiv.org/abs/0811.0468

---------------------------------------------------------------

7710. GAUSSIAN CORRELATION CONJECTURE FOR SYMMETRIC CONVEX SETS

He-Jing Hong and  Ze-Chun Hu

Gaussian correlation conjecture states that the Gaussian measure of the
intersection of two symmetric convex sets is greater or equal to the  
product of
the measures. In this paper, firstly we prove that the inequality  
holds when
one of the two convex sets is the intersection of finite centered  
ellipsoids
and the other one is simply symmetric. Then we prove that any  
symmetric convex
set can be approximated by the intersection of finite centered  
ellipsoids, and
thus the inequality holds for any two symmetric convex sets in any  
dimensional
$\mathbb{R}^n$, i.e. Gaussian correlation conjecture is true.


  http://arxiv.org/abs/0811.0488

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7711. FIRST HITTING TIME OF THE BOUNDARY OF THE WEYL CHAMBER BY RADIAL  
DUNKL  PROCESSES

Nizar Demni

We provide two equivalent approaches for computing the tail  
distribution of
the first hitting time of the boundary of the Weyl chamber by a radial  
Dunkl
process. The first approach is based on a spectral problem with  
initial value.
The second one expresses the tail distribution by means of the $W$- 
invariant
Dunkl-Hermite polynomials. Illustrative examples are given by the  
irreducible
root systems of types $A$, $B$, $D$. The paper ends with an interest  
in the
case of Brownian motions for which our formulae take determinantal  
forms.


  http://arxiv.org/abs/0811.0504

---------------------------------------------------------------

7712. GENERALIZED BESSEL FUNCTION OF TYPE D

Nizar Demni

We write down the generalized Bessel function associated with the root  
system
of type $D$ by means of multivariate hypergeometric series. Our hint  
comes from
the particular case of the Brownian motion in the Weyl chamber of type  
$D$.


  http://arxiv.org/abs/0811.0507

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7713. DIFFERENCES OF RANDOM CANTOR SETS AND LOWER SPECTRAL RADII

F. Michel Dekking and Bram Kuijvenhoven

We investigate the question under which conditions the algebraic  
difference
between two independent random Cantor sets $C_1$ and $C_2$ almost surely
contains an interval, and when not. The natural condition is whether  
the sum
$d_1+d_2$ of the Hausdorff dimensions of the sets is smaller (no  
interval) or
larger (an interval) than 1. Palis conjectured that \emph{generically}  
it
should be true that $d_1+d_2>1$ should imply that $C_1-C_2$ contains an
interval. We prove that for 2-adic random Cantor sets generated by a  
vector of
probabilities $(p_0,p_1)$ the interior of the region where the Palis  
conjecture
does not hold is given by those $p_0,p_1$ which satisfy $p_0+p_1> 
\sqrt{2}$ and
$p_0p_1(1+p_0^2+p_1^2)<1$. We furthermore prove a general result which
characterizes the interval/no interval property in terms of the lower  
spectral
radius of a set of $2\times 2$ matrices.


  http://arxiv.org/abs/0811.0525

---------------------------------------------------------------

7714. ADVERSARIAL SCHEDULING ANALYSIS OF GAME THEORETIC MODELS OF  
NORM  DIFFUSION

Gabriel Istrate and  Madhav V. Marathe and S.S.Ravi

In (Istrate, Marathe, Ravi SODA 2001) we advocated the investigation of
robustness of results in the theory of learning in games under  
adversarial
scheduling models. We provide evidence that such an analysis is  
feasible and
can lead to nontrivial results by investigating, in an adversarial  
scheduling
setting, Peyton Young's model of diffusion of norms. In particular,  
our main
result incorporates into Peyton Young's model.


  http://arxiv.org/abs/0803.2495

---------------------------------------------------------------

7715. ON THE DYNAMICS OF SOCIAL BALANCE ON GENERAL NETWORKS (WITH AN   
APPLICATION TO XOR-SAT)

Gabriel Istrate

We study nondeterministic and probabilistic versions of a discrete  
dynamical
system (due to T. Antal, P. L. Krapivsky, and S. Redner) inspired by  
Heider's
social balance theory. We investigate the convergence time of this  
dynamics on
several classes of graphs. Our contributions include:
   1. We point out the connection between the triad dynamics and a
generalization of annihilating walks to hypergraphs. In particular, this
connection allows us to completely characterize the recurrent states  
in graphs
where each edge belongs to at most two triangles.
   2. We also solve the case of hypergraphs that do not contain edges  
consisting
of one or two vertices.
   3. We show that on the so-called "triadic cycle" graph, the  
convergence time
is linear.
   4. We obtain a cubic upper bound on the convergence time on 2- 
regular triadic
simplexes G. This bound can be further improved to a quantity that  
depends on
the Cheeger constant of G. In particular this provides some rigorous
counterparts to previous experimental observations.
   We also point out an application to the analysis of the random walk  
algorithm
on certain instances of the 3-XOR-SAT problem.


  http://arxiv.org/abs/0811.0381

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7716. STOCHASTIC CAHN-HILLIARD EQUATION WITH SINGULAR NONLINEARITY  
AND  REFLECTION

Ludovic Gouden\`ege (IRMAR)

We consider a stochastic partial differential equation with  
logarithmic (or
negative power) nonlinearity, with one reflection at 0 and with a  
constraint of
conservation of the space average. The equation, driven by the  
derivative in
space of a space-time white noise, contains a bi-Laplacian in the  
drift. The
lack of the maximum principle for the bi-Laplacian generates  
difficulties for
the classical penalization method, which uses a crucial monotonicity  
property.
Being inspired by the works of Debussche and Zambotti, we use a method  
based on
infinite dimensional equations, approximation by regular equations and
convergence of the approximated semi-group. We obtain existence and  
uniqueness
of solution for nonnegative intial conditions, results on the invariant
measures, and on the reflection measures.


  http://arxiv.org/abs/0811.0580

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7717. ASYMPTOTIC ANALYSIS AND DIFFUSION LIMIT OF THE PERSISTENT  
TURNING WALKER  MODEL

Patrick Cattiaux (IMT) and  Djalil Chafai (IMT and  UPTE) and  S 
\'ebastien Motsch  (IMT)

The Persistent Turning Walker Model (PTWM) was introduced by Gautrais  
et al
in Mathematical Biology for the modelling of fish motion. It involves a
nonlinear pathwise functional of a non-elliptic hypo-elliptic  
diffusion. This
diffusion solves a kinetic Fokker-Planck equation based on an
Ornstein-Uhlenbeck Gaussian process. The long time "diffusive"  
behavior of this
model was recently studied by Degond & Motsch using partial differential
equations techniques. This model is however intrinsically  
probabilistic. In the
present paper, we show how the long time diffusive behavior of this  
model can
be essentially recovered and extended by using appropriate tools from
stochastic analysis. The approach can be adapted to many other kinetic
"probabilistic" models. Beyond the mathematical results, the aim of  
this short
paper is also to contribute to the diffusion of stochastic techniques  
in the
domain of partial differential equations. Also, the text aims to be very
accessible for non probabilists.


  http://arxiv.org/abs/0811.0600

---------------------------------------------------------------

7718. CLOSENESS OF CONVOLUTIONS OF PROBABILITY MEASURES

Bero Roos

We derive new explicit bounds for the total variation distance between  
two
convolution products of $n$ probability distributions, one of which  
having
identical convolution factors. Approximations by finite signed  
measures of
arbitrary order are considered as well. We are interested in bounds  
with magic
factors, i.e. roughly speaking $n$ also appears in the denominator.  
Special
emphasis is given to the approximation by the $n$-fold convolution of  
the
arithmetic mean of the distributions under consideration. As an  
application, we
consider the multinomial approximation of the generalized multinomial
distribution. It turns out that here the order of some bounds given in  
Roos
(2001) and Loh (1992) can significantly be improved. In particular, it  
follows
that a dimension factor can be dropped. Moreover, better accuracy is  
achieved
in the context of symmetric distributions with finite support. In the  
course of
proof, we use a basic Banach algebra technique for measures on a  
measurable
Abelian group. Though this method was already used by Le Cam (1960), our
central arguments seem to be new. We also derive new smoothness bounds  
for
convolutions of probability distributions, which might be of independent
interest.


  http://arxiv.org/abs/0811.0622

---------------------------------------------------------------

7719. AN ASYMPTOTIC THEORY FOR RANDOMLY-FORCED DISCRETE NONLINEAR  
HEAT  EQUATIONS

Mohammud Foondun and Davar Khoshnevisan

We study discrete nonlinear parabolic stochastic heat equations of the  
form,
$u_{n+1}(x) - u_n(x) = (\sL u_n)(x) + \sigma(u_n(x))\xi_n(x)$, for $n 
\in \Z_+$
and $x\in \Z^d$, where $\bm\xi:=\{\xi_n(x)\}_{n\ge 0,x\in\Z^d}$  
denotes random
forcing and $\sL$ the generator of a random walk on $\Z^d$. Under mild
conditions, we prove that the preceding stochastic PDE has a unique  
solution
that grows at most exponentially in time. And that, under natural  
conditions,
it is "weakly intermittent." Along the way, we establish a comparison  
principle
as well as a finite-support property.


  http://arxiv.org/abs/0811.0643

---------------------------------------------------------------

7720. ASYMPTOTICS FOR KOTZ TYPE III ELLIPTICAL DISTRIBUTIONS

Enkelejd Hashorva

In this paper we derive the tail asymptotics of a Kotz Type III  
elliptical
random vector. As an application of our asymptotic expansion we derive  
an
approximation for the conditional excess distribution. Furthermore, we  
discuss
the asymptotic dependence of Kotz Type III triangular arrays and  
provide some
details on the estimation of conditional excess distribution and  
survivor
function.


  http://arxiv.org/abs/0811.0662

---------------------------------------------------------------

7721. PROBABILITY MEASURES, L\'{E}VY MEASURES AND ANALYTICITY IN TIME

Ole E. Barndorff-Nielsen and  Friedrich Hubalek

We investigate the relation of the semigroup probability density of an
infinite activity L\'{e}vy process to the corresponding L\'{e}vy  
density. For
subordinators, we provide three methods to compute the former from the  
latter.
The first method is based on approximating compound Poisson  
distributions, the
second method uses convolution integrals of the upper tail integral of  
the
L\'{e}vy measure and the third method uses the analytic continuation  
of the
L\'{e}vy density to a complex cone and contour integration. As a by- 
product, we
investigate the smoothness of the semigroup density in time. Several  
concrete
examples illustrate the three methods and our results.


  http://arxiv.org/abs/0811.0678

---------------------------------------------------------------

7722. DIFFUSION LIMIT FOR MANY PARTICLES IN A PERIODIC STOCHASTIC  
ACCELERATION  FIELD

Yves Elskens (PIIM) and  Etienne Pardoux (LATP)

The one-dimensional motion of any number $\cN$ of particles in the  
field of
many independent waves (with strong spatial correlation) is formulated  
as a
second-order system of stochastic differential equations, driven by  
two Wiener
processes. In the limit of vanishing particle mass ${\mathfrak{m}} \to  
0$, or
equivalently of large noise intensity, we show that the momenta of all  
$N$
particles converge weakly to $N$ independent Brownian motions, and this
convergence holds even if the noise is periodic. This justifies the  
usual
application of the diffusion equation to a family of particles in a  
unique
stochastic force field. The proof rests on the ergodic properties of the
relative velocity of two particles in the scaling limit.


  http://arxiv.org/abs/0811.0801

---------------------------------------------------------------

7723. AN ELEMENTARY APPROACH TO EXTREME VALUES THEORY

Philippe Barbe (CNRS)

This note presents a rather intuitive approach to extreme value  
theory. This
approach was devised mostly for pedagogical reason.


  http://arxiv.org/abs/0811.0753

---------------------------------------------------------------

7724. CONFLUENCE OF GEODESIC PATHS AND SEPARATING LOOPS IN LARGE  
PLANAR  QUADRANGULATIONS

J. Bouttier and E. Guitter

We consider planar quadrangulations with three marked vertices and  
discuss
the geometry of triangles made of three geodesic paths joining them.  
We also
study the geometry of minimal separating loops, i.e. paths of minimal  
length
among all closed paths passing by one of the three vertices and  
separating the
two others in the quadrangulation. We concentrate on the universal  
scaling
limit of large quadrangulations, also known as the Brownian map, where  
pairs of
geodesic paths or minimal separating loops have common parts of non-zero
macroscopic length. This is the phenomenon of confluence, which  
distinguishes
the geometry of random quadrangulations from that of smooth surfaces. We
characterize the universal probability distribution for the lengths of  
these
common parts.


  http://arxiv.org/abs/0811.0509

---------------------------------------------------------------

7725. MARGINAL RELEVANCE OF DISORDER FOR PINNING MODELS

Giambattista Giacomin and  Hubert Lacoin and Fabio Lucio Toninelli

The effect of disorder on pinning and wetting models has attracted much
attention in theoretical physics. In particular, it has been predicted  
on the
basis of the Harris criterion that disorder is relevant (annealed and  
quenched
model have different critical points and critical exponents) if the  
return
probability exponent alpha, a positive number that characterizes the  
model, is
larger than 1/2. Weak disorder has been predicted to be irrelevant (i.e.
coinciding critical points and exponents) if alpha < 1/2. Recent  
mathematical
work has put these predictions on firm grounds. In renormalization  
group terms,
the case alpha = 1/2 is a 'marginal case' and there is no agreement in  
the
literature as to whether one should expect disorder relevance or  
irrelevance at
marginality. The question is particularly intriguing also because the  
case
alpha = 1/2 includes the classical models of two-dimensional wetting  
of a rough
substrate, of pinning of directed polymers on a defect line in  
dimension (3+1)
or (1+1) and of pinning of an heteropolymer by a point potential in
three-dimensional space. Here we prove disorder relevance both for the  
general
alpha = 1/2 pinning model and for the hierarchical version of the model
proposed by B. Derrida, V. Hakim and J. Vannimenus (JSP, 1992), in the  
sense
that we prove a shift of the quenched critical point with respect to the
annealed one. In both cases we work with Gaussian disorder and we show  
that the
shift is at least of order exp(-1/\beta^4) for beta small, if beta is  
the
standard deviation of the disorder.


  http://arxiv.org/abs/0811.0723

---------------------------------------------------------------

7726. COGNITIVE OFDM NETWORK SENSING: A FREE PROBABILITY APPROACH

Romain Couillet and  Merouane Debbah

In this paper, a practical power detection scheme for OFDM terminals,  
based
on recent free probability tools, is proposed. The objective is for the
receiving terminal to determine the transmission power and the number  
of the
surrounding base stations in the network. However, thesystem  
dimensions of the
network model turn energy detection into an under-determined problem.  
The focus
of this paper is then twofold: (i) discuss the maximum amount of  
information
that an OFDM terminal can gather from the surrounding base stations in  
the
network, (ii) propose a practical solution for blind cell detection  
using the
free deconvolution tool. The efficiency of this solution is measured  
through
simulations, which show better performance than the classical power  
detection
methods.


  http://arxiv.org/abs/0811.0731

---------------------------------------------------------------

7727. MULTIPLE ANTENNA COGNITIVE RECEIVERS AND SIGNAL DETECTION

Romain Couillet and  Merouane Debbah

A Bayesian inference learning process for cognitive receivers is  
provided in
this paper. We focus on the particular case of signal detectionas an
explanatory example to the learning framework. Under any prior state of
knowledge on the communication channel, an information theoretic  
criterion is
presented to decide on the presence of informative data in a noisy  
wireless
MIMO communication. We detail the particular cases of knowledge, or  
absence of
knowledge at the receiver, of the number of transmit antennas and  
noise power.
The provided method is instrumental to provide intelligence to the  
receiver and
gives birth to a novel Bayesian signal detector. The detector is  
compared to
the classical power detector and provides detection performance upper  
bounds.
Simulations corroborate the theoretical results and quantify the gain  
achieved
using the proposed Bayesian framework.


  http://arxiv.org/abs/0811.0764

---------------------------------------------------------------

7728. A MAXIMUM ENTROPY APPROACH TO OFDM CHANNEL ESTIMATION

Romain Couillet and  Merouane Debbah

In this work, a new Bayesian framework for OFDM channel estimation is
proposed. Using Jaynes' maximum entropy principle to derive prior  
information,
we successively tackle the situations when only the channel delay  
spread is a
priori known, then when it is not known. Exploitation of the time- 
frequency
dimensions are also considered in this framework, to derive the  
optimal channel
estimation associated to some performance measure under any state of  
knowledge.
Simulations corroborate the optimality claim and always prove as good  
or better
in performance than classical estimators.


  http://arxiv.org/abs/0811.0778

---------------------------------------------------------------

7729. DISTRIBUTION OF THE BROWNIAN MOTION ON ITS WAY TO HITTING ZERO

P.Chigansky and F.C.Klebaner

For the one-dimensional Brownian motion $B=(B_t)_{t\ge 0}$, started at  
$x>0$,
and the first hitting time $\tau=\inf\{t\ge 0:B_t=0\}$, we find the  
probability
density of $B_{u\tau}$ for a $u\in(0,1)$, i.e. of the Brownian motion  
on its
way to hitting zero.


  http://arxiv.org/abs/0811.0909

---------------------------------------------------------------

7730. ASYMPTOTIC INDEPENDENCE IN THE SPECTRUM OF THE GAUSSIAN UNITARY  
ENSEMBLE

P. Bianchi and M. Debbah and J. Najim

Consider a $n \times n$ matrix from the Gaussian Unitary Ensemble (GUE).
Given a finite collection of bounded disjoint real Borel sets $ 
(\Delta_{i,n},\
1\leq i\leq p)$, properly rescaled, and eventually included in any
neighbourhood of the support of Wigner's semi-circle law, we prove  
that the
related counting measures $({\mathcal N}_n(\Delta_{i,n}), 1\leq i\leq  
p)$,
where ${\mathcal N}_n(\Delta)$ represents the number of eigenvalues  
within
$\Delta$, are asymptotically independent as the size $n$ goes to  
infinity, $p$
being fixed.
   As a consequence, we prove that the largest and smallest eigenvalues,
properly centered and rescaled, are asymptotically independent; we  
finally
describe the fluctuations of the condition number of a matrix from the  
GUE.


  http://arxiv.org/abs/0811.0979

---------------------------------------------------------------

7731. A STOCHASTIC EPIDEMIOLOGICAL MODEL AND A DETERMINISTIC LIMIT  
FOR  BITTORRENT-LIKE PEER-TO-PEER FILE-SHARING NETWORKS

George Kesidis and  Takis Konstantopoulos and  Perla Sousi

In this paper, we propose a stochastic model for a file-sharing peer- 
to-peer
network which resembles the popular BitTorrent system: large files are  
split
into chunks and a peer can download or swap from another peer only one  
chunk at
a time. We prove that the fluid limits of a scaled Markov model of  
this system
are of the coagulation form, special cases of which are well-known
epidemiological (SIR) models. In addition, Lyapunov stability and  
settling-time
results are explored. We derive conditions under which the BitTorrent
incentives under consideration result in shorter mean file-acquisition  
times
for peers compared to client-server (single chunk) systems. Finally, a
diffusion approximation is given and some open questions are discussed.


  http://arxiv.org/abs/0811.1003

---------------------------------------------------------------

7732. ON PERCOLATION AND THE BUNKBED CONJECTURE

Svante Linusson

We study a problem on percolation on product graphs G x K_2. Here G is  
any
finite graph and K_2 consists of two vertices {0,1} connected by an  
edge. In
edge percolation every edge in G x K_2 is present with probability p.  
In [3]
Olle H\"aggstr\"om stated a conjecture (which he claimed to be  
folklore) that
for all G and p the probability that (u,0) is in the same component as  
(v,0) is
greater than the probability that (u,0) is in the same component as (v, 
1) for
every pair of vertices u,v in G.
   We generalize this conjecture and formulate and prove similar  
statements for
randomly directed graphs. The methods lead to a proof of the original
conjecture for special classes of graphs $G$, in particular  
outerplanar graphs.


  http://arxiv.org/abs/0811.0949

---------------------------------------------------------------

7733. SPECTRUM OF LARGE RANDOM REVERSIBLE MARKOV CHAINS

Charles Bordenave (IMT) and  Pietro Caputo and  Djalil Chafai (IMT  
and  UPTE)

In this work, we adopt a Random Matrix Theory point of view to study the
spectrum of large reversible Markov chains in random environment. As  
the number
of states tends to infinity, we consider both the almost sure global  
behavior
of the spectrum, and the local behavior at the edge including the so  
called
spectral gap. We study presently two simple models. The first one is  
on the
complete graph while the second is on the chain graph (birth-and-death
dynamics). These two models exhibit different scalings and limiting  
objects.
The first model is related to the semi--circle law and Wigner's  
theorem. It
contains as a special case a natural reversible Dirichlet Markov  
Ensemble. The
second model is related to homogenization and also to asymptotics for  
the roots
of random orthogonal polynomials. A special case gives rise to the  
arc--sine
law as in a theorem by Erdos & Turan. This work raises several open  
problems.


  http://arxiv.org/abs/0811.1097

---------------------------------------------------------------

7734. ISOTROPIC ORNSTEIN-UHLENBECK FLOWS

Georgi Dimitroff and  Holger van Bargen

Isotropic Brownian flows (IBFs) are a fairly natural class of stochastic
flows which has been studied extensively by various authors. Their rich
structure allows for explicit calculations in several situations and  
makes them
a natural object to start with if one wants to study more general  
stochastic
flows. Often the intuition gained by understanding the problem in the  
context
of IBFs transfers to more general situations. However, the obvious  
link between
stochastic flows, random dynamical systems and ergodic theory cannot be
exploited in its full strength as the IBF does not have an invariant
probability measure but rather an infinite one. Isotropic Ornstein- 
Uhlenbeck
flows are in a sense localized IBFs and do have an invariant probability
measure. The imposed linear drift destroys the translation invariance  
of the
IBF, but many other important structure properties like the Markov  
property of
the distance process remain valid and allow for explicit calculations in
certain situations. The fact that isotropic Ornstein-Uhlenbeck flows  
have
invariant probability measures allows one to apply techniques from  
random
dynamical systems theory. We demonstrate this by applying the results of
Ledrappier and Young to calculate the Hausdorff dimension of the  
statistical
equilibrium of an isotropic Ornstein-Uhlenbeck flow.


  http://arxiv.org/abs/0811.1107

---------------------------------------------------------------

7735. RECONSTRUCTION OF SYMMETRIC POTTS MODELS

Allan Sly

The reconstruction problem on the tree has been studied in numerous  
contexts
including statistical physics, information theory and computational  
biology.
However, rigorous reconstruction thresholds have only been established  
in a
small number of models. We prove the first exact reconstruction  
threshold in a
non-binary model establishing the Kesten-Stigum bound for the 3-state  
Potts
model on regular trees of large degree. We further establish that the
Kesten-Stigum bound is not tight for the $q$-state Potts model when $q  
\geq 5$.
Moreover, we determine asymptotics for the reconstruction thresholds.


  http://arxiv.org/abs/0811.1208

---------------------------------------------------------------

7736. A SELF-REGULATING AND PATCH SUBDIVIDED POPULATION

Lamia Belhadji and  Daniela Bertacchi and  Fabio Zucca

We consider an interacting particle process on a graph which, from a
macroscopic point of view, looks like $\Z^d$ and, at a microscopic  
level, is a
complete graph of degree $N$ (called a patch). There are two birth  
rates: an
inter-patch one $\lambda$ and an intra-patch one $\phi$. Once a site is
occupied, there is no breeding from outside the patch and the  
probability
$c(i)$ of success of an intra-patch breeding decreases with the size $i 
$ of the
population in the site. We prove the existence of a critical value
$\lambda_{cr}(\phi, c, N)$ and a critical value $\phi_{cr}(\lambda, c,  
N)$. We
consider a sequence of processes generated by the families of control  
functions
$\{c_i\}_{i \in \N}$ and degrees $\{N_i\}_{i \in \N}$; we prove, under  
mild
assumptions, the existence of a critical value $i_{cr}$. Roughly  
speaking we
show that, in the limit, these processes behave as the branching  
random walk on
$\Z^d$ with external birth rate $\lambda$ and internal birth rate $\phi 
$. Some
examples of models that can be seen as particular cases are given.


  http://arxiv.org/abs/0811.1279

---------------------------------------------------------------

7737. MATRIX VALUED BROWNIAN MOTION AND A PAPER BY POLYA

Philippe Biane (IGM)

We give a geometric description of the motion of eigenvalues of a  
Brownian
motion with values in some matrix spaces. In the second part we  
consider a
paper by Polya where he introduced a function close to the Riemann zeta
function, which satisfies Riemann hypothesis. We show that each of  
these two
functions can be related to Brownian motion on a symmetric space.


  http://arxiv.org/abs/0811.1490

---------------------------------------------------------------

7738. ROBUST ADAPTIVE IMPORTANCE SAMPLING FOR NORMAL RANDOM VECTORS

Benjamin Jourdain and Jerome Lelong

Adaptive Monte Carlo methods are very efficient techniques designed to  
tune
simulation estimators on-line. In this work, we present an alternative  
to
stochastic approximation to tune the optimal change of measure in the  
context
of importance sampling for normal random vectors. Unlike stochastic
approximation, which requires very fine tuning in practice, we propose  
to use
sample average approximation and deterministic optimization techniques  
to
devise a robust and fully automatic variance reduction methodology.  
The same
samples are used in the sample optimization of the importance sampling
parameter and in the Monte Carlo computation of the expectation of  
interest
with the optimal measure computed in the previous step. We prove that  
this
highly non independent Monte Carlo estimator is convergent and  
satisfies a
central limit theorem with the optimal limiting variance. Numerical  
experiments
confirm the performance of this estimator : in comparison with the  
crude Monte
Carlo method, the computation time needed to achieve a given precision  
is
divided by a factor going from 2 to 10.


  http://arxiv.org/abs/0811.1496

---------------------------------------------------------------

7739. OPTIMAL SEQUENTIAL MULTIPLE HYPOTHESIS TESTS

Andrey Novikov

This work deals with a general problem of testing multiple hypotheses  
about
the distribution of a discrete-time stochastic process. Both the  
Bayesian and
the conditional settings are considered. The structure of optimal  
sequential
tests is characterized.


  http://arxiv.org/abs/0811.1297

---------------------------------------------------------------

7740. THE FUNDAMENTAL GROUP OF RANDOM 2-COMPLEXES

Eric Babson and  Christopher Hoffman and Matthew Kahle

The random 2-complex Y=Y(n,p) is the probability space of all simplicial
complexes on vertex set [n] and edge set [n] \choose 2, with each 2- 
dimensional
face included with probability p independently. Nathan Linial and Roy  
Meshulam
showed that if p >> 2\log{n}/n then the probability that H_{1}(Y,F_2) is
trivial goes to 1 as n approaches infinity. This is an analogue of the  
phase
transition for connectivity of the Erd\H{o}s-R\'enyi random graph  
G(n,p).
   We show here that if p >> n^{-1/2}, then the probability that Y is  
simply
connected goes to 1 as n approaches infinity, but if p << n^{-1/2}  
then the
probability that Y is simply connected goes to 0. This implies in  
particular
that vanishing of H_{1}(Y,F_2) and \pi_1(Y) have distinct thresholds.  
Finding
the threshold for vanishing of H_{1}(Y,Z}) is still an open problem.


  http://arxiv.org/abs/0711.2704

---------------------------------------------------------------

7741. DIRICHLET FORMS ON LAAKSO AND BARLOW-EVANS FRACTALS OF  
ARBITRARY  DIMENSION

Benjamin Steinhurst

In this paper we explore the metric-measure spaces introduced by  
Laakso in
2000. Building upon the work of Barlow and Evans we are able to show the
existence of a large supply of Dirichlet forms, or alternatively Markov
Processes, on these spaces. The construction of Barlow and Evans  
allows us to
justify the use of a quantum graph perspective to identify and  
describe a
Laplacian operator generated by minimal generalized upper gradients on  
any of
the Laakso spaces


  http://arxiv.org/abs/0811.1378

---------------------------------------------------------------

7742. SPECTRAL MEASURE OF HEAVY TAILED BAND AND COVARIANCE RANDOM  
MATRICES

Serban Belinschi and  Amir Dembo and  Alice Guionnet

We study the asymptotic behavior of the appropriately scaled and  
possibly
perturbed spectral measure $\mu$ of large random real symmetric  
matrices with
heavy tailed entries. Specifically, consider the N by N symmetric matrix
$Y_N^\sigma$ whose (i,j) entry is $\sigma(i/N,j/N)X_{ij}$ where $ 
(X_{ij},
0<i<j+1<\infty)$ is an infinite array of i.i.d real variables with  
common
distribution in the domain of attraction of an $\alpha$-stable law,
$0<\alpha<2$, and $\sigma$ is a deterministic function. For a random  
diagonal
$D_N$ independent of $Y_N^\sigma$ and with appropriate rescaling $a_N 
$, we
prove that the distribution $\mu$ of $a_N^{-1}Y_N^\sigma + D_N$  
converges in
mean towards a limiting probability measure which we characterize. As  
a special
case, we derive and analyze the almost sure limiting spectral density  
for
empirical covariance matrices with heavy tailed entries.


  http://arxiv.org/abs/0811.1587

---------------------------------------------------------------

7743. MEASURE CHANGES WITH EXTINCTION

Simon Harris and  Matthew Roberts

We consider a change of measure by a martingale $Z_t$ and clarify that  
in
general $1/Z_t$ is only a supermartingale under the changed measure.  
We then
give a necessary and sufficient condition for the event that the limit  
of the
martingale is zero to coincide with the event that the martingale hits  
zero in
finite time (up to a set of zero probability).


  http://arxiv.org/abs/0811.1696

---------------------------------------------------------------

7744. BRANCHING BROWNIAN MOTION: ALMOST SURE GROWTH ALONG UNSCALED PATHS

Simon Harris and  Matthew Roberts

We give new results on the growth of the number of particles in a dyadic
branching Brownian motion which follow within a fixed distance of a path
$f:[0,\infty)\to \mathbb{R}$. We show that it is possible to count the  
number
of particles without rescaling the paths. Our results reveal that the  
number of
particles along certain paths can oscillate dramatically. The methods  
used are
entirely probabilistic, taking advantage of the spine technique  
developed by,
amongst others, Lyons et al, Kyprianou, and Hardy & Harris.


  http://arxiv.org/abs/0811.1704

---------------------------------------------------------------

7745. SLOWDOWN ESTIMATES FOR BALLISTIC RANDOM WALK IN RANDOM ENVIRONMENT

Noam Berger

We consider random walk in an elliptic i.i.d. random environment in  
dimension
greater than or equal to 4, satisfying the ballisticity condition  
(T'). We show
that for every $\alpha< d$ and $n$ large enough, the probability of  
linear
slowdown is bounded above by $\exp(-(\log n)^\alpha)$. This is almost  
matching
the known lower bound of $\exp(-C(\log n)^d)$, and significantly  
improves
previously known upper bounds. As a tool for the main result, we show  
an almost
local version of the quenched central limit theorem under the  
assumption of
condition (T').


  http://arxiv.org/abs/0811.1710

---------------------------------------------------------------

7746. MOMENTS, CUMULANTS AND DIAGRAM FORMULAE FOR NON-LINEAR  
FUNCTIONALS OF  RANDOM MEASURES

Giovanni Peccati (LSTA and  Modal'x) and  Murad S. Taqqu (BOSTON  
University)

This survey provides a unified discussion of multiple integrals,  
moments,
cumulants and diagram formulae associated with functionals of  
completely random
measures. Our approach is combinatorial, as it is based on the algebraic
formalism of partition lattices and M\"obius functions. Gaussian and  
Poisson
measures are treated in great detail. We also present several  
combinatorial
interpretations of some recent CLTs involving sequences of random  
variables
belonging to a fixed Wiener chaos.


  http://arxiv.org/abs/0811.1726

---------------------------------------------------------------

7747. SURVIVAL OF BRANCHING RANDOM WALKS IN RANDOM ENVIRONMENT

Nina Gantert and  Sebastian M\"uller and  Serguei Popov and  Marina  
Vachkovskaia

We study survival of nearest-neighbour branching random walks in random
environment (BRWRE) on $\mathbb{Z}$. A priori there are three  
different regimes
of survival: global survival, local survival, and strong local  
survival. We
show that local and strong local survival regimes coincide for BRWRE  
and that
they can be characterized with the spectral radius of the first moment  
matrix
of the process. These results are generalizations of the  
classification of
BRWRE in recurrent and transient regimes. Our main result is a  
characterization
of global survival that is given in terms of Lyapunov exponents of an  
infinite
product of i.i.d. $2\times 2$ random matrices.


  http://arxiv.org/abs/0811.1748

---------------------------------------------------------------

7748. THE DISTRIBUTION OF FST AND OTHER GENETIC STATISTICS FOR A CLASS  
OF  POPULATION STRUCTURE MODELS

Sivan Leviyang

We examine genetic statistics used in the study of structured  
populations. In
a 1999 paper, Wakeley observed that the coalescent process associated  
with the
finite island model can be decomposed into a scattering phase and a  
collecting
phase. In this paper, we introduce a class of population structure  
models,
which we refer to as G/KC models, that obey such a decomposition. In a  
large
population, large sample limit we derive the distribution of the  
statistic Fst
for all G/KC models under the assumptions of strong or weak mutation.  
We show
that in the large population, large sample limit the island and two  
dimensional
stepping stone models are members of the G/KC class of models, thereby  
deriving
the distributions of Fst for these two well known models as a special  
case of a
general formula. We show that our analysis of Fst can be extended to  
an entire
class of genetic statistics, and we use our approach to examine  
homozygosity
measures. Our analysis uses coalescent based methods.


  http://arxiv.org/abs/0811.1553

---------------------------------------------------------------

7749. ON $K$-FREE-LIKE GROUPS

A.Yu. Olshanskii and M. V. Sapir

A $k$-free like group is a $k$-generated group $G$ with a sequence of
$k$-element generating sets $Z_n$ such that the girth of $G$ relative  
to $Z_n$
is unbounded and the Cheeger constant of $G$ relative to $Z_n$ is  
bounded away
from 0. By a recent result of Benjamini-Nachmias-Peres, this implies  
that the
critical bond percolation probability of the Cayley graph of $G$  
relative to
$Z_n$ tends to $1/(2k-1)$ as $n\to \infty$. Answering a question of  
Benjamini,
we construct many non-free groups that are $k$-free like for all  
sufficiently
large $k$.


  http://arxiv.org/abs/0811.1607

---------------------------------------------------------------

7750. COMBINATORICS OF TRIPARTITE BOUNDARY CONNECTIONS FOR TREES AND  
DIMERS

Richard W. Kenyon and David B. Wilson

A grove is a spanning forest of a planar graph in which every  
component tree
contains at least one of a special subset of vertices on the outer  
face called
nodes. For the natural probability measure on groves, we compute various
connection probabilities for the nodes in a random grove. In  
particular, for
``tripartite'' pairings of the nodes, the probability can be computed  
as a
Pfaffian in the entries of the Dirichlet-to-Neumann matrix (discrete  
Hilbert
transform) of the graph. These formulas generalize the determinant  
formulas
given by Curtis, Ingerman, and Morrow, and by Fomin, for parallel  
pairings.
These Pfaffian formulas are used to give exact expressions for  
reconstruction:
reconstructing the conductances of a planar graph from boundary  
measurements.
We prove similar theorems for the double-dimer model on bipartite planar
graphs.


  http://arxiv.org/abs/0811.1766

---------------------------------------------------------------

7751. STOCHASTIC INTEGRALS AND CONDITIONAL FULL SUPPORT

Mikko S. Pakkanen

We show that any continuous stochastic process having the form Z_t =  
H_t +
\int_0^t K_s dW_s, t \in [0,T], where (H_t) and (K_t) are continuous  
processes,
independent of the driving Brownian motion (W_t), has the conditional  
full
support property, introduced by Guasoni, R\'asonyi, and Schachermayer  
[Ann.
Appl. Probab. 18 (2) (2008) 491-520] in connection pricing models with
transaction costs. Using this result, we show that several stochastic
volatility (SV) models (e.g. Heston, Hull-White, log-OU, fractional  
SV) have
the conditional full support property.


  http://arxiv.org/abs/0811.1847

---------------------------------------------------------------

7752. ON OPTIMALITY OF THE BARRIER STRATEGY IN DE FINETTI'S DIVIDEND  
PROBLEM  FOR SPECTRALLY NEGATIVE L\'{E}VY PROCESSES

R. L. Loeffen

We consider the classical optimal dividend control problem which was  
proposed
by de Finetti [Trans. XVth Internat. Congress Actuaries 2 (1957)  
433--443].
Recently Avram, Palmowski and Pistorius [Ann. Appl. Probab. 17 (2007)  
156--180]
studied the case when the risk process is modeled by a general  
spectrally
negative L\'{e}vy process. We draw upon their results and give  
sufficient
conditions under which the optimal strategy is of barrier type,  
thereby helping
to explain the fact that this particular strategy is not optimal in  
general. As
a consequence, we are able to extend considerably the class of  
processes for
which the barrier strategy proves to be optimal.


  http://arxiv.org/abs/0811.1862

---------------------------------------------------------------

7753. AIRY PROCESSES WITH WANDERERS AND NEW UNIVERSALITY CLASSES

Mark Adler and  Patrik L. Ferrari and  Pierre van Moerbeke

Consider n+m non-intersecting Brownian bridges, with n of them leaving  
from 0
at time t=-1 and returning to 0 at time t=1, while the m remaining ones
(wanderers) go from m points a_i to m points b_i. First we keep m  
fixed and we
scale a_i,b_i appropriately with n. In the large-n limit we obtain a  
new Airy
process with wanderers, in the neighborhood of (2n)^(1/2), the  
approximate
location of the rightmost particle in the absence of wanderers. This new
process is governed by an Airy-type kernel, with a rational  
perturbation.
   Letting the number m of wanderers tend to infinity as well, leads  
to two
Pearcey processes about two cusps, a closing and an opening cusp, the  
location
of the tips being related by an elliptic curve. Upon tuning the  
starting and
target points, one can let the two tips of the cusps grow very close;  
this
leads to a new process, which we conjecture to be governed by a kernel,
represented as a double integral involving the exponential of a quintic
polynomial in the integration variables.


  http://arxiv.org/abs/0811.1863

---------------------------------------------------------------

7754. CENTRAL LIMIT THEOREM AND THE BOOTSTRAP FOR U-STATISTICS OF  
STRONGLY  MIXING DATA

Herold Dehling and Martin Wendler

The asymptotic normality of U-statistics has so far been proved for  
iid data
and under various mixing conditions such as absolute regularity, but  
not for
strong mixing. We use a coupling technique introduced 1983 by Bradley  
to prove
a new generalized covariance inequality similar to Yoshihara's. It  
follows from
the Hoeffding-decomposition and this inequality, that U-statistics of  
strong
mixing observations converge to a normal limit if the kernel of the U- 
statistic
fulfills some moment and continuity conditions.
   The validity of the bootstrap for U-statistics has until now only  
been
established in the case of iid data (see Bickel and Freedman). For  
mixing data,
Politis and Romano proposed the circular block bootstrap, which leads  
to a
consistent estimation of the sample mean's distribution. We extend these
results to U-statistics of weakly dependent data and prove a CLT for the
circular block bootstrap version of U-statistics under absolute  
regularity and
strong mixing. We also calculate a rate of convergence for the bootstrap
variance estimator of a U-statistic.


  http://arxiv.org/abs/0811.1888

---------------------------------------------------------------

7755. BINOMIAL APPROXIMATIONS OF SHORTFALL RISK FOR GAME OPTIONS

Yan Dolinsky and  Yuri Kifer

We show that the shortfall risk of binomial approximations of game  
(Israeli)
options converges to the shortfall risk in the corresponding Black-- 
Scholes
market considering Lipschitz continuous path-dependent payoffs for both
discrete- and continuous-time cases. These results are new also for  
usual
American style options. The paper continues and extends the study of  
Kifer
[Ann. Appl. Probab. 16 (2006) 984--1033] where estimates for binomial
approximations of prices of game options were obtained. Our arguments  
rely, in
particular, on strong invariance principle type approximations via the
Skorokhod embedding, estimates from Kifer [Ann. Appl. Probab. 16 (2006)
984--1033] and the existence of optimal shortfall hedging in the  
discrete time
established by Dolinsky and Kifer [Stochastics 79 (2007) 169--195].


  http://arxiv.org/abs/0811.1896

---------------------------------------------------------------

7756. EQUALITY OF CRITICAL POINTS FOR POLYMER DEPINNING TRANSITIONS  
WITH LOOP  EXPONENT ONE

Kenneth S. Alexander and  Nikos Zygouras

We consider a polymer with configuration modeled by the trajectory of a
Markov chain, interacting with a potential of form $u+V_n$ when it  
visits a
particular state 0 at time $n$, with $\{V_n\}$ representing i.i.d.  
quenched
disorder. There is a critical value of $u$ above which the polymer is  
pinned by
the potential. A particular case not covered in a number of previous  
studies is
that of loop exponent one, in which the probability of an excursion of  
length
$n$ takes the form $\varphi(n)/n$ for some slowly varying $\varphi$;  
this
includes simple random walk in two dimensions. We show that in this  
case, at
all temperatures, the critical values of $u$ in the quenched and  
annealed
models are equal, in contrast to all other loop exponents, for which  
these
critical values are known to differ at least at low temperatures.


  http://arxiv.org/abs/0811.1902

---------------------------------------------------------------

7757. AN ELEMENTARY PROOF OF HAWKES'S CONJECTURE ON GALTON-WATSON TREES

Thomas Duquesne

In 1981, J. Hawkes conjectured the exact form of the Hausdorff gauge  
function
for the boundary of supercritical Galton-Watson trees under a certain
assumption on the tail at the infinity of the total mass of the  
branching
measure. Hawkes's conjecture has been proved by T. Watanabe in 2007 as  
well as
other other precise results on fractal properties of the boundary of
Galton-Watson trees. The goal of this paper is to provide an  
elementary proof
of Hawkes's conjecture under a less restrictive assumption than in T.
Watanabe's paper, by use of size-biased Galton-Watson trees introduced  
by
Lyons, Pemantle and Peres in 1995.


  http://arxiv.org/abs/0811.1935

---------------------------------------------------------------

7758. AGGREGATION OF AUTOREGRESSIVE PROCESSES AND LONG MEMORY

Didier Dacunha-Castelle (1) and Lisandro J. Ferm\'in (1 and 2) ((1)   
Universit\'e Paris Sud, (2) Universit\'e Paris Descartes)

We study the aggregation of AR processes and generalized Ornstein- 
Uhlenbeck
(OU) processes. Mixture of spectral densities with random poles are  
the main
tool. In this context, we apply our results for the aggregation of  
doubly
stochastic interactives processes, see Dacunha-Castelle and Fermin  
(2006).
Thus, we study the relationship between aggregation of autoregressive  
processes
and long memory considering complex interaction structures. We precise  
a very
interesting qualitative phenomena: how the long memory creation  
depends on the
poles concentration near to the boundary of stability (measured in the
Prokhorov sense). Our results extends the results given by Oppenheim  
and Viano
(2004), and highlight the importance of the angular dispersion measure  
of poles
in the appearance of the long memory.


  http://arxiv.org/abs/0811.1917

---------------------------------------------------------------

7759. BROWNIAN MOVING AVERAGES HAVE CONDITIONAL FULL SUPPORT

Alexander Cherny

We prove that any Brownian moving average
\[X_t=\int_{-\infty}^t\bigl(f(s-t)-f(s)\bigr) dB_s,\qquad t\ge0,\]  
satisfies
the conditional full support condition introduced by Guasoni, R 
\'{a}sonyi and
Schachermayer [Ann. Appl. Probab. 18 (2008) 491--520].


  http://arxiv.org/abs/0811.2040

---------------------------------------------------------------

7760. ON THE LAWS OF FIRST HITTING TIMES OF POINTS FOR ONE- 
DIMENSIONAL  SYMMETRIC STABLE L\'EVY PROCESSES

Kouji Yano and  Yuko Yano and Marc Yor

Several aspects of the laws of first hitting times of points are  
investigated
for one-dimensional symmetric stable L\'evy processes. It\^o's  
excursion theory
plays a key role in this study.


  http://arxiv.org/abs/0811.2046

---------------------------------------------------------------

7761. A CONNECTION BETWEEN EXTREME VALUE THEORY AND LONG TIME  
APPROXIMATION OF  SDE'S

Fabien Panloup (LSProba)

We consider a sequence $(\xi_n)_{n\ge1}$ of $i.i.d.$ random values  
living in
the domain of attraction of an extreme value distribution. For such  
sequence,
there exists $(a_n)$ and $(b_n)$, with $a_n>0$ and $b_n\in\ER$ for  
every $n\ge
1$, such that the sequence $(X_n)$ defined by
$X_n=(\max(\xi_1,...,\xi_n)-b_n)/a_n$ converges in distribution to a non
degenerated distribution. In this paper, we show that $(X_n)$ can be  
viewed as
an Euler scheme with decreasing step of an ergodic Markov process  
solution to a
SDE with jumps and we derive a functional limit theorem for the sequence
$(X_n)$ from some methods used in the long time numerical  
approximation of
ergodic SDE's.


  http://arxiv.org/abs/0811.2052

---------------------------------------------------------------

7762. SINGULAR STOCHASTIC EQUATIONS ON HILBERT SPACES: HARNACK  
INEQUALITIES  FOR THEIR TRANSITION SEMIGROUPS

Giuseppe Da Prato and  Michael R\"ockner and Feng-Yu Wang

We consider stochastic equations in Hilbert spaces with singular drift  
in the
framework of [Da Prato, R\"ockner, PTRF 2002]. We prove a Harnack  
inequality
(in the sense of [Wang, PTRF 1997]) for its transition semigroup and  
exploit
its consequences. In particular, we prove regularizing and  
ultraboundedness
properties of the transition semigroup as well as that the corresponding
Kolmogorov operator has at most one infinitesimally invariant measure $ 
\mu$
(satisfying some mild integrability conditions). Finally, we prove  
existence of
such a measure $\mu$ for non-continuous drifts.


  http://arxiv.org/abs/0811.2061

---------------------------------------------------------------

7763. FINITE TIME EXTINCTION FOR SOLUTIONS TO FAST DIFFUSION  
STOCHASTIC POROUS  MEDIA EQUATIONS

Viorel Barbu and  Giuseppe Da Prato and Michael R\"ockner

We prove that the solutions to fast diffusion stochastic porous media
equations have finite time extinction with strictly positive  
probability.


  http://arxiv.org/abs/0811.2064

---------------------------------------------------------------

7764. ON UNIVERSAL ESTIMATES FOR BINARY RENEWAL PROCESSES

Guszt\'av Morvai and  Benjamin Weiss

A binary renewal process is a stochastic process $\{X_n\}$ taking  
values in
$\{0,1\}$ where the lengths of the runs of 1's between successive  
zeros are
independent. After observing ${X_0,X_1,...,X_n}$ one would like to  
predict the
future behavior, and the problem of universal estimators is to do so  
without
any prior knowledge of the distribution. We prove a variety of results  
of this
type, including universal estimates for the expected time to renewal  
as well as
estimates for the conditional distribution of the time to renewal.  
Some of our
results require a moment condition on the time to renewal and we show  
by an
explicit construction how some moment condition is necessary.


  http://arxiv.org/abs/0811.2076

---------------------------------------------------------------

7765. SELF-ORGANIZED CRITICALITY VIA STOCHASTIC PARTIAL DIFFERENTIAL  
EQUATIONS

Viorel Barbu and  Philippe Blanchard and  Giuseppe Da Prato and  
Michael  R\"ockner

Models of self-organized criticality, which can be described as singular
diffusions with or without (multiplicative) Wiener forcing term (as  
e.g. the
Bak/Tang/Wiesenfeld- and Zhang-models), are analyzed. Existence and  
uniqueness
of nonnegative strong solutions are proved. Previously numerically  
predicted
transition to the critical state in 1-D is confirmed by a rigorous  
proof that
this indeed happens in finite time with high probability.


  http://arxiv.org/abs/0811.2093

---------------------------------------------------------------

7766. A CENTRAL LIMIT THEOREM, AND RELATED RESULTS, FOR A TWO-COLOR  
RANDOMLY  REINFORCED URN

G. Aletti and  C. May and  and P. Secchi

We prove a Central Limit Theorem for the sequence of random  
compositions of a
two-color randomly reinforced urn. As a consequence, we are able to  
show that
the distribution of the urn limit composition has no point masses.


  http://arxiv.org/abs/0811.2097

---------------------------------------------------------------

7767. LATTICE GAS MODEL IN RANDOM MEDIUM AND OPEN BOUNDARIES:  
HYDRODYNAMIC AND  RELAXATION TO THE STEADY STATE

Mustapha Mourragui and  Enza Orlandi

We consider a lattice gas interacting by the exclusion rule in the  
presence
of a random field given by i.i.d. bounded random variables in a  
bounded domain
in contact with particles reservoir at different densities. We show, in
dimensions $d \ge 3$, that the rescaled empirical density field almost  
surely,
with respect to the random field, converges to the unique weak  
solution of a
non linear parabolic equation having the diffusion matrix determined  
by the
statistical properties of the external random field and boundary  
conditions
determined by the density of the reservoir. Further we show that the  
rescaled
empirical density field, in the stationary regime, almost surely with  
respect
to the random field, converges to the solution of the associated  
stationary
transport equation.


  http://arxiv.org/abs/0811.2121

---------------------------------------------------------------

7768. VARIATION AND ROUGH PATH PROPERTIES OF LOCAL TIMES OF L\'EVY  
PROCESSES

Chunrong Feng and  Huaizhong Zhao

In this paper, we will prove that the local time of a L\'evy process  
is of
finite $p$-variation in the space variable in the classical sense,  
a.s. for any
$p>2$, $t\geq 0$, and is a rough path of roughness $p$ a.s. for any  
$2<p<3$.
Then for any function $g$ of finite $q$-variation ($1\leq q <3$), we  
establish
the integral $\int_{-\infty}^{\infty}g(x)dL_t^x$ as a Young integral  
when
$1\leq q<2$ and a Lyons' rough path integral when $2\leq q<3$. We  
therefore
apply these path integrals to extend the Tanaka-Meyer formula for a  
continuous
function $f$ if $\nabla ^-f$ exists and is of finite $q$-variation  
when $1\leq
q<3$, for both continuous semi-martingales and a class of L\'evy  
processes.


  http://arxiv.org/abs/0811.2179

---------------------------------------------------------------

7769. ON THE LONG TIME BEHAVIOR OF THE TCP WINDOW SIZE PROCESS

Djalil Chafai (IMT and  UPTE) and  Florent Malrieu (IRMAR) and  Katy  
Paroux  (LM-Besan\c{c}on, IRISA)

The TCP window size process appears in the modeling of the famous
Transmission Control Protocol used over the Internet. This continuous  
time
Markov process takes its values in $[0,\infty)$, is ergodic and  
irreversible.
It belongs to the Additive Increase Multiplicative Decrease class of  
processes.
The sample paths are piecewise linear deterministic and the whole  
randomness of
the dynamics comes from the jump mechanism. Several aspects of this  
process
have been already investigated in the literature. In the present  
paper, we
mainly get quantitative estimates for the exponential convergence to
equilibrium, in terms of the $W_1$ Wasserstein coupling distance, for  
the
process and also for its embedded chain.


  http://arxiv.org/abs/0811.2180

---------------------------------------------------------------

7770. MEASURES AND DIMENSIONS OF JULIA SETS OF SEMI-HYPERBOLIC  
RATIONAL  SEMIGROUPS

Hiroki Sumi and Mariusz Urbanski

We consider the dynamics of semi-hyperbolic semigroups generated by  
finitely
many rational maps on the Riemann sphere. Assuming that the nice open  
set
condition holds it is proved that there exists a geometric measure on  
the Julia
set with exponent $h$ equal to the Hausdorff dimension of the Julia  
set. Both
$h$-dimensional Hausdorff and packing measures are finite and positive  
on the
Julia set and are mutually equivalent with Radon-Nikodym derivatives  
uniformly
separated from zero and infinity. All three fractal dimensions,  
Hausdorff,
packing and box counting are equal. It is also proved that for the  
canonically
associated skew-product map there exists a unique $h$-conformal measure.
Furthermore, it is shown that this conformal measure admits a unique  
Borel
probability absolutely continuous invariant (under the skew-product map)
measure. In fact these two measures are equivalent, and the invariant  
measure
is metrically exact, hence ergodic.


  http://arxiv.org/abs/0811.1809

---------------------------------------------------------------

7771. REFLECTED AND DOUBLY REFLECTED BSDES WITH JUMPS: A PRIORI  
ESTIMATES AND  COMPARISON

St\'ephane Cr\'epey and  Anis Matoussi

It is now established that under quite general circumstances,  
including in
models with jumps, the existence of a solution to a reflected BSDE is
guaranteed under mild conditions, whereas the existence of a solution  
to a
doubly reflected BSDE is essentially equivalent to the so-called  
Mokobodski
condition. As for uniqueness of solutions, this holds under mild  
integrability
conditions. However, for practical purposes, existence and uniqueness  
are not
enough. In order to further develop these results in Markovian set- 
ups, one
also needs a (simply or doubly) reflected BSDE to be well posed, in  
the sense
that the solution satisfies suitable bound and error estimates, and  
one further
needs a suitable comparison theorem. In this paper, we derive such  
estimates
and comparison results. In the last section, applicability of the  
results is
illustrated with a pricing problem in finance.


  http://arxiv.org/abs/0811.2276

---------------------------------------------------------------

7772. A CONTINUOUS SEMIGROUP OF NOTIONS OF INDEPENDENCE BETWEEN THE  
CLASSICAL  AND THE FREE ONE

Florent Benaych-Georges (PMA and  CMAP) and  Thierry L\'evy (DMA)

In this paper, we investigate a continuous family of notions of  
independence
which interpolates between the classical and free ones for non- 
commutative
random variables. These notions are related to the liberation process
introduced by D. Voiculescu. To each notion of independence correspond  
new
convolutions of probability measures, for which we establish formulae  
and of
which we compute simple examples. We prove that there exists no  
reasonable
analogue of classical and free cumulants associated to these notions of
independence.


  http://arxiv.org/abs/0811.2335

---------------------------------------------------------------

7773. SHIFTED SMALL DEVIATIONS AND CHUNG LIL FOR SYMMETRIC ALPHA- 
STABLE  PROCESSES

Elena Shmileva

Consider a symmetric $\alpha$-stable L\'evy process with $\alpha\in  
(1,2)$.
We study shifted small ball probabilities for these processes in the  
uniform
topology, when the shift function is an arbitrary continuous function  
which
starts at 0. We obtain the exact rate of decrease for these  
probabilities
including constants.
   Using these small ball estimates, we obtain a functional LIL for
$\alpha$-stable L\'evy process with attracting functions that are  
continuous.
It occurs that the limit set for the family of renormalized $\alpha$- 
stable
L\'evy processes is equal to the set of all continuous functions on  
$[0,1]$
which start at 0, under certain choice of normalizing functions.


  http://arxiv.org/abs/0811.2583

---------------------------------------------------------------

7774. ON BOUNDARY CROSSING PROBABILITIES FOR DIFFUSION PROCESSES

Konstantin A. Borovkov and  Andrew N. Downes

In this paper, we establish a relationship between the asymptotic form  
of
conditional boundary crossing probabilities and first passage time  
densities
for diffusion processes. Namely, we show that, under broad  
assumptions, the
first crossing time density of a general curvilinear boundary by a  
general
time-homogeneous diffusion process has a product-form, the factors  
being the
transition density of the process and the coefficient of the leading  
term in
the asymptotic representation of the non-crossing probability of the  
boundary
by the respective diffusion bridge (as the end-point of the bridge  
approaches
the boundary). Using a similar technique, we also demonstrate that the  
boundary
crossing probability is a Gateaux differentiable function of the  
boundary and
give an explicit representation of its derivative.


  http://arxiv.org/abs/0811.2629

---------------------------------------------------------------

7775. SLOW DECAY OF GIBBS MEASURES WITH HEAVY TAILS

Cyril Roberto (LAMA)

We consider Glauber dynamics reversible with respect to Gibbs measures  
with
heavy tails. Spins are unbounded. The interactions are bounded and  
finite
range. The self potential enters into two classes of measures, $\kappa 
$-concave
probability measure and sub-exponential laws, for which it is known  
that no
exponential decay can occur. We prove, using coercive inequalities,  
that the
associated infinite volume semi-group decay to equilibrium  
polynomially and
stretched exponentially, respectively. Thus improving and extending  
previous
results by Bobkov and Zegarlinski.


  http://arxiv.org/abs/0811.2733

---------------------------------------------------------------

7776. LIMIT THEOREMS AND COEXISTENCE PROBABILITIES FOR THE CURIE-WEISS  
POTTS  MODEL WITH AN EXTERNAL FIELD

Daniel Gandolfo (CPT) and  Jean Ruiz (CPT) and  Marc Wouts (LAGA)

The Curie-Weiss Potts model is a mean field version of the well-known  
Potts
model. In this model, the critical line $\beta = \beta_c (h)$ is  
explicitly
known and corresponds to a first order transition when $q > 2$. In the  
present
paper we describe the fluctuations of the density vector in the whole  
domain
$\beta \geqslant 0$ and $h \geqslant 0$, including the conditional  
fluctuations
on the critical line and the non-Gaussian fluctuations at the  
extremity of the
critical line. The probabilities of each of the two thermodynamically  
stable
states on the critical line are also computed. Similar results are  
inferred for
the Random-Cluster model on the complete graph.


  http://arxiv.org/abs/0811.2735

---------------------------------------------------------------

7777. QUANTITATIVE ASYMPTOTICS OF GRAPHICAL PROJECTION PURSUIT

Elizabeth Meckes

There is a result of Diaconis and Freedman which says that, in a  
limiting
sense, for large collections of high-dimensional data most one- 
dimensional
projections of the data are approximately Gaussian. This paper gives
quantitative versions of that result. For a set of deterministic vectors
$\{x_i\}_{i=1}^n$ in $\R^d$ with $n$ and $d$ fixed, let $\theta\in 
\s^{d-1}$ be
a random point of the sphere and let $\mu_n^\theta$ denote the random  
measure
which puts mass $\frac{1}{n}$ at each of the points
$\inprod{x_1}{\theta},\ldots,\inprod{x_n}{\theta}$. For a fixed bounded
Lipschitz test function $f$, $Z$ a standard Gaussian random variable and
$\sigma^2$ a suitable constant, an explicit bound is derived for the  
quantity
$\ds\P\left[\left|\int f d\mu_n^\theta-\E f( \sigma Z)\right|>\epsilon 
\right]$.
A bound is also given for $\ds\P\left[d_{BL}(\mu_n^\theta,
N(0,\sigma^2))>\epsilon\right]$, where $d_{BL}$ denotes the bounded- 
Lipschitz
distance.


  http://arxiv.org/abs/0811.2769

---------------------------------------------------------------

7778. BRUNET-DERRIDA BEHAVIOR OF BRANCHING-SELECTION PARTICLE SYSTEMS  
ON THE  LINE

Jean B\'erard (ICJ) and  Jean-Baptiste Gou\'er\'e (MAPMO)

The term Brunet-Derrida behavior refers to the 1997 paper by E. Brunet  
and B.
Derrida "Shift in the velocity of a front due to a cutoff" (see the  
reference
within the paper), where it is shown, based on numerical simulations and
heuristic arguments, that a certain branching-selection particle  
system on the
line exhibits the following behavior: as N goes to infinity, the  
asymptotic
velocity of the system with N particles converges to a limiting value  
at the
surprisingly slow rate $(\log N)^{-2}$. In this paper, we consider a  
class of
branching-selection particle systems on $\R$ with N particles, defined  
through
iterated branching-selection steps of the following type. During a  
branching
step, each particle is replaced by two new particles, whose positions  
are
shifted from that of the original particle by independently performing  
two
random walk steps, according to some distribution $p$. During the  
selection
step that follows, only the N rightmost particles are kept among the 2N
particles obtained at the branching step, to form a new population of N
particles. Under generic assumptions on $p$, it is shown that Brunet- 
Derrida
behavior holds for the corresponding particle system. The proofs are  
based on
ideas and results by R. Pemantle, and by N. Gantert, Y. Hu and Z. Shi,  
and rely
on a comparison of the particle system with a family of N independent  
branching
random walks killed below a linear space-time barrier. The results  
presented
here both improve and generalize upon previous work by the first  
author of this
paper, which was completed just before the results by Gantert, Hu and  
Shi
became publicly available.


  http://arxiv.org/abs/0811.2782

---------------------------------------------------------------

7779. A WAVELET ANALYSIS OF THE ROSENBLATT PROCESS: CHAOS EXPANSION  
AND  ESTIMATION OF THE SELF-SIMILARITY PARAMETER

Jean-Marc Bardet (CES and  Matisse and  Samos) and  Ciprian Tudor (CES  
and  Matisse and   Samos)

The purpose of this paper is to make a wavelet analysis of self-similar
stochastic processes by using the techniques of the Malliavin calculus  
and the
chaos expansion into multiple stochastic integrals. Our examples are the
fractional Brownian motion and the Rosenblatt process. We study the  
asymptotic
behavior of the statistics based on the wavelet coefficients of these
processes. We find that, in the case when driven process is the  
Rosenblatt
process, this statistics satisfy a non-central limit theorem although  
a part of
it converges to a Gaussian limit. We also construct estimators for the
self-similarity index and we illustrate our results by numerical  
simulations.


  http://arxiv.org/abs/0811.2664

---------------------------------------------------------------

7780. THE "NORTH POLE PROBLEM" AND RANDOM ORTHOGONAL MATRICES

Morris L. Eaton and  Robb J. Muirhead

This paper is motivated by the following observation. Take a 3 x 3  
random
(Haar distributed) orthogonal matrix $\Gamma$, and use it to "rotate"  
the north
pole, $x_0$ say, on the unit sphere in $R^3$. This then gives a point  
$u=\Gamma
x_0$ that is uniformly distributed on the unit sphere. Now use the same
orthogonal matrix to transform u, giving $v=\Gamma u=\Gamma^2 x_0$.  
Simulations
reported in Marzetta et al (2002) suggest that v is more likely to be  
in the
northern hemisphere than in the southern hemisphere, and, morever, that
$w=\Gamma^3 x_0$ has higher probability of being closer to the poles $ 
\pm x_0$
than the uniformly distributed point u. In this paper we prove these  
results,
in the general setting of dimension $p\ge 3$, by deriving the exact
distributions of the relevant components of u and v. The essential  
questions
answered are the following. Let x be any fixed point on the unit  
sphere in
$R^p$, where $p\ge 3$. What are the distributions of $U_2=x'\Gamma^2 x 
$ and
$U_3=x'\Gamma^3 x$? It is clear by orthogonal invariance that these
distribution do not depend on x, so that we can, without loss of  
generality,
take x to be $x_0=(1,0,...,0)'\in R^p$. Call this the "north pole". Then
$x_0'\Gamma^ k x_0$ is the first component of the vector $\Gamma^k  
x_0$. We
derive stochastic representations for the exact distributions of $U_2$  
and
$U_3$ in terms of random variables with known distributions.


  http://arxiv.org/abs/0811.2678

---------------------------------------------------------------

7781. PARTICLE APPROXIMATION OF SOME LANDAU EQUATIONS

Nicolas Fournier

We consider a class of nonlinear partial-differential equations,  
including
the spatially homogeneous Fokker-Planck-Landau equation for Maxwell (or
pseudo-Maxwell) molecules. Continuing the work of
Fontbona-Gu\'erin-M\'el\'eard, we propose a probabilistic  
interpretation of
such a P.D.E. in terms of a nonlinear stochastic differential equation  
driven
by a standard Brownian motion. We derive a numerical scheme, based on  
a system
of $n$ particles driven by $n$ Brownian motions, and study its rate of
convergence. We finally deal with the possible extension of our  
numerical
scheme to the case of the Landau equation for soft potentials, and  
give some
numerical results.


  http://arxiv.org/abs/0811.2688

---------------------------------------------------------------

7782. AN UPPER BOUND ON THE CRITICAL DENSITY FOR ACTIVATED RANDOM  
WALKS ON  EUCLIDEAN LATTICES

Eric Shellef

We show the critical density for activated random walks on Euclidean  
lattices
is at most one.


  http://arxiv.org/abs/0811.2892

---------------------------------------------------------------

7783. RANDOM COMPLEXES AND L^2-BETTI NUMBERS

Russell Lyons

Uniform spanning trees on finite graphs and their analogues on infinite
graphs are a well-studied area. On a Cayley graph of a group, we show  
that they
are related to the first $\ell^2$-Betti number of the group. Our main  
aim,
however, is to present the basic elements of a higher-dimensional  
analogue on
finite and infinite CW-complexes, which relate to the higher $\ell^2$- 
Betti
numbers. One consequence is a uniform isoperimetric inequality  
extending work
of Lyons, Pichot, and Vassout. We also present an enumeration similar  
to recent
work of Duval, Klivans, and Martin.


  http://arxiv.org/abs/0811.2933

---------------------------------------------------------------

7784. THE STRUCTURE OF TYPICAL CLUSTERS IN LARGE SPARSE RANDOM  
CONFIGURATIONS

Jean Bertoin (DMA and  PMA) and  Vladas Sidoravicius (UCI and  CWI)

The initial purpose of this work is to provide a probabilistic  
explanation of
a recent result on a version of Smoluchowski's coagulation equations  
in which
the number of aggregations is limited. The latter models the  
deterministic
evolution of concentrations of particles in a medium where particles  
coalesce
pairwise as time passes and each particle can only perform a given  
number of
aggregations. Under appropriate assumptions, the concentrations of  
particles
converge as time tends to infinity to some measure which bears a  
striking
resemblance with the distribution of the total population of a Galton- 
Watson
process started from two ancestors. Roughly speaking, the  
configuration model
is a stochastic construction which aims at producing a typical graph  
on a set
of vertices with pre-described degrees. Specifically, one attaches to  
each
vertex a certain number of stubs, and then join pairwise the stubs  
uniformly at
random to create edges between vertices. In this work, we use the  
configuration
model as the stochastic counterpart of Smoluchowski's coagulation  
equations
with limited aggregations. We establish a hydrodynamical type limit  
theorem for
the empirical measure of the shapes of clusters in the configuration  
model when
the number of vertices tends to $\infty$. The limit is given in terms  
of the
distribution of a Galton-Watson process started with two ancestors.


  http://arxiv.org/abs/0811.2988

---------------------------------------------------------------

7785. ON REALATIONS BETWEEN URBANIK NAD MEHLER SEMIGROUPS

Zbigniew J. Jurek

It is shown that operator-selfdecomposable measures, or more precisely  
their
Urbanik decomposability semigroups, induce generalized Mehler  
semigroups of
bounded linear operators. Moreover, those semigroups can be  
represented as
random integrals of operator valued functions with respect to  
stochastic L\'evy
processes. Our Banach space setting is in the contrast with the  
Hilbert spaces
on which so far and most often the generalized Mehler semigroups were  
studied.
Furthermore, we give new proofs of the random integral representation.


  http://arxiv.org/abs/0811.2989

---------------------------------------------------------------

7786. INFORMATION PERCOLATION WITH EQUILIBRIUM SEARCH DYNAMICS

Darrell Duffie and  Semyon Malamud and  Gustavo Manso

We solve for the equilibrium dynamics of information sharing in a large
population. Each agent is endowed with signals regarding the likely  
outcome of
a random variable of common concern. Individuals choose the effort  
with which
they search for others from whom they can gather additional  
information. When
two agents meet, they share their information. The information  
gathered is
further shared at subsequent meetings, and so on. Equilibria exist in  
which
agents search maximally until they acquire sufficient information  
precision,
and then minimally. A tax whose proceeds are used to subsidize the  
costs of
search improves information sharing and can in some cases increase  
welfare. On
the other hand, endowing agents with public signals reduces  
information sharing
and can in some cases decrease welfare.


  http://arxiv.org/abs/0811.3023

---------------------------------------------------------------

7787. INFORMATION PERCOLATION

Darrell Duffie and  Gaston Giroux and  Gustavo Manso

For a setting in which a large number of asymmetrically informed  
agents are
randomly matched into groups over time, exchanging their information  
with each
other when matched, we provide an explicit solution for the dynamics  
of the
cross-sectional distribution of posterior beliefs. We also show that
convergence of the cross-sectional distribution of beliefs to a common
posterior is exponential and that the rate of convergence does not  
depend on
the size of the groups of agents that meet. The rate of convergence is  
merely
the mean rate at which an individual agent is matched.


  http://arxiv.org/abs/0811.3024

---------------------------------------------------------------

7788. EXACT AND ASYMPTOTIC $N$-TUPLE LAWS AT FIRST AND LAST PASSAGE

A. Kyprianou and  J.C. Pardo and V. Rivero

Understanding the space-time features of how a L\'evy process crosses a
constant barrier for the first time, and indeed the last time, is a  
problem
which is central to many models in applied probability such as  
queueing theory,
financial and actuarial mathematics, optimal stopping problems, the  
theory of
branching processes to name but a few. In \cite{KD} a new quintuple  
law was
established for a general L\'evy process at first passage above a  
fixed level.
In this article we use the quintuple law to establish a family of  
related joint
laws, which we call $n$-tuple laws, for L\'evy processes, L\'evy  
processes
conditioned to stay positive and positive self-similar Markov  
processes at both
first and last passage over a fixed level. Here the integer $n$  
typically
ranges from three to seven. Moreover, we look at asymptotic overshoot  
and
undershoot distributions and relate them to overshoot and undershoot
distributions of positive self-similar Markov processes issued from  
the origin.
Although the relation between the $n$-tuple laws for L\'evy processes  
and
positive self-similar Markov processes are straightforward thanks to the
Lamperti transformation, by inter-playing the role of a (conditioned)  
stable
processes as both a (conditioned) L\'evy processes and a positive self- 
similar
Markov processes, we obtain a suite of completely explicit first and  
last
passage identities for so-called Lamperti-stable L\'evy processes.  
This leads
further to the introduction of a more general family of L\'evy  
processes which
we call hypergeometric L\'evy processes, for which similar explicit  
identities
may be considered.


  http://arxiv.org/abs/0811.3075

---------------------------------------------------------------

7789. RANDOM TREE GROWTH BY VERTEX SPLITTING

Francois David and  Mark Dukes and  Thordur Jonsson and  Sigurdur Orn  
Stefansson

We study a model of growing planar tree graphs where in each time step  
we
separate the tree into two components by splitting a vertex and then  
connect
the two pieces by inserting a new link between the daughter vertices. We
develop a mean field theory for the vertex degree distribution, prove  
that the
mean field theory is exact in some special cases and check that it  
agrees with
numerical simulations in general. We calculate various correlation  
functions
and show that the intrinsic Hausdorff dimension can vary from one to  
infinity,
depending on the parameters of the model.


  http://arxiv.org/abs/0811.3183

---------------------------------------------------------------

7790. LAGRANGIAN STRUCTURES FOR THE STOKES, NAVIER-STOKES AND EULER  
EQUATIONS

Jacky Cresson (IMCCE and  LMA-PAU) and  S\'ebastien Darses (BU)

We prove that the Navier-Stokes, the Euler and the Stokes equations  
admit a
Lagrangian structure using the stochastic embedding of Lagrangian  
systems.
These equations coincide with extremals of an explicit stochastic  
Lagrangian
functional, i.e. they are stochastic Lagrangian systems in the sense of
[Cresson-Darses, J. Math. Phys. 48, 072703 (2007]


  http://arxiv.org/abs/0811.3286

---------------------------------------------------------------

7791. THE FIRST DIGIT FREQUENCIES OF PRIMES AND RIEMANN ZETA ZEROS  
TEND TO  UNIFORMITY FOLLOWING A SIZE-DEPENDENT GENERALIZED BENFORD'S LAW

Bartolo Luque and  Lucas Lacasa

Prime numbers seem to distribute among the natural numbers with no  
other law
than that of chance, however its global distribution presents a quite
remarkable smoothness. Such interplay between randomness and  
regularity has
motivated sci- entists of all ages to search for local and global  
patterns in
this distribution that eventually could shed light into the ultimate  
nature of
primes. In this work we show that a generalization of the well known
first-digit Benford's law, which addresses the rate of appearance of a  
given
leading digit d in data sets, describes with astonishing precision the
statistical distribution of leading digits in the prime numbers  
sequence.
Moreover, a reciprocal version of this pattern also takes place in the  
sequence
of the nontrivial Riemann zeta zeros. We prove that the prime number  
theorem
is, in the last analysis, the responsible of these patterns. Some new  
relations
concerning the prime numbers distribution are also deduced, including  
a new
approximation to the counting function pi(n). Furthermore, some  
relations
concerning the statistical conformance to this generalized Benford's  
law are
derived. Some applications are finally discussed.


  http://arxiv.org/abs/0811.3302

---------------------------------------------------------------

7792. DISTRIBUTION OF NORMALIZED ZERO-SETS OF RANDOM ENTIRE FUNCTIONS

Weihong Yao

This paper is concerned with the distribution of normalized zero-sets of
random entire functions. The normalization of the zero-set is  
performed in the
same way as that of the counting function for an entire function in  
Nevanlinna
theory. The result generalizes the Shiffman and Zelditch theory on the
distribution of the zeroes of random holomorphic sections of powers for
positive Hermitian holomorphic line bundles from polynomial functions  
to entire
functions. Our result can also be viewed as the analogy of  
Nevanlinna's First
Main Theorem in the theory of the distribution of zero-sets of random  
entire
functions.


  http://arxiv.org/abs/0811.3365

---------------------------------------------------------------

7793. AN EXTENSION OF A LOGARITHMIC FORM OF CRAMER'S RUIN THEOREM TO  
SOME  FARIMA AND RELATED PROCESSES

Ph. Barbe (CNRS) and W.P. McCormick (UGA)

Cramer's theorem provides an estimate for the tail probability of the  
maximum
of a random walk with negative drift and increments having a moment  
generating
function finite in a neighborhood of the origin. The class of (g,F)- 
processes
generalizes in a natural way random walks and fractional ARIMA models  
used in
time series analysis. For those (g,F)-processes with negative drift,  
we obtain
a logarithmic estimate of the tail probability of their maximum, under
conditions comparable to Cramer's. Furthermore, we exhibit the most  
likely
paths as well as the most likely behavior of the innovations leading  
to a large
maximum.


  http://arxiv.org/abs/0811.3460

---------------------------------------------------------------

7794. ON THE RESIDUAL DEPENDENCE INDEX ELLIPTICAL DISTRIBUTIONS

Enkelejd Hashorva

The residual dependence index of bivariate Gaussian distributions is
determined by the correlation coefficient. This tail index is of certain
statistical importance when extremes and related rare events of  
bivariate
samples with asymptotic independent components are being modeled. In  
this paper
we calculate the partial residual dependence indices of a multivariate
elliptical random vector assuming that the associated random radius is  
in the
Gumbel max-domain of attraction. Furthermore, we discuss the  
estimation of
these indices when the associated random radius possesses a Weibull-tail
distribution.


  http://arxiv.org/abs/0811.3552

---------------------------------------------------------------

7795. DISCRETE RANDOM WALK WITH BARRIERS ON A LOCALLY INFINITE GRAPH

Theo van Uem

We obtain expected number of arrivals, absorption probabilities and  
expected
time before absorption for an asymmetric discrete random walk on a  
locally
infinite graph in the presence of multiple function barriers


  http://arxiv.org/abs/0811.3682

---------------------------------------------------------------

7796. THE TASEP SPEED PROCESS

Gideon Amir and  Omer Angel and Benedek Valko

In a multi-type totally asymmetric simple exclusion process (TASEP) on  
the
line, each site of Z is occupied by a particle labeled with a number  
and two
neighboring particles are interchanged at rate one if their labels are  
in
increasing order. Consider the process with the initial configuration  
where
each particle is labeled by its position. It is known that in this  
case a.s.
each particle has an asymptotic speed which is distributed uniformly  
on [-1,1].
We study the joint distribution of these speeds: the TASEP speed  
process.
   We prove that the TASEP speed process is stationary with respect to  
the
multi-type TASEP dynamics. Consequently, every ergodic stationary  
measure is
given as a projection of the speed process measure.
   By relating this form to the known stationary measures for multi- 
type TASEPs
with finitely many types we compute several marginals of the speed  
process,
including the joint density of two and three consecutive speeds. One  
striking
property of the distribution is that two speeds are equal with positive
probability and for any given particle there are infinitely many  
others with
the same speed.
   We also study the (partially) asymmetric simple exclusion process  
(ASEP). We
prove that the ASEP with the above initial configuration has a certain
symmetry. This allows us to extend some of our results, including the
stationarity and description of all ergodic stationary measures, also  
to the
ASEP.


  http://arxiv.org/abs/0811.3706

---------------------------------------------------------------

7797. QUANTILE HEDGING FOR AN INSIDER

Przemyslaw Klusik and  Zbigniew Palmowski and  Jakub Zwierz

In this paper we consider the problem of the quantile hedging from the  
point
of view of a better informed agent acting on the market. The additional
knowledge of the agent is modelled by a filtration initially enlarged  
by some
random variable. By using equivalent martingale measures introduced in
Amendinger (2000) and Amendinger, Imkeller and Schweizer (1998) we  
solve the
problem for the complete case, by extending the results obtained in  
F{\"o}llmer
and Leukert (1999) to the insider context. Finally, we consider the  
examples
with the explicit calculations within the standard Black-Scholes model.


  http://arxiv.org/abs/0811.3749

---------------------------------------------------------------

7798. A NOTE ON A COMPOSITION OF TWO RANDOM INTEGRAL MAPPINGS $\J^\BE$  
AND  SOME EXAMPLES

Agnieszka Czyzewska-Jankowska and Zbigniew J. Jurek

A method of random integral representation, that is, a method of  
representing
a given probability measure as the probability distribution of some  
random
integral, was quite successful in the past few decades. In this note  
we show
that a composition of two random integral mappings $\J^\be$ is again a  
random
integral mapping. We illustrate our results on some examples.


  http://arxiv.org/abs/0811.3750

---------------------------------------------------------------

7799. A CALCULUS ON L\'EVY EXPONENTS AND SELFDECOMPOSABILITY ON BANACH  
SPACES

Zbigniew J. Jurek

In infinite dimensional Banach spaces there is no complete  
characterization
of the L\'evy exponents of infinitely divisible probability measures.  
Here we
propose \emph{a calculus on L\'evy exponents} that is derived from  
some random
integrals. As a consequence we prove that \emph{each} selfdecomposable  
measure
can by factorized as another selfdecomposable measure and its background
driving measure that is s-selfdecomposable. This complements a result  
from the
paper of Iksanov-Jurek-Schreiber in the Annals of Probability  
\textbf{32},
2004.}


  http://arxiv.org/abs/0811.3752

---------------------------------------------------------------

7800. LIMIT THEOREMS FOR P-VARIATIONS OF SOLUTIONS OF SDES DRIVEN BY  
ADDITIVE  NON-GAUSSIAN STABLE LEVY NOISE

C. Hein and  P. Imkeller and I. Pavlyukevich

In this paper we study the asymptotic properties of the power  
variations of
stochastic processes of the type X=Y+L, where L is an alpha-stable Levy
process, and Y a perturbation which satisfies some mild Lipschitz  
continuity
assumptions. We establish local functional limit theorems for the power
variation processes of X. In case X is a solution of a stochastic  
differential
equation driven by L, these limit theorems provide estimators of the  
stability
index alpha. They are applicable for instance to model fitting  
problems for
paleo-climatic temperature time series taken from the Greenland ice  
core.


  http://arxiv.org/abs/0811.3769

---------------------------------------------------------------

7801. THE REGULARIZING EFFECTS OF RESETTING IN A PARTICLE SYSTEM FOR  
THE  BURGERS' EQUATION

Gautam Iyer and  Alexei Novikov

We study the dissipation mechanism of a stochastic particle system for  
the
Burgers' equation. The velocity field of the viscous Burgers' and  
Navier-Stokes
equations can be expressed as an expected value of a stochastic  
process based
on noisy particle trajectories (Constantin, Iyer, Comm. Pure Appl.  
Math, 2008).
In this paper we study a particle system for the viscous Burgers'  
equations
using a Monte-Carlo version of the above; we consider $N$ copies of  
the above
stochastic flow, each driven by independent Wiener processes, and  
replace the
expected value with $\frac{1}{N}$ times the sum over these copies. A  
similar
construction for the Navier-Stokes equations was studied by J.  
Mattingly and
the first author (\texttt{arXiv:0803.1222}, to appear in Nonlinearity).
   Surprisingly, for any finite $N$, the particle system for the  
Burgers'
equations shocks almost surely in finite time. In contrast to the full  
expected
value, the empirical mean $\frac{1}{N} \sum_1^N$ does not regularize  
the system
enough to ensure a time global solution. To avoid these shocks, we  
consider a
resetting procedure, which at first sight should have no regularizing  
effect at
all. We however prove that this procedure prevents the formation of  
shocks for
any $N \geq 2$, and consequently as $N \to \infty$ we get convergence  
to the
solution of the viscous Burgers' equations on long time intervals.


  http://arxiv.org/abs/0811.3799

---------------------------------------------------------------

7802. MULTIVARIATE UTILITY MAXIMIZATION WITH PROPORTIONAL TRANSACTION  
COSTS

Luciano Campi and Mark P. Owen

We present an optimal investment theorem for a currency exchange model  
with
random and possibly discontinuous proportional transaction costs. The
investor's preferences are represented by a multivariate utility  
function,
allowing for simultaneous consumption of any prescribed selection of the
currencies at a given terminal date. We prove the existence of an  
optimal
portfolio process under the assumption of asymptotic satiability of  
the value
function. Sufficient conditions for asymptotic satiability of the value
function include reasonable asymptotic elasticity of the utility  
function, or a
growth condition on its dual function. We show that the portfolio  
optimization
problem can be reformulated in terms of maximization of a terminal  
liquidation
utility function, and that both problems have a common optimizer.


  http://arxiv.org/abs/0811.3889

---------------------------------------------------------------

7803. PREDICTABILITY IN NONLINEAR DYNAMICAL SYSTEMS WITH MODEL  
UNCERTAINTY

Jinqiao Duan

Nonlinear systems with model uncertainty are often described by  
stochastic
differential equations. Some techniques from random dynamical systems  
are
discussed. They are relevant to better understanding of solution  
processes of
stochastic differential equations and thus may shed lights on  
predictability in
nonlinear systems with model uncertainty.


  http://arxiv.org/abs/0811.3697

---------------------------------------------------------------

7804. THE RELATIONSHIP BETWEEN TSALLIS STATISTICS, THE FOURIER  
TRANSFORM, AND  NONLINEAR COUPLING

Kenric P. Nelson and Sabir Umarov

Tsallis statistics (or q-statistics) in nonextensive statistical  
mechanics is
a one-parameter description of correlated states. In this paper we use a
translated entropic index: $1 - q \to q$ . The essence of this  
translation is
to improve the mathematical symmetry of the q-algebra and make q  
directly
proportional to the nonlinear coupling. A conjugate transformation is  
defined
$\hat q \equiv \frac{{- 2q}}{{2 + q}}$ which provides a dual mapping  
between
the heavy-tail q-Gaussian distributions, whose translated q parameter is
between $ - 2 < q < 0$, and the compact-support q-Gaussians, between  
$0 < q <
\infty $ . This conjugate transformation is used to extend the  
definition of
the q-Fourier transform to the domain of compact support. A conjugate  
q-Fourier
transform is proposed which transforms a q-Gaussian into a conjugate $ 
\hat q$
-Gaussian, which has the same exponential decay as the Fourier  
transform of a
power-law function. The nonlinear statistical coupling is defined such  
that the
conjugate pair of q-Gaussians have equal strength but either couple
(compact-support) or decouple (heavy-tail) the statistical states.  
Many of the
nonextensive entropy applications can be shown to have physical  
parameters
proportional to the nonlinear statistical coupling.


  http://arxiv.org/abs/0811.3777

---------------------------------------------------------------

7805. RESCALED LEVY-LOEWNER HULLS AND RANDOM GROWTH

Fredrik Johansson and Alan Sola

We consider radial Loewner evolution driven by unimodular L\'evy  
processes.
We rescale the hulls of the evolution by capacity, and prove that the  
weak
limit of the rescaled hulls exists. We then study a random growth model
obtained by driving the Loewner equation with a compound Poisson  
process. The
process involves two real parameters: the intensity of the underlying  
Poisson
process and a localization parameter of the Poisson kernel which  
determines the
jumps. A particular choice of parameters yields a growth process  
similar to the
Hastings-Levitov $\rm{HL}(0)$ model. We describe the asymptotic  
behavior of the
hulls with respect to the parameters, showing that growth tends to  
become
localized as the jump parameter increases. We obtain deterministic  
evolutions
in one limiting case, and Loewner evolution driven by a unimodular  
Cauchy
process in another. We show that the Hausdorff dimension of the limiting
rescaled hulls is equal to 1. Using a different type of compound Poisson
process, where the Poisson kernel is replaced by the heat kernel, as  
driving
function, we recover one case of the aforementioned model and
$\rm{SLE}(\kappa)$ as limits.


  http://arxiv.org/abs/0811.3857

---------------------------------------------------------------

7806. HEDGING OF DEFAULTABLE CONTINGENT CLAIMS USING BSDE WITH  
UNCERTAIN TIME  HORIZON

Christophette Blanchet-Scalliet (ICJ) and  Anne Eyraud-Loisel (SAF -   
EA2429), Manuela Royer-Carenzi (LATP)

This article focuses on the mathematical problem of existence and  
uniqueness
of BSDE with a random terminal time which is a general random variable  
but not
a stopping time, as it has been usually the case in the previous  
literature of
BSDE with random terminal time. The main motivation of this work is a  
financial
or actuarial problem of hedging of defaultable contingent claims or life
insurance contracts, for which the terminal time is a default time or  
a death
time, which are not stopping times. We have to use progressive  
enlargement of
the Brownian filtration, and to solve the obtained BSDE under this  
enlarged
filtration. This work gives a solution to the mathematical problem and  
proves
the existence and uniqueness of solutions of such BSDE under certain  
general
conditions. This approach is applied to the financial problem of  
hedging of
defaultable contingent claims, and an expression of the hedging  
strategy is
given for a defaultable contingent claim or a life insurance contract.


  http://arxiv.org/abs/0811.4039

---------------------------------------------------------------

7807. BROWNIAN MOTION CONDITIONED TO STAY IN A CONE

Rodolphe Garbit (LMJL)

A result of R. Durrett, D. Iglehart and D. Miller states that Brownian
meander is Brownian motion conditioned to stay positive for a unit of  
time, in
the sense that it is the weak limit, as $x$ goes to 0, of Brownian  
motion
started at $x>0$ and conditioned to stay positive for a unit of time.  
We extend
this limit theorem to the case of multidimensional Brownian motion  
conditioned
to stay in a smooth convex cone. Properties of the limit process are  
obtained
and applications to random walks are given.


  http://arxiv.org/abs/0811.4079

---------------------------------------------------------------

7808. ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE FRAGMENTATION EQUATION  
WITH  SHATTERING: AN APPROACH VIA SELF-SIMILAR MARKOV PROCESSES

B\'en\'edicte Haas (CEREMADE)

The subject of this paper is a fragmentation equation with non- 
conservative
solutions, some mass being lost to a dust of zero-mass particles as a
consequence of an intensive splitting. Under some assumptions of regular
variation on the fragmentation rate, we describe the large-time  
behavior of
solutions. Our approach is based on probabilistic tools: the solutions  
to the
fragmentation equation are constructed via non-increasing self-similar  
Markov
processes that reach continuously 0 in finite time. Our main  
probabilistic
result describes the asymptotic behavior of these processes  
conditioned on
non-extinction and is then used for the solutions to the fragmentation
equation. We notice that two parameters influence significantly these
large-time behaviors: the rate of formation of "nearly-1 relative  
masses" (this
rate is related to the behavior near 0 of the L\'evy measure  
associated to the
corresponding self-similar Markov process) and the distribution of large
initial particles. Correctly rescaled, the solutions then converge to a
non-trivial limit which is related to the quasi-stationary solutions  
to the
equation. Besides, these quasi-stationary solutions, or equivalently the
quasi-stationary distributions of the self-similar Markov processes, are
entirely described.


  http://arxiv.org/abs/0811.4267

---------------------------------------------------------------

7809. RANDOM WALKS IN RANDOM DIRICHLET ENVIRONMENT ARE TRANSIENT IN  
DIMENSION  $D\GE 3$

Christophe Sabot (ICJ)

We consider random walks in random Dirichlet environment (RWDE) which  
is a
special type of random walks in random environment where the exit  
probabilities
at each site are i.i.d. Dirichlet random variables. On $Z^d$, RWDE are
parameterized by a 2d-uplet of positive reals. We prove that for all  
values of
the parameters, RWDE are transient in dimension $d\ge 3$. We also  
prove that
the Green function has some finite moments and, on $Z^d$, $d\ge 3$, we
explicitly compute the critical integrability exponent. Our result is  
more
general and applies forexample to finitely generated transient Cayley  
graphs.
In terms of reinforced random walks it implies that linearly edge- 
oriented
reinforced random walks are transient for $d\ge 3$.


  http://arxiv.org/abs/0811.4285

---------------------------------------------------------------

7810. LINEAR STOCHASTIC SYSTEMS: A WHITE NOISE APPROACH

Daniel Alpay and David Levanony

Using the white noise setting, in particular the Wick product, the  
Hermite
transform, and the Kondratiev space, we present a new approach to  
study linear
stochastic systems, where randomness is also included in the transfer  
function.
We prove BIBO type stability theorems for these systems, both in the  
discrete
and continuous time cases. We also consider the case of dissipative  
systems for
both discrete and continuous time systems. We further study $\ell_1$-$ 
\ell_2$
stability in the discrete time case, and ${\mathbf L}_2$-${\mathbf L}_ 
\infty$
stability in the continuous time case.


  http://arxiv.org/abs/0811.4321

---------------------------------------------------------------

7811. CONVERGENCE TO WEIGHTED FRACTIONAL BROWNIAN SHEETS

Johanna Garz\'on

We define weighted fractional Brownian sheets, which are a class of  
Gaussian
random fields with four parameters that include fractional Brownian  
sheets as
special cases, and we give some of their properties. We show that for  
certain
values of the parameters the weighted fractional Brownian sheets are  
obtained
as limits in law of occupation time fluctuations of a stochastic  
particle
model. In contrast with some known approximations of fractional  
Brownian sheets
which use a kernel in a Volterra type integral representation of  
fractional
Brownian motion with respect to ordinary Brownian motion, our  
approximation
does not make use of a kernel.


  http://arxiv.org/abs/0811.4455

---------------------------------------------------------------

7812. SECOND ORDER POINCAR\'E INEQUALITIES AND CLTS ON WIENER SPACE

Ivan Nourdin (PMA) and  Giovanni Peccati (MODAL'X) and  Gesine Reinert

We prove infinite-dimensional second order Poincar\'e inequalities on  
Wiener
space, thus closing a circle of ideas linking limit theorems for  
functionals of
Gaussian fields, Stein's method and Malliavin calculus. We provide two
applications: (i) to a new "second order" characterization of CLTs on  
a fixed
Wiener chaos, and (ii) to linear functionals of Gaussian-subordinated  
fields.


  http://arxiv.org/abs/0811.4485

---------------------------------------------------------------

7813. THE 2D ISING MODEL NEAR CRITICALITY: A FK PERCOLATION ANALYSIS

Raphael Cerf and Reda Messikh

We study the 2d-Ising model defined on finite boxes at temperatures  
that are
below but very close from the critical point. When the temperature  
approaches
the critical point and the size of the box grows fast enough, we  
establish
large deviations estimates on FK-percolation events that concern the  
phenomenon
of phase coexistence.


  http://arxiv.org/abs/0811.4507

---------------------------------------------------------------

7814. COAGULATION, DIFFUSION AND THE CONTINUOUS SMOLUCHOWSKI EQUATION

Mohammad Reza Yaghouti and  Fraydoun Rezakhanlou and  Alan Hammond

The Smoluchowski equation is a system of partial differential equations
modelling the diffusion and binary coagulation of a large collection  
of tiny
particles. The mass parameter may be indexed either by positive  
integers, or by
positive reals, these corresponding to the discrete or the continuous  
form of
the equations. In dimension at least 3, we derive the continuous  
Smoluchowski
PDE as a kinetic limit of a microscopic model of Brownian particles  
liable to
coalesce, using a similar method to that used to derive the discrete  
form of
the equations in Hammond and Rezakhanlou [4]. The principal innovation  
is a
correlation-type bound on particle locations that permits the  
derivation in the
continuous context while simplifying the arguments of [4]. We also  
comment on
the scaling satisfied by the continuous Smoluchowski PDE, and its  
potential
implications for blow-up of solutions of the equations.


  http://arxiv.org/abs/0811.4601

---------------------------------------------------------------

7815. RECURRENCE AND TRANSIENCE FOR LONG-RANGE REVERSIBLE RANDOM WALKS  
ON A  RANDOM POINT PROCESS

P. Caputo and  A. Faggionato and  A. Gaudilliere

We consider reversible random walks in random environment obtained from
symmetric long--range jump rates on a random point process. We prove  
almost
sure transience and recurrence results under suitable assumptions on  
the point
process and the jump rate function. For recurrent models we obtain  
almost sure
estimates on effective resistances in finite boxes. For transient  
models we
construct explicit fluxes with finite energy on the associated  
electrical
network.


  http://arxiv.org/abs/0811.4623

---------------------------------------------------------------

7816. THE FALLING APPART OF THE TAGGED FRAGMENT AND THE ASYMPTOTIC   
DISINTEGRATION OF THE BROWNIAN HEIGHT FRAGMENTATION

Ger\'onimo Uribe Bravo

We present a further analysis of the fragmentation at heights of the
normalized Brownian excursion. Specifically we study a representation  
for the
mass of a tagged fragment in terms of a Doob transformation of the 1/2- 
stable
subordinator and use it to study its jumps; this accounts for a  
description of
how a typical fragment falls apart. These results carry over to the  
height
fragmentation of the stable tree. Additionally, the sizes of the  
fragments in
the Brownian fragmentation when it is about to reduce to dust are  
described in
a limit theorem.


  http://arxiv.org/abs/0811.4754

---------------------------------------------------------------

7817. CONVERGENCE RATES OF POSTERIOR DISTRIBUTIONS FOR OBSERVATIONS  
WITHOUT  THE IID STRUCTURE

Yang Xing

The classical condition on the existence of uniformly exponentially
consistent tests for testing the true density against the complement  
of its
arbitrary neighborhood has been widely adopted in study of asymptotics  
of
Bayesian nonparametric procedures. Because we follow a Bayesian  
approach, it
seems to be more natural to explore alternative and appropriate  
conditions
which incorporate the prior distribution. In this paper we supply a new
prior-dependent integration condition to establish general posterior
convergence rate theorems for observations which may not be  
independent and
identically distributed. The posterior convergence rates for such  
observations
have recently studied by Ghosal and van der Vaart \cite{ghv1}. We  
moreover
adopt the Hausdorff $\alpha$-entropy given by Xing and Ranneby
\cite{xir1}\cite{xi1}, which is also prior-dependent and smaller than  
the
widely used metric entropies. These lead to extensions of several  
existing
theorems. In particular, we establish a posterior convergence rate  
theorem for
general Markov processes and as its application we improve on the  
currently
known posterior rate of convergence for a nonlinear autoregressive  
model.


  http://arxiv.org/abs/0811.4677

---------------------------------------------------------------

7818. PRICING FINANCIAL DERIVATIVES BY A MINIMIZING METHOD

Eduard Rotenstein

We shall study backward stochastic differential equations and we will  
present
a new approach for the existence of the solution. This type of  
equation appears
very often in the valuation of financial derivatives in complete  
markets.
Therefore, the identification of the solution as the unique element in a
certain Banach space where a suitably chosen functional attains its  
minimum
becomes interesting for numerical computations.


  http://arxiv.org/abs/0811.4613

---------------------------------------------------------------

7819. UTILITY MAXIMIZATION IN INCOMPLETE MARKETS WITH DEFAULT

Thomas Lim (PMA) and  Marie-Claire Quenez (PMA)

We adress the maximization problem of expected utility from terminal  
wealth.
The special feature of this paper is that we consider a financial  
market where
the price process of risky assets can have a default time. Using dynamic
programming, we characterize the value function with a backward  
stochastic
differential equation and the optimal portfolio policies. We  
separately treat
the cases of exponential, power and logarithmic utility.


  http://arxiv.org/abs/0811.4715

---------------------------------------------------------------

7820. SOME EXAMPLES OF DYNAMICS FOR GELFAND TSETLIN PATTERNS

Jon Warren and  Peter Windridge

We give examples of stochastic processes in the Gelfand Tsetlin cone  
in which
each component evolves independently apart from a blocking and pushing
interaction. The processes give couplings to certain conditioned Markov
processes, last passage times and asymetric exclusion processes. An  
example of
a cone valued process whose components cannot escape past a wall at  
the origin
is also considered.


  http://arxiv.org/abs/0812.0022

---------------------------------------------------------------

7821. CUTOFF PHENOMENA FOR RANDOM WALKS ON RANDOM REGULAR GRAPHS

Eyal Lubetzky and  Allan Sly

The cutoff phenomenon describes a sharp transition in the convergence  
of a
family of ergodic finite Markov chains to equilibrium. Many natural  
families of
chains are believed to exhibit cutoff, and yet establishing this fact  
is often
extremely challenging. An important such family of chains is the  
random walk on
$\G(n,d)$, a random $d$-regular graph on $n$ vertices. It is well  
known that
the spectral gap of this class of chains for $d \geq 3$ fixed is  
constant,
implying a mixing-time of $O(\log n)$. According to a conjecture of  
Peres, the
simple random walk on $\G(n,d)$ for such $d$ should then exhibit  
cutoff whp. As
a special case of this, Durrett conjectured that the mixing time of  
the lazy
random walk on a random 3-regular graph is whp $(6+o(1))\log_2 n$.
   In this work we confirm the above conjectures, and establish cutoff  
in
total-variation, its location and its optimal window, both for simple  
and for
non-backtracking random walks on $\G(n,d)$. Namely, for any fixed $d  
\geq 3$,
the simple random walk on $\G(n,d)$ whp has cutoff at $\frac{d} 
{d-2}\log_{d-1}
n$ with window order $\sqrt{\log n}$. Surprisingly, the non- 
backtracking random
walk on $\G(n,d)$ whp has cutoff already at $\log_{d-1} n$ with  
constant window
order. We further extend these results to $\G(n,d)$ for any $d=n^{o(1)}$
(beyond which the mixing time is O(1)), provide efficient algorithms for
testing cutoff, as well as give explicit constructions where cutoff  
occurs.


  http://arxiv.org/abs/0812.0060

---------------------------------------------------------------

7822. EXTENDING THE SET OF QUADRATIC EXPONENTIAL VECTORS

Luigi Accardi and  Ameur Dhahri and  Michael Skeide

We extend the square of white noise algebra over the step functions on  
R to
the test function space of bounded square-integrable functions on R^d,  
and we
show that in the Fock representation the exponential vectors exist for  
all test
functions bounded by 1/2.


  http://arxiv.org/abs/0812.0089

---------------------------------------------------------------

7823. FIXATION PROBABILITY FOR COMPETING SELECTIVE SWEEPS

Feng Yu and  Alison Etheridge and  Charles Cuthbertson

We consider a biological population in which a beneficial mutation is
undergoing a selective sweep when a second beneficial mutation arises  
at a
linked locus and we investigate the probability that both mutations will
eventually fix in the population. Previous work has dealt with the  
case where
the second mutation to arise confers a smaller benefit than the first.  
In that
case population size plays almost no role. Here we consider the  
opposite case
and observe that, by contrast, the probability of both mutations  
fixing can be
heavily dependent on population size. Indeed the key parameter is $ 
\rho N$, the
product of the population size and the recombination rate between the  
two
selected loci. If $\rho N$ is small, the probability that both  
mutations fix
can be reduced through interference to almost zero while for large $ 
\rho N$ the
mutations barely influence one another. The main rigorous result is a  
method
for calculating the fixation probability of a double mutant in the large
population limit.


  http://arxiv.org/abs/0812.0104

---------------------------------------------------------------

7824. BOUNDS FOR THE RETURN PROBABILITY OF THE DELAYED RANDOM WALK ON  
FINITE  PERCOLATION CLUSTERS IN THE CRITICAL CASE

Florian Sobieczky

By an eigenvalue comparison-technique, the expected return probability  
of the
delayed random walk on the finite clusters of critical Bernoulli bond
percolation on the two-dimensional Euclidean lattice is estimated. The  
results
are generalised to invariant percolations on unimodular graphs with  
almost
surely finite clusters. A similar method has been used elsewhere to  
derive
bounds for invariant percolation of finite clusters on unimodular  
transitive
graphs. It is adapted here to match the special situation of  
criticality. The
approach followed here involves using the special property of Cartesian
Products of finite graphs with cycles of a certain minimal size to be
Hamiltonian.


  http://arxiv.org/abs/0812.0117

---------------------------------------------------------------

7825. MULTIPLE INTERSECTION EXPONENTS

Achim Klenke and  Peter M\"orters

Let $p\ge2$, $n_1\le...\le n_p$ be positive integers and $B_1^1, ...,
B_{n_1}^1; ...; B_1^p, ..., B_{n_p}^{p}$ be independent planar  
Brownian motions
started uniformly on the boundary of the unit circle. We define a $p$- 
fold
intersection exponent $\varsigma(n_1,..., n_p)$, as the exponential  
rate of
decay of the probability that the packets $\bigcup_{j=1}^{n_i}  
B_j^i[0,t^2]$,
$i=1,...,p$, have no joint intersection. The case $p=2$ is well-known  
and,
following two decades of numerical and mathematical activity, Lawler,  
Schramm
and Werner (2001) rigorously identified precise values for these  
exponents. The
exponents have not been investigated so far for $p>2$. We present an  
extensive
mathematical and numerical study, leading to an exact formula in the  
case
$n_1=1$, $n_2=2$, and several interesting conjectures for other cases.


  http://arxiv.org/abs/0812.0131

---------------------------------------------------------------

7826. COMPLETE CONVERGENCE OF MESSAGE PASSING ALGORITHMS FOR SOME   
SATISFIABILITY PROBLEMS

Uriel Feige and  Elchanan Mossel and Dan Vilenchik

Experimental results show that certain message passing algorithms,  
namely,
Survey Propagation, are very effective in finding satisfying  
assignments for
random satisfiable 3CNF formulas which are considered hard for other SAT
heuristics. Unfortunately, rigorous understanding of this phenomena is  
still
lacking. In this paper we make a modest step towards providing rigorous
explanation for the effectiveness of message passing algorithms. We  
analyze the
performance of Warning Propagation, a popular message passing  
algorithm that is
simpler than Survey Propagation. We show that for 3CNF formulas drawn  
from a
certain distribution over random satisfiable 3CNF formulas, commonly  
referred
to as the planted-assignment distribution, running Warning Propagation  
in the
standard way (run message passing until convergence, simplify the  
formula
according to the resulting assignment, and satisfy the remaining  
subformula, if
necessary, using a simple "off the shelf" heuristic) works when the
clause-variable ratio is a sufficiently large constant. We are not  
aware of
previous rigorous analysis of message passing algorithms for  
satisfiability
instances, though such analysis was performed for decoding of Low  
Density
Parity Check (LDPC) Codes. We discuss some of the differences between  
results
for the LDPC setting and our results.


  http://arxiv.org/abs/0812.0147

---------------------------------------------------------------

7827. UNIFORM TIME AVERAGE CONSISTENCY OF MONTE CARLO PARTICLE FILTERS

Ramon van Handel

We prove that bootstrap type Monte Carlo particle filters approximate  
the
optimal nonlinear filter in a time average sense uniformly with  
respect to the
time horizon when the signal is ergodic and the particle system  
satisfies a
tightness property. The latter is satisfied without further  
assumptions when
the signal state space is compact, as well as in the noncompact  
setting when
the signal is geometrically ergodic and the observations satisfy  
additional
regularity assumptions.


  http://arxiv.org/abs/0812.0350

---------------------------------------------------------------

7828. OPTIMAL SEQUENTIAL PROCEDURES WITH BAYES DECISION RULES

Andrey Novikov

In this article, a general problem of sequential statistical inference  
for
general discrete-time stochastic processes is considered. The problem  
is to
minimize an average sample number given that Bayesian risk due to  
incorrect
decision does not exceed some given bound. We characterize the form of  
optimal
sequential stopping rules in this problem. In particular, we have a
characterization of the form of optimal sequential decision procedures  
when the
Bayesian risk includes both the loss due to incorrect decision and the  
cost of
observations.


  http://arxiv.org/abs/0812.0159

---------------------------------------------------------------

7829. DYNAMICS OF POSTCRITICALLY BOUNDED POLYNOMIAL SEMIGROUPS III:   
CLASSIFICATION OF SEMI-HYPERBOLIC SEMIGROUPS AND RANDOM JULIA SETS  
WHICH ARE
   JORDAN CURVES BUT NOT QUASICIRCLES

Hiroki Sumi

We investigate the dynamics of polynomial semigroups (semigroups  
generated by
a family of polynomial maps on the Riemann sphere) and the random  
dynamics of
polynomials on the Riemann sphere. Combining the dynamics of  
semigroups and the
fiberwise (random) dynamics, we give a classification of polynomial  
semigroups
$G$ such that $G$ is generated by a compact family $\Gamma $, the planar
postcritical set of $G$ is bounded, and $G$ is (semi-) hyperbolic. In  
one of
the classes, we have that for almost every sequence $\gamma \in \Gamma
^{\Bbb{N}}$, the Julia set $J_{\gamma}$ of $\gamma $ is a Jordan curve  
but not
a quasicircle, the unbounded component of the Fatou set $F_{\gamma}$ of
$\gamma$ is a John domain, and the bounded component of $F_{\gamma}$  
is not a
John domain. Note that this phenomenon does not hold in the usual  
iteration of
a single polynomial. Moreover, we consider the dynamics of polynomial
semigroups $G$ such that the planar postcritical set of $G$ is bounded  
and the
Julia set is disconnected. Those phenomena of polynomial semigroups  
and random
dynamics of polynomials that do not occur in the usual dynamics of  
polynomials
are systematically investigated.


  http://arxiv.org/abs/0811.4536

---------------------------------------------------------------

7830. MULTIPLICATIVE APPROXIMATION OF WEALTH PROCESSES INVOLVING NO- 
SHORT-SALE  STRATEGIES VIA SIMPLE TRADING

Constantinos Kardaras and  Eckhard Platen

A financial market model with general semimartingale asset-price  
processes
and where agents can only trade using no-short-sale strategies is  
considered.
We show that wealth processes using continuous trading can be  
approximated very
closely by wealth processes using simple combinations of buy-and-hold  
trading.
This approximation is based on controlling the proportions of wealth  
invested
in the assets. As an application, the utility maximization problem is
considered and it is shown that optimal utilities and wealth processes
resulting from continuous trading can be approximated arbitrarily well  
by the
use of simple combinations of buy-and-hold strategies.


  http://arxiv.org/abs/0812.0033

---------------------------------------------------------------

7831. ANNEALED LARGE DEVIATION ESTIMATES FOR THE ENERGY OF A POLYMER

Amine Asselah

We consider the energy of a randomly charged random walk. We assume  
that only
charges on the same site interact. We study the upper and lower tails  
of the
energy, when averaged over both randomness, in dimension three or more.


  http://arxiv.org/abs/0812.0443

---------------------------------------------------------------

7832. ON NEAR OPTIMAL TRAJECTORIES FOR A GAME ASSOCIATED WITH THE   
\INFTY-LAPLACIAN

Rami Atar and  Amarjit Budhiraja

A two-player stochastic differential game representation has recently  
been
obtained for solutions of the equation -\Delta_\infty u=h in a \calC^2  
domain
with Dirichlet boundary condition, where h is continuous and takes  
values in
\RR\setminus\{0\}. Under appropriate assumptions, including smoothness  
of u,
the vanishing \delta limit law of the state process, when both players  
play
\delta-optimally, is identified as a diffusion process with  
coefficients given
explicitly in terms of derivatives of the function u.


  http://arxiv.org/abs/0812.0496

---------------------------------------------------------------

7833. SPATIAL RANDOM PERMUTATIONS WITH SMALL CYCLE WEIGHTS

Volker Betz and  Daniel Ueltschi

We consider the distribution of cycles in two models of random  
permutations,
that are related to one another. In the first model, cycles receive a  
weight
that depends on their length. The second model deals with permutations  
of
points in the space and there is an additional weight that involves  
the length
of permutation jumps. We prove the occurrence of infinite macroscopic  
cycles
above a certain critical density.


  http://arxiv.org/abs/0812.0569

---------------------------------------------------------------

7834. FAST APPROXIMATION OF SOLUTIONS OF SDE'S WITH OBLIQUE REFLECTION  
ON AN  ORTHANT

Krzysztof Czarkowski

We consider the discrete "fast" penalization scheme for SDE's driven by
general semimartingale on orthant $\mathbb{R}_{+}^{d}$ with oblique  
reflection.


  http://arxiv.org/abs/0812.0619

---------------------------------------------------------------

7835. CENSORED GLAUBER DYNAMICS FOR THE MEAN FIELD ISING MODEL

Jian Ding and  Eyal Lubetzky and  Yuval Peres

We study Glauber dynamics for the Ising model on the complete graph on  
$n$
vertices, known as the Curie-Weiss Model. It is well known that at high
temperature ($\beta < 1$) the mixing time is $\Theta(n\log n)$,  
whereas at low
temperature ($\beta > 1$) it is $\exp(\Theta(n))$. Recently, Levin,  
Luczak and
Peres considered a censored version of this dynamics, which is  
restricted to
non-negative magnetization. They proved that for fixed $\beta > 1$, the
mixing-time of this model is $\Theta(n\log n)$, analogous to the
high-temperature regime of the original dynamics. Furthermore, they  
showed
\emph{cutoff} for the original dynamics for fixed $\beta<1$. The  
question
whether the censored dynamics also exhibits cutoff remained unsettled.
   In a companion paper, we extended the results of Levin et al. into  
a complete
characterization of the mixing-time for the Currie-Weiss model.  
Namely, we
found a scaling window of order $1/\sqrt{n}$ around the critical  
temperature
$\beta_c=1$, beyond which there is cutoff at high temperature. However,
determining the behavior of the censored dynamics outside this  
critical window
seemed significantly more challenging.
   In this work we answer the above question in the affirmative, and  
establish
the cutoff point and its window for the censored dynamics beyond the  
critical
window, thus completing its analogy to the original dynamics at high
temperature. Namely, if $\beta = 1 + \delta$ for some $\delta > 0$ with
$\delta^2 n \to \infty$, then the mixing-time has order $(n /
\delta)\log(\delta^2 n)$. The cutoff constant is $(1/2+[2(\zeta^2  
\beta /
\delta - 1)]^{-1})$, where $\zeta$ is the unique positive root of
$g(x)=\tanh(\beta x)-x$, and the cutoff window has order $n / \delta$.


  http://arxiv.org/abs/0812.0633

---------------------------------------------------------------

7836. AN ALMOST SURE LIMIT THEOREM FOR SUPER-BROWNIAN MOTION

Li Wang

We establish an almost sure scaling limit theorem for super-Brownian  
motion
on $\mathbb{R}^d$ associated with the semi-linear equation $u_t =  
{1/2}\Delta u
+\beta u-\alpha u^2$, where $\alpha$ and $\beta$ are positive  
constants. In
this case, the spectral theoretical assumptions that required in Chen  
et al
(2008) are not satisfied. An example is given to show that the main  
results
also hold for some sub-domains in $\mathbb{R}^d$.


  http://arxiv.org/abs/0812.0642

---------------------------------------------------------------

7837. ASKEY-WILSON POLYNOMIALS, QUADRATIC HARNESSES AND MARTINGALES

Wlodek Bryc and Jacek Wesolowski

We use orthogonality measures of Askey-Wilson polynomials to construct  
Markov
processes with linear regressions and quadratic conditional variances.
Askey-Wilson polynomials are orthogonal martingale polynomials for these
processes.


  http://arxiv.org/abs/0812.0657

---------------------------------------------------------------

7838. HOW LONG DOES IT TAKE TO CATCH A WILD KANGAROO?

Ravi Montenegro and Prasad Tetali

The discrete logarithm problem asks to solve for the exponent $x$,  
given the
generator $g$ of a cyclic group $G$ and an element $h\in G$ such that  
$g^x=h$.
We give the first rigorous proof that Pollard's Kangaroo method finds  
the
discrete logarithm in expected time $(3+o(1))\sqrt{b-a}$ when the  
logarithm
$x\in[a,b]$, and $(2+o(1))\sqrt{b-a}$ when $x\in_{uar}[a,b]$. This  
matches the
conjectured time complexity and, rare among the analysis of algorithms  
based on
Markov chains, even the lead constants 2 and 3 are correct.


  http://arxiv.org/abs/0812.0789

---------------------------------------------------------------

7839. PHI-ENTROPY INEQUALITIES FOR DIFFUSION SEMIGROUPS

Fran\c{c}ois Bolley (CEREMADE) and  Ivan Gentil (CEREMADE)

We obtain and study new $\Phi$-entropy inequalities for diffusion  
semigroups,
with Poincar\'e or logarithmic Sobolev inequalities as particular  
cases. From
this study we derive the asymptotic behaviour of a large class of linear
Fokker-Plank type equations under simple conditions, widely extending  
previous
results. Nonlinear diffusion equations are also studied by means of  
these
inequalities. The $\Gamma_2$ criterion of D. Bakry and M. Emery  
appears as a
main tool in the analysis, in local or integral forms.


  http://arxiv.org/abs/0812.0800

---------------------------------------------------------------

7840. QUALITATIVE PROPERTIES OF LOCAL RANDOM INVARIANT MANIFOLDS FOR  
SPDES  WITH QUADRATIC NONLINEARITY

Dirk Blomker and  Wei Wang

The qualitative properties of local random invariant manifolds for  
stochastic
partial differential equations with quadratic nonlinearities and  
multiplicative
noise is studied by a cut off technique. By a detail estimates on the  
Perron
fixed point equation describing the local random invariant manifold, the
structure near a bifurcation is given.


  http://arxiv.org/abs/0812.0390

---------------------------------------------------------------

7841. POISSON BOUNDARY OF THE DISCRETE QUANTUM GROUP A_U(F)^

Stefaan Vaes and Nikolas Vander Vennet

We identify the Poisson boundary of the dual of the universal compact  
quantum
group A_u(F) with a measurable field of ITPFI factors.


  http://arxiv.org/abs/0812.0804

---------------------------------------------------------------

7842. THE VANISHING APPROACH FOR THE AVERAGE CONTINUOUS CONTROL OF  
PIECEWISE  DETERMINISTIC MARKOV PROCESSES

O.L.V. Costa and  F. Dufour

The main goal of this paper is to derive sufficient conditions for the
existence of an optimal control strategy for the long run average  
continuous
control problem of piecewise deterministic Markov processes (PDMP's)  
taking
values in a general Borel space and with compact action space  
depending on the
state variable. In order to do that we apply the so-called vanishing  
discount
approach to obtain a solution to an average cost optimality inequality
associated to the long run average cost problem. Our main assumptions  
are
written in terms of some integro-differential inequalities related to  
the
so-called expected growth condition, and geometric convergence of the  
post-jump
location kernel associated to the PDMP.


  http://arxiv.org/abs/0812.0820

---------------------------------------------------------------

7843. STOCHASTIC VOLTERRA EQUATIONS IN BANACH SPACES AND STOCHASTIC  
PARTIAL  DIFFERENTIAL EQUATIONS

Xicheng Zhang

In this paper, we first study the existence-uniqueness and large  
deviation
estimate of solutions for stochastic Volterra integral equations with  
singular
kernels in 2-smooth Banach spaces. Then, we apply them to a large  
class of
semilinear stochastic partial differential equations (SPDE) driven by  
Brownian
motions as well as by fractional Brownian motions, and obtain the  
existence of
unique maximal strong solutions (in the sense of SDE and PDE) under  
local
Lipschitz conditions. Lastly, high order SPDEs in a bounded domain of  
Euclidean
space, second order SPDEs on complete Riemannian manifolds, as well as
stochastic Navier-Stokes equations are investigated.


  http://arxiv.org/abs/0812.0834

---------------------------------------------------------------

7844. TWO-PARAMETER HEAVY-TRAFFIC LIMITS FOR INFINITE-SERVER QUEUES

Guodong Pang and  Ward Whitt

In order to obtain Markov heavy-traffic approximations for infinite- 
server
queues with general non-exponential service-time distributions and  
general
arrival processes, possibly with time-varying arrival rates, we  
establish
heavy-traffic limits for two-parameter stochastic processes. We  
consider the
random variables $Q^e(t,y)$ and $Q^r(t,y)$ representing the number of  
customers
in the system at time $t$ that have elapsed service times less than or  
equal to
time $y$, or residual service times strictly greater than $y$. We also  
consider
$W^r(t,y)$ representing the total amount of work in service time  
remaining to
be done at time $t+y$ for customers in the system at time $t$. The
two-parameter stochastic-process limits in the space $D([0,\infty),D)$  
of
$D$-valued functions in $D$ draw on, and extend, previous heavy- 
traffic limits
by Glynn and Whitt (1991), where the case of discrete service-time
distributions was treated, and Krichagina and Puhalskii (1997), where  
it was
shown that the variability of service times is captured by the Kiefer  
process
with second argument set equal to the service-time c.d.f.


  http://arxiv.org/abs/0812.0877

---------------------------------------------------------------

7845. DISTRIBUTION AND ASYMPTOTICS UNDER BETA RANDOM SCALING

Enkelejd Hashorva

Let X,Y,B be three independent random variables such that $X$ has the  
same
distribution function as Y B. Assume that B is a Beta random variable  
with
positive parameters a,b and Y has distribution function H. Pakes and  
Navarro
(2007) show under some mild conditions that the distribution function  
H_{a,b}
of X determines H. Based on that result we derive in this paper a  
recursive
formula for calculation of H, if H_{a,b} is known. Furthermore, we  
investigate
the relation between the tail asymptotic behaviour of X and Y. We  
present three
applications of our asymptotic results concerning the extremes of two  
random
samples with underlying distribution functions H and H_{a,b},  
respectively, and
the conditional limiting distribution of bivariate elliptical  
distributions.


  http://arxiv.org/abs/0812.0881

---------------------------------------------------------------

7846. MEIXNER CLASS OF NON-COMMUTATIVE GENERALIZED STOCHASTIC  
PROCESSES WITH  FREELY INDEPENDENT VALUES I. A CHARACTERIZATION

Marek Bozejko and Eugene Lytvynov

Let $T$ be an underlying space with a non-atomic measure $\sigma$ on  
it (e.g.
$T=\mathbb R^d$ and $\sigma$ is the Lebesgue measure). We introduce  
and study a
class of non-commutative generalized stochastic processes, indexed by  
points of
$T$, with freely independent values. Such a process (field),
$\omega=\omega(t)$, $t\in T$, is given a rigorous meaning through  
smearing out
with test functions on $T$, with $\int_T \sigma(dt)f(t)\omega(t)$  
being a
(bounded) linear operator in a full Fock space. We define a set $ 
\mathbf{CP}$
of all continuous polynomials of $\omega$, and then define a con- 
commutative
$L^2$-space $L^2(\tau)$ by taking the closure of $\mathbf{CP}$ in the  
norm
$\|P\|_{L^2(\tau)}:=\|P\Omega\|$, where $\Omega$ is the vacuum in the  
Fock
space. Through procedure of orthogonalization of polynomials, we  
construct a
unitary isomorphism between $L^2(\tau)$ and a (Fock-space-type)  
Hilbert space
$\mathbb F=\mathbb R\oplus\bigoplus_{n=1}^\infty L^2(T^n,\gamma_n)$,  
with
explicitly given measures $\gamma_n$. We identify the Meixner class as  
those
processes for which the procedure of orthogonalization leaves the set $ 
\mathbf
{CP}$ invariant. (Note that, in the general case, the projection of a
continuous monomial of oder $n$ onto the $n$-th chaos need not remain a
continuous polynomial.) Each element of the Meixner class is  
characterized by
two continuous functions $\lambda$ and $\eta\ge0$ on $T$, such that,  
in the
$\mathbb F$ space, $\omega$ has representation
$\omega(t)=\di_t^\dag+\lambda(t)\di_t^\dag\di_t+\di_t+\eta(t)\di_t^\dag 
\di^2_t$,
where $\di_t^\dag$ and $\di_t$ are the usual creation and annihilation
operators at point $t$.


  http://arxiv.org/abs/0812.0895

---------------------------------------------------------------

7847. LOWERING AND RAISING OPERATORS FOR THE FREE MEIXNER CLASS OF  
ORTHOGONAL  POLYNOMIALS

Eugene Lytvynov and  Irina Rodionova

We compare some properties of the lowering and raising operators for the
classical and free classes of Meixner polynomials on the real line.


  http://arxiv.org/abs/0812.0896

---------------------------------------------------------------

7848. THE TWO UNIFORM INFINITE QUADRANGULATIONS OF THE PLANE HAVE THE  
SAME LAW

Laurent Menard

We prove that the uniform infinite random quadrangulations defined
respectively by Chassaing-Durhuus and Krikun have the same distribution.


  http://arxiv.org/abs/0812.0965

---------------------------------------------------------------

7849. SHARP ERROR TERMS FOR RETURN TIME STATISTICS UNDER MIXING  
CONDITIONS

Miguel Abadi Nicolas Vergne

We describe the statistics of repetition times of a string of symbols  
in a
stochastic process. Denote by T(A) the time elapsed until the process  
spells
the finite string A and by S(A) the number of consecutive repetitions  
of A. We
prove that, if the length of the string grows unbondedly, (1) the  
distribution
of T(A), when the process starts with A, is well aproximated by a  
certain
mixture of the point measure at the origin and an exponential law, and  
(2) S(A)
is approximately geometrically distributed. We provide sharp error  
terms for
each of these approximations. The errors we obtain are point-wise and  
allow to
get also approximations for all the moments of T(A) and S(A). To  
obtain (1) we
assume that the process is phi-mixing while to obtain (2) we assume the
convergence of certain contidional probabilities.


  http://arxiv.org/abs/0812.1016

---------------------------------------------------------------

7850. REGULARITY RESULTS FOR STABLE-LIKE OPERATORS

Richard F. Bass

For $\alpha\in [1,2)$ we consider operators of the form $$L f(x)= 
\int_{R^d}
[f(x+h)-f(x)-1_{(|h|\leq 1)} \nabla f(x)\cdot h]
\frac{A(x,h)}{|h|^{d+\alpha}}$$ and for $\alpha\in (0,1)$ we consider  
the same
operator but where the $\nabla f$ term is omitted. We prove, under  
appropriate
conditions on $A(x,h)$, that the solution $u$ to $L u=f$ will be in
$C^{\alpha+\beta}$ if $f\in C^\beta$.


  http://arxiv.org/abs/0812.0982

---------------------------------------------------------------

7851. ASYMPTOTICS OF ONE-DIMENSIONAL FOREST FIRE PROCESSES

Xavier Bressaud and  Nicolas Fournier

We consider the so-called one-dimensional forest-fire process. At each  
site
of $\mathbb{Z}$, a tree appears at rate 1. At each site of $\mathbb{Z} 
$ a fire
starts at rate $\lambda>0$, destroying immediately the whole  
corresponding
connected component of trees. We show that when making $\lambda$ tend  
to 0,
with a correct normalization, the forest-fire process tends to an  
uniquely
defined process, of which we describe precisely the dynamics. The  
normalization
consists of accelerating time by a factor $\log (1/\lambda)$ and of  
compressing
space by a factor $\lambda \log(1/\lambda)$. The limit process is  
quite simple:
it can be built using a graphical construction, and can be perfectly  
simulated.
Finally, we derive some asymptotic estimates (when $\lambda\to 0$) for  
the
cluster-size distribution of the forest-fire process.


  http://arxiv.org/abs/0812.1099

---------------------------------------------------------------

7852. DYNAMICS OF THE TIME TO THE MOST RECENT COMMON ANCESTOR IN A  
LARGE  BRANCHING POPULATION

Steven N. Evans and Peter L. Ralph

If we follow an asexually reproducing population through time, then the
amount of time that has passed since the most recent common ancestor  
(MRCA) of
all current individuals lived will change as time progresses. The  
resulting
stochastic process has been studied previously when the population has a
constant large size and evolves via the diffusion limit of standard
Wright-Fisher dynamics. We investigate cases in which the population  
varies in
size and evolves according to a class of models that includes suitably
conditioned $(1+\beta)$-stable continuous state branching processes (in
particular, it includes the conditioned Feller continuous state  
branching
process). We also consider the discrete time Markov chain that tracks  
the MRCA
age just before and after its successive jumps. We find transition
probabilities for both the continuous and discrete time processes,  
determine
when these processes are transient and recurrent, and compute stationary
distributions when they exist. We also introduce a new family of Markov
processes that stand in a relation with respect to the $(1+\beta)$- 
stable
continuous state branching process that is similar to the one between  
the
Bessel-squared diffusions and the Feller continuous state branching  
process.


  http://arxiv.org/abs/0812.1302

---------------------------------------------------------------

7853. A SIMPLE PROOF OF EXPONENTIAL DECAY IN THE TWO DIMENSIONAL  
PERCOLATION  MODEL

Yu Zhang

Kesten showed the exponential decay of percolation probability in the
subcritical phase for the two-dimensional percolation model. This result
implies his celebrated computation that $p_c=0.5$ for bond percolation  
in the
square lattice, and site percolation in the triangular lattice,  
respectively.
In this paper, we present a simpler proof for Kesten's theorem.


  http://arxiv.org/abs/0812.1384

---------------------------------------------------------------

7854. ON THE LARGEST-EIGENVALUE PROCESS FOR GENERALIZED WISHART  
RANDOM  MATRICES

A.B. Dieker and J. Warren

Using a change-of-measure argument, we prove an equality in law  
between the
process of largest eigenvalues in a generalized Wishart random-matrix  
process
and a last-passage percolation process. This equality in law was  
conjectured by
Borodin and Peche.


  http://arxiv.org/abs/0812.1504

---------------------------------------------------------------

7855. OPTIMAL SEQUENTIAL TESTING OF TWO SIMPLE HYPOTHESES IN PRESENCE  
OF  CONTROL VARIABLES

Andrey Novikov

Suppose that at any stage of a statistical experiment a control  
variable $X$
that affects the distribution of the observed data $Y$ can be used. The
distribution of $Y$ depends on some unknown parameter $\theta$, and we  
consider
the classical problem of testing a simple hypothesis $H_0: \theta= 
\theta_0$
against a simple alternative $H_1: \theta=\theta_1$ allowing the data  
to be
controlled by $X$, in the following sequential context. The experiment  
starts
with assigning a value $X_1$ to the control variable and observing  
$Y_1$ as a
response. After some analysis, we choose another value $X_2$ for the  
control
variable, and observe $Y_2$ as a response, etc. It is supposed that the
experiment eventually stops, and at that moment a final decision in  
favour of
$H_0$ or $H_1$ is to be taken.
   In this article, our aim is to characterize the structure of optimal
sequential procedures, based on this type of data, for testing a simple
hypothesis against a simple alternative.


  http://arxiv.org/abs/0812.1395

---------------------------------------------------------------

7856. ON VERTEX, EDGE, AND VERTEX-EDGE RANDOM GRAPHS

Elizabeth Beer and  James Allen Fill and  Svante Janson and  and  
Edward R.  Scheinerman

We consider three classes of random graphs: edge random graphs, vertex  
random
graphs, and vertex-edge random graphs. Edge random graphs are Erdos- 
Renyi
random graphs, vertex random graphs are generalizations of geometric  
random
graphs, and vertex-edge random graphs generalize both. The names of  
these three
types of random graphs describe where the randomness in the models  
lies: in the
edges, in the vertices, or in both. We show that vertex-edge random  
graphs,
ostensibly the most general of the three models, can be approximated
arbitrarily closely by vertex random graphs, but that the two  
categories are
distinct.


  http://arxiv.org/abs/0812.1410

---------------------------------------------------------------

7857. LARGE DEVIATIONS FOR INTERSECTION LOCAL TIMES IN CRITICAL  
DIMENSION

Castell Fabienne

We prove a large deviations principle for the q-fold (q>1) self- 
intersection
local time of a continuous time simple random walk on the d-dimensional
lattice, in the critical dimension d=(2q)/(q-1). When q is integer, we  
obtain
similar results for the intersection local times of q independent  
simple random
walks.


  http://arxiv.org/abs/0812.1639

---------------------------------------------------------------

7858. POLYMORPHIC EVOLUTION SEQUENCE AND EVOLUTIONARY BRANCHING

Nicolas Champagnat (INRIA Sophia Antipolis / INRIA Lorraine / IECN)  
and   Sylvie M\'el\'eard (CMAP)

We are interested in the study of models describing the evolution of a
polymorphic population with mutation and selection in the specific  
scales of
the biological framework of adaptive dynamics. The population size is  
assumed
to be large and the mutation rate small. We prove that under a good  
combination
of these two scales, the population process is approximated in the  
long time
scale of mutations by a Markov pure jump process describing the  
successive
trait equilibria of the population. This process, which generalizes the
so-called trait substitution sequence, is called polymorphic evolution
sequence. Then we introduce a scaling of the size of mutations and we  
study the
polymorphic evolution sequence in the limit of small mutations. From  
this study
in the neighborhood of evolutionary singularities, we obtain a full
mathematical justification of a heuristic criterion for the phenomenon  
of
evolutionary branching. To this end we finely analyze the asymptotic  
behavior
of 3-dimensional competitive Lotka-Volterra systems.


  http://arxiv.org/abs/0812.1655

---------------------------------------------------------------

7859. TRANSPORT DIFFUSION COEFFICIENT FOR A KNUDSEN GAS IN A RANDOM TUBE

Francis Comets and  Serguei Popov and  Gunter M. Sch\"utz and  Marina  
Vachkovskaia

We consider transport diffusion in a stochastic billiard in a random  
tube
which is elongated in the direction of the first coordinate (the tube  
axis).
Inside the random tube, which is stationary and ergodic, non-interacting
particles move straight with constant speed. Upon hitting the tube  
walls, they
are reflected randomly, according to the cosine law: the density of the
outgoing direction is proportional to the cosine of the angle between  
this
direction and the normal vector. Steady state transport is studied by
introducing an open tube segment as follows: We cut out a large finite  
segment
of the tube with segment boundaries perpendicular to the tube axis.  
Particles
which leave this piece through the segment boundaries disappear from the
system. Through stationary injection of particles at one boundary of the
segment a steady state with non-vanishing stationary particle current is
maintained. We prove (i) that in the thermodynamic limit of an  
infinite open
piece the coarse-grained density profile inside the segment is linear,  
and (ii)
that the transport diffusion coefficient obtained from the ratio of  
stationary
current and effective boundary density gradient equals the diffusion
coefficient of a tagged particle in an infinite tube. Thus we prove  
equality of
transport diffusion and self-diffusion coefficients for quite generic  
rough
(random) tubes.


  http://arxiv.org/abs/0812.1659

---------------------------------------------------------------

7860. SUPERMARTINGALE DEOMPOSITION WITH GENERAL INDEX SET

Gianluca Casseses

We prove results on the existence of Dol\'{e}ans-Dade measures and of  
the
Doob-Meyer decomposition for supermartingales indexed by a general  
index set


  http://arxiv.org/abs/0812.1664

---------------------------------------------------------------

7861. EXPONENTIAL INEQUALITIES FOR MARTINGALES AND ASYMPTOTIC  
PROPERTIES OF  THE FREE ENERGY OF DIRECTED POLYMERS IN RANDOM  
ENVIRONMENT

Quansheng Liu (LMAM) and  Fr\'ed\'erique Watbled (LMAM)

The objective of the present paper is to establish exponential large
deviation inequalities, and to use them to show exponential  
concentration
inequalities for the free energy of a polymer in general random  
environment,
its rate of convergence, and an expression of its limit value in terms  
of those
of some multiplicative cascades.


  http://arxiv.org/abs/0812.1719

---------------------------------------------------------------

7862. EXISTENCE AND UNIQUENESS OF SOLUTIONS OF STOCHASTIC FUNCTIONAL   
DIFFERENTIAL EQUATIONS

Max-K. von Renesse and  Michael Scheutzow

We provide sufficient conditions on the coefficients of a stochastic
functional differential equation with bounded memory driven by  
Brownian motion
which guarantee existence and uniqueness of a maximal local and global  
strong
solution for each initial condition. Our results extend those of  
previous
works. For local existence and uniqueness, we only require the  
coefficients to
be continuous and to satisfy a one-sided local Lipschitz (or  
monotonicity)
condition. In an appendix we formulate and prove four lemmas which may  
be of
independent interest: three of them can be viewed as rather general  
stochastic
versions of Gronwall's Lemma, the final one - which we call Dereich- 
Lemma -
provides tail bounds for Hoelder norms of stochastic integrals.


  http://arxiv.org/abs/0812.1726

---------------------------------------------------------------

7863. PALINDROMIC RANDOM TRIGONOMETRIC POLYNOMIALS

J. Brian Conrey and  David W. Farmer and  and \"Ozlem Imamoglu

We show that if a real trigonometric polynomial has few real roots,  
then the
trigonometric polynomial obtained by writing the coefficients in  
reverse order
must have many real roots. This is used to show that a class of random
trigonometric polynomials has, on average, many real roots. In the  
case that
the coefficients of a real trigonometric polynomial are independently  
and
identically distributed, but with no other assumptions on the  
distribution, the
expected fraction of real zeros is at least one-half. This result is  
best
possible.


  http://arxiv.org/abs/0812.1752

---------------------------------------------------------------

7864. OCCUPATION TIMES VIA BESSEL FUNCTIONS

Yevgeniy Kovchegov and  Nick Meredith and  Eyal Nir

This study of occupation time densities for continuous-time Markov  
processes
was inspired by the work of E.Nir et al (2006) in the field of Single  
Molecule
FRET spectroscopy. There, a single molecule fluctuates between two or  
more
states, and the experimental observable depends on the state's  
occupation time
distribution. To mathematically describe the observable there was a  
need to
calculate a single state occupation time distribution.
   In this paper, we consider a Markov process with countably many  
states. In
order to find a one-stete occupation time density, we use a  
combination of
Fourier and Laplace transforms in the way that allows for inversion of  
the
Fourier transform. We derive an explicit expression for an occupation  
time
density in the case of a simple continuous time random walk on Z. Also  
we
examine the spectral measures in Karlin-McGregor diagonalization in an  
attempt
to represent occupation time densities via modified Bessel functions.


  http://arxiv.org/abs/0812.1775

---------------------------------------------------------------

7865. ORTHOGONALITY AND PROBABILITY: BEYOND NEAREST NEIGHBOR TRANSITIONS

Yevgeniy Kovchegov

In this article, we will explore why Karlin-McGregor method of using
orthogonal polynomials in the study of Markov processes was so  
successful for
one dimensional nearest neighbor processes, but failed beyond nearest  
neighbor
transitions. We will proceed by suggesting and testing possible  
fixtures.


  http://arxiv.org/abs/0812.1779

---------------------------------------------------------------

7866. COMPLETENESS OF BOND MARKET DRIVEN BY L\'EVY PROCESS

Michal Baran and  Jerzy Zabczyk

The completeness problem of the bond market model with noise given by  
the
independent Wiener process and Poisson random measure is studied.  
Hedging
portfolios are assumed to have maturities in a countable, dense subset  
of a
finite time interval. It is shown that under some assumptions the  
market is not
complete unless the support of the Levy measure consists of a finite  
number of
points. Explicit constructions of contingent claims which can not be  
replicated
are provided.


  http://arxiv.org/abs/0812.1796

---------------------------------------------------------------

7867. BROWNIAN MOTION ON THE SIERPINSKI CARPET

Martin T. Barlow and  Richard F. Bass and  Takashi Kumagai and  and  
Alexander  Teplyaev

We prove that, up to scalar multiples, there exists only one local  
regular
Dirichlet form on a generalized Sierpinski carpet that is invariant with
respectto the local symmetries of the carpet. Consequently for each  
suchfractal
the law of Brownian motion is uniquely determined and theLaplacian is  
well
defined.


  http://arxiv.org/abs/0812.1802

---------------------------------------------------------------

7868. ALTERNATING I-DIVERGENCE MINIMIZATION IN FACTOR ANALYSIS

Lorenzo Finesso and Peter Spreij

In this paper we attempt at understanding how to build an optimal normal
factor analysis model. The criterion we have chosen to evaluate the  
distance
between different models is the I-divergence between the corresponding  
normal
laws. The algorithm that we propose for the construction of the best
approximation is of an the alternating minimization kind.


  http://arxiv.org/abs/0812.1804

---------------------------------------------------------------

7869. STOCHASTIC HEAT EQUATION WITH MULTIPLICATIVE FRACTIONAL-COLORED  
NOISE

Raluca Balan and  Ciprian Tudor (CES and  SAMOS)

We consider the stochastic heat equation with multiplicative noise
$u_t={1/2}\Delta u+ u \diamond \dot{W}$ in $\bR_{+} \times \bR^d$, where
$\diamond$ denotes the Wick product, and the solution is interpreted  
in the
mild sense. The noise $\dot W$ is fractional in time (with Hurst index  
$H \geq
1/2$), and colored in space (with spatial covariance kernel $f$). We  
prove that
if $f$ is the Riesz kernel of order $\alpha$, or the Bessel kernel of  
order
$\alpha<d$, then the sufficient condition for the existence of the  
solution is
$d \leq 2+\alpha$ (if $H>1/2$), respectively $d<2+\alpha$ (if  
$H=1/2$), whereas
if $f$ is the heat kernel or the Poisson kernel, then the equation has a
solution for any $d$. We give a representation of the $k$-th order  
moment of
the solution, in terms of an exponential moment of the "convoluted  
weighted"
intersection local time of $k$ independent $d$-dimensional Brownian  
motions.


  http://arxiv.org/abs/0812.1913

---------------------------------------------------------------

7870. A CENTRAL LIMIT THEOREM FOR RANDOM WALK IN RANDOM ENVIRONMENT ON  
MARKED  GALTON-WATSON TREES

Gabriel Faraud

We study a very general model of random walk in random environment on  
trees,
for which we present a recurrence criterion and a functional central  
limit
theorem. This last result is a generalization of a result of Y. Peres  
and O.
Zeitouni (2006).


  http://arxiv.org/abs/0812.1948

---------------------------------------------------------------

7871. PHYLOGENETIC DISTANCES FOR NEIGHBOUR DEPENDENT SUBSTITUTION  
PROCESSES

Mikael Falconnet (IF)

We consider models of nucleotidic substitution processes where the  
rate of
substitution at a given site depends on the state of its neighbours.  
For a wide
class of such nonreversible models, we show how to compute consistent,
mathematically exact, estimators of the time elapsed between aligned  
sequences,
for an ancestral sequence and a present one, and also for two present
sequences. In both cases, we provide asymptotic confidence intervals,  
valid for
nucleotidic sequences of finite length. We compute explicit formulas  
for the
estimators and for their confidence intervals in the simplest  
nontrivial case,
the Jukes-Cantor model with CpG influence.


  http://arxiv.org/abs/0812.1962

---------------------------------------------------------------

7872. COMPLEX WISHART ENSEMBLE AND KP $\TAU$ FUNCTIONS

Dong Wang

In this paper we prove that the integral of the Wishart ensemble is,  
in a
certain sense, a KP $\tau$ function, and generalize the result to  
other random
matrix models, especially the Hermitian matrix model with external  
source.
Besides potential application in multivariate statistics, we obtain some
interesting combinatorial results.


  http://arxiv.org/abs/0810.0280

---------------------------------------------------------------

7873. GIBBS-NON-GIBBS PROPERTIES FOR N-VECTOR LATTICE AND MEAN-FIELD  
MODELS

A. C. D. van Enter and  C. Kuelske and  A. A. Opoku and  W. M. Ruszel

We review some recent developments in the study of Gibbs and non-Gibbs
properties of transformed n-vector lattice and mean-field models under  
various
transformations. Also, some new results for the loss and recovery of  
the Gibbs
property of planar rotor models during stochastic time evolution are  
presented.


  http://arxiv.org/abs/0812.1751

---------------------------------------------------------------

7874. MACROSCOPIC REDUCTION FOR STOCHASTIC REACTION-DIFFUSION EQUATIONS

Wei Wang and A. J. Roberts

The macroscopic behavior of dissipative stochastic partial differential
equations usually can be described by a finite dimensional system.  
This article
proves that a macroscopic reduced model may be constructed for  
stochastic
reaction-diffusion equations with cubic nonlinearity by artificial  
separating
the system into two distinct slow-fast time parts. An averaging method  
and a
deviation estimate show that the macroscopic reduced model should be a
stochastic ordinary equation which includes the random effect  
transmitted from
the microscopic timescale due to the nonlinear interaction. Numerical
simulations of an example stochastic heat equation confirms the  
predictions of
this stochastic modelling theory. This theory empowers us to better  
model the
long time dynamics of complex stochastic systems.


  http://arxiv.org/abs/0812.1837

---------------------------------------------------------------

7875. A UNIVERSALITY RESULT FOR THE SMALLEST EIGENVALUES OF CERTAIN  
SAMPLE  COVARIANCE MATRICES

Ohad N. Feldheim and  Sasha Sodin

After proper rescaling and under some technical assumptions, the  
smallest
eigenvalue of a sample covariance matrix with aspect ratio bounded  
away from 1
converges to the Tracy--Widom distribution. This complements the  
results on the
largest eigenvalue, due to Soshnikov and Peche.


  http://arxiv.org/abs/0812.1961

---------------------------------------------------------------

7876. HARMONIC FUNCTIONS FOR A CLASS OF INTEGRO-DIFFERENTIAL OPERATORS

Mohammud Foondun

We consider the operator $\sL$ defined on $C^2(\bR^d)$ functions by \sL
f(x)&=&{1/2}\sum_{i,j=1}^d a_{ij}(x)\frac{\partial^2f(x)}{\partial x_i 
\partial
x_j}+\sum_{i=1}^d b_i(x)\frac{\partial f(x)}{\partial x_i}
&+&\int_{\bR^d\backslash\{0\}}[f(x+h)-f(x)-1_{(|h|\leq1)}h\cdot \grad
f(x)]n(x,h)dh. Under the assumption that the local part of the  
operator is
uniformly elliptic and with suitable conditions on $n(x,h)$, we  
establish a
Harnack inequality for functions that are nonnegative in $\bR^d$ and  
harmonic
in a domain. We also show that the Harnack inequality can fail without  
suitable
conditions on $n(x,h)$. A regularity theorem for those nonnegative  
harmonic
functions is also proved


  http://arxiv.org/abs/0812.2082

---------------------------------------------------------------

7877. RANDOM INTEGRAL REPRESENTATION OF THE CLASS $L^F$ DISTRIBUTIONS  
AND SOME  RELATED PROPERTIES

Agnieszka Czyzewska-Jankowska and Zbigniew J. Jurek

The method of \emph{random integral representation}, that is, the  
method of
representing given probability measure as the probability distribution  
of some
random integral, was quite successful in the past few decades. In this  
note we
will find such a representation for the class $L^f$ of selfdecomposable
distributions that posses the factorization property. The class $L^f$  
was
introduced in the paper of Iksanov, Jurek and Schreiber, \textbf{Ann.  
Probab.}
vol. 32, 2004. In addition, we also study composition of some random  
integral
mappings.


  http://arxiv.org/abs/0812.2129

---------------------------------------------------------------

7878. CHAOS IN A SPATIAL EPIDEMIC MODEL

Richard Durrett and  Daniel Remenik

We investigate an interacting particle system inspired by the gypsy  
moth,
whose populations grow until they become sufficiently dense so that an  
epidemic
reduces them to a low level. We consider this process on a random 3- 
regular
graph and on the $d$-dimensional lattice and torus, with $d\geq2$. On  
the
finite graphs with global dispersal or with a dispersal radius that  
grows with
the number of sites, we prove convergence to a dynamical system that  
is chaotic
for some parameter values. We conjecture that on the infinite lattice  
with a
fixed finite dispersal distance, distant parts of the lattice  
oscillate out of
phase so there is a unique non-trivial stationary distribution.


  http://arxiv.org/abs/0812.2248

---------------------------------------------------------------

7879. MIXING TIME OF EXPONENTIAL RANDOM GRAPHS

Shankar Bhamidi and  Guy Bresler and  and Allan Sly

Exponential random graphs are used extensively in the sociology  
literature.
This model seeks to incorporate in random graphs the notion of  
reciprocity,
that is, the larger than expected number of triangles and other small
subgraphs. Sampling from these distributions is crucial for parameter
estimation hypothesis testing, and more generally for understanding  
basic
features of the network model itself. In practice sampling is  
typically carried
out using Markov chain Monte Carlo, in particular either the Glauber  
dynamics
or the Metropolis-Hasting procedure.
   In this paper we characterize the high and low temperature regimes  
of the
exponential random graph model. We establish that in the high  
temperature
regime the mixing time of the Glauber dynamics is $\Theta(n^2 \log n) 
$, where
$n$ is the number of vertices in the graph; in contrast, we show that  
in the
low temperature regime the mixing is exponentially slow for any local  
Markov
chain. Our results, moreover, give a rigorous basis for criticisms  
made of such
models. In the high temperature regime, where sampling with MCMC is  
possible,
we show that any finite collection of edges are asymptotically  
independent;
thus, the model does not possess the desired reciprocity property, and  
is not
appreciably different from the Erd\H{o}s-R\'enyi random graph.


  http://arxiv.org/abs/0812.2265

---------------------------------------------------------------

7880. GIBBSIANNESS AND NON-GIBBSIANNESS IN GENERALISED FK MODELS

Andras Balint

For parameters p and q such that the random-cluster measure \phi for  
Z^d with
parameters p and q is unique, the q-divide and colour (DaC(q)) model  
on Z^d is
defined as follows. First we draw a bond configuration distributed  
according to
\phi. Then to each FK cluster (i.e., to every vertex in the FK cluster),
independently for different FK clusters, we assign a spin from the set
{1,2,...,s} in such a way that spin i has probability a_i.
   In this paper we prove that the resulting measure on the spin  
configurations
is a Gibbs measure for small values of p, and it is not a Gibbs  
measure for
large p, except in the special case of a_1=a_2=...=a_s=1/q, when the  
DaC(q)
model coincides with the random-cluster representation of the q-state  
Potts
model.
   Our analysis is based on Haggstrom's methods developed for the  
fuzzy Potts
model.


  http://arxiv.org/abs/0812.2399

---------------------------------------------------------------

7881. SPECTRAL NORM OF PRODUCTS OF RANDOM AND DETERMINISTIC MATRICES

Roman Vershynin

We study the spectral norm of matrices M that can be factored as M=BA,  
where
A is a random matrix with independent mean zero entries, and B is a  
fixed
matrix. Under the (4+epsilon)-th moment assumption on the entries of  
A, we show
that the spectral norm of such an m by n matrix M is bounded by  
\sqrt{m} +
\sqrt{n}, which is sharp. In other words, in regard to the spectral  
norm,
products of random and deterministic matrices behave similarly to random
matrices with independent entries. This result along with the previous  
work of
M. Rudelson and the author implies that the smallest singular value of  
a random
m times n matrix with i.i.d. mean zero entries and bounded (4+epsilon)- 
th
moment is bounded below by \sqrt{m} - \sqrt{n-1} with high probability.


  http://arxiv.org/abs/0812.2432

---------------------------------------------------------------

7882. LIQUIDITY RISK, PRICE IMPACTS AND THE REPLICATION PROBLEM

Alexandre F. Roch

We extend the model of liquidity risk of Cetin et al. [5] to allow for  
price
impacts. Starting from simple principles, we show that the impact of a  
trade on
prices is directly proportional to the size of the transaction and the  
amount
of liquidity of the asset. This leads to a new characterization of
self-financing trading strategies and a sufficient condition for no  
arbitrage.
We show that, with the use of volatility swaps, contingent claims  
whose payoffs
depend on the value of the asset can be approximately replicated. The
replicating costs of such payoffs are obtained from the solutions of  
BSDEs with


  http://arxiv.org/abs/0812.2440

---------------------------------------------------------------

7883. VISCOSITY SOLUTIONS AND AMERICAN OPTION PRICING IN A STOCHASTIC   
VOLATILITY MODEL OF THE ORNSTEIN-UHLENBECK TYPE

Alexandre F. Roch

In this paper, we study the valuation of American type derivatives in  
the
stochastic volatility model of Barndorff-Nielsen and Shephard (2001). We
characterize the value of such derivatives as the unique viscosity  
solution of
an integral-partial differential equation when the payoff function  
satisfies a
Lipschitz condition.


  http://arxiv.org/abs/0812.2444

---------------------------------------------------------------

7884. GENERALIZED HAMMERSLEY PROCESS AND PHASE TRANSITION FOR  
ACTIVATED RANDOM  WALK MODELS

Leonardo T. Rolla

* ACTIVATED RANDOM WALK MODEL * This is a conservative particle system  
on the
lattice, with a Markovian continuous-time evolution. Active particles  
perform
random walks without interaction, and they may as well change their  
state to
passive, then stopping to jump. When particles of both types occupy  
the same
site, they all become active. This model exhibits phase transition in  
the sense
that for low initial densities the system locally fixates and for high
densities it keeps active. Though extensively studied in the physics
literature, the matter of giving a mathematical proof of such phase  
transition
remained as an open problem for several years. In this work we  
identify some
variables that are sufficient to characterize fixation and at the same  
time are
stochastically monotone in the model's parameters. We employ an explicit
graphical representation in order to obtain the monotonicity. With  
this method
we prove that there is a unique phase transition for the one-dimensional
finite-range random walk. Joint with V. Sidoravicius.
   * BROKEN LINE PROCESS * We introduce the broken line process and  
derive some
of its properties. Its discrete version is presented first and a natural
generalization to the continuum is then proposed and studied. The  
broken lines
are related to the Young diagram and the Hammersley process and are  
useful for
computing last passage percolation values and finding maximal oriented  
paths.
For a class of passage time distributions there is a family of boundary
conditions that make the process stationary and reversible. One  
application is
a simple proof of the explicit law of large numbers for last passage
percolation with exponential and geometric distributions. Joint with V.
Sidoravicius, D. Surgailis, and M. E. Vares.


  http://arxiv.org/abs/0812.2473

---------------------------------------------------------------

7885. FLUCTUATION THEORY AND EXIT SYSTEMS FOR POSITIVE SELF-SIMILAR  
MARKOV  PROCESSES

Loic Chaumont and  Andreas Eos Kyprianou and  Juan Carlos Pardo and   
Victor Rivero

For a positive self-similar Markov process, $X$, we construct a local  
time
for the random set, $\Theta,$ of times where the process reaches its  
past
supremum. Using this local time we describe an exit system for the  
excursions
of $X$ out of its past supremum. Next, we define and study the ladder  
process
$(R,H)$ associated to a positive self-similar Markov process $X$, viz. a
bivariate Markov process with a scaling property whose coordinates are  
the
right inverse of the local time of the random set $\Theta$ and the  
process $X$
sampled on the local time scale. The process $(R,H)$ is described in  
terms of
ladder process associated to the Levy process associated to $X$ via  
Lamperti's
transformation. In the case where $X$ never hits 0 and the upward  
ladder height
process is not arithmetic and has finite mean we prove the finite  
dimensional
convergence of $(R,H)$ as the starting point of $X$ tends to $0.$  
Finally, we
use these results to provide an alternative proof to the weak  
convergence of
$X$ as the starting point tends to $0.$ Our approach allows us to  
address two
issues that remained open in \cite{CCh}, namely to remove a redundant
hypothesis and to provide a formula for the entrance law of $X$ in the  
case
where the underlying Levy process oscillates.


  http://arxiv.org/abs/0812.2506

---------------------------------------------------------------

7886. THE SPEED OF A BIASED RANDOM WALK ON A PERCOLATION CLUSTER AT  
HIGH  DENSITY

Alexander Fribergh (ICJ)

We study the speed of a biased random walk on a percolation cluster on  
$\Z^d$
in function of the percolation parameter $p$. We obtain a first order  
expansion
of the speed at $p=1$ which proves that percolating slows down the  
random walk
at least in the case where the drift is along a component of the  
lattice.


  http://arxiv.org/abs/0812.2532

---------------------------------------------------------------

7887. COPULAS FOR MARKOVIAN DEPENDENCE

Andreas Nordvall Lager{\aa}s

Copulas have been popular to model dependence for multivariate  
distributions,
but have not been used much in modelling temporal dependence of  
univariate time
series. This paper shows some difficulties with using copulas even for  
Markov
processes: some tractable copulas such as mixtures between copulas of  
complete
co- and countermonotonicity and independence (Fr{\'e}chet copulas) are  
shown to
imply quite a restricted type of Markov process, and Archimedean  
copulas are
shown to be incompatible with Markov chains. We also investigate  
Markov chains
that are spreadable, or equivalently, conditionally i.i.d.


  http://arxiv.org/abs/0812.2548

---------------------------------------------------------------

7888. SPARSE GRAPHS: METRICS AND RANDOM MODELS

Bela Bollobas and Oliver Riordan

Recently, Bollob\'as, Janson and Riordan have introduced a family of  
random
graph models producing inhomogeneous graphs with $n$ vertices and $ 
\Theta(n)$
edges whose distribution is characterized by a kernel, i.e., a symmetric
measurable function $\ka:[0,1]^2 \to [0,\infty)$. To understand these  
models,
we should like to know when different kernels $\ka$ give rise to  
`similar'
graphs, and, given a real-world network, how `similar' is it to a  
typical graph
$G(n,\ka)$ derived from a given kernel $\ka$.
   The analogous questions for dense graphs, with $\Theta(n^2)$ edges,  
are
answered by recent results of Borgs, Chayes, Lov\'asz, S\'os, Szegedy  
and
Vesztergombi, who showed that several natural metrics on graphs are  
equivalent,
and moreover that any sequence of graphs converges in each metric to a  
graphon,
i.e., a kernel taking values in $[0,1]$.
   Possible generalizations of these results to graphs with $o(n^2)$ but
$\omega(n)$ edges are discussed in a companion paper [arXiv: 
0708.1919]; here we
focus only on graphs with $\Theta(n)$ edges, which turn out to be much  
harder
to handle. Many new phenomena occur, and there are a host of plausible  
metrics
to consider; many of these metrics suggest new random graph models,  
and vice
versa.


  http://arxiv.org/abs/0812.2656

---------------------------------------------------------------

7889. ANOMALOUS HEAT-KERNEL DECAY FOR RANDOM WALK AMONG POLYNOMIAL  
LOWER TAIL  RANDOM CONDUCTANCES

Omar Boukhadra

We consider the nearest-neighbor simple random walk on $\Z^{d}$, $d 
\geq 4$,
driven by a field of i.i.d. random nearest-neighbor conductances
$\omega_{xy}\in[0,1]$. Our aim is to derive estimates of the heat- 
kernel decay
in a case where ellipticity assumption is absent. We consider the case  
of
independant conductances with polynomial tail near 0 and obtain for  
almost
every environment an anomalous lower bound on the heat-kernel.


  http://arxiv.org/abs/0812.2669

---------------------------------------------------------------

7890. COMMUTING BIRTH-AND-DEATH PROCESSES

Steven N. Evans and  Bernd Sturmfels and Caroline Uhler

We use methods from combinatorics and algebraic statistics to study  
analogues
of birth-and-death processes that have as their state space a finite  
subset of
the $m$-dimensional lattice and for which the $m$ matrices that record  
the
transition probabilities in each of the lattice directions commute  
pairwise.
One reason such processes are of interest is that the transition  
matrix is
straightforward to diagonalize, and hence it is easy to compute $n$ step
transition probabilities. The set of commuting birth-and-death processes
decomposes as a union of toric varieties, with the main component  
being the
closure of all processes whose nearest neighbor transition  
probabilities are
positive. We exhibit an explicit monomial parametrization for this main
component, and we explore the boundary components using primary  
decomposition.


  http://arxiv.org/abs/0812.2724

---------------------------------------------------------------

7891. MEAN FIELD FROZEN PERCOLATION

Balazs Rath

We define a modification of the Erdos-Renyi random graph process which  
can be
regarded as the mean field frozen percolation process. We describe the  
behavior
of the process using differential equations and investigate their  
solutions in
order to show the self-organized critical and extremum properties of the
critical frozen percolation model. We prove two limit theorems about the
distribution of the size of the component of a typical frozen vertex.


  http://arxiv.org/abs/0812.2750

---------------------------------------------------------------

7892. AN EMPIRICAL CENTRAL LIMIT THEOREM IN L^1 FOR STATIONARY SEQUENCES

Sophie Dede (PMA)

In this paper, we derive asymptotic results for L^1-Wasserstein distance
between the distribution function and the corresponding empirical  
distribution
function of a stationary sequence. Next, we give some applications to  
dynamical
systems and causal linear processes. To prove our main result, we give a
Central Limit Theorem for ergodic stationary sequences of random  
variables with
values in L^1. The conditions obtained are expressed in terms of
projective-type conditions. The main tools are martingale  
approximations.


  http://arxiv.org/abs/0812.2839

---------------------------------------------------------------

7893. UNIVERSALITY IN COMPLEX WISHART ENSEMBLES: THE 1 CUT CASE

M. Y. Mo

We studied universality of Wishart ensembles whose covariance matrix  
has 2
distinct eigenvalues and the number of each of these eigenvalue goes to
infinity in the asymptotic limit. In this case, the limiting eigenvalue
distribution can be supported on 1 or 2 disjoint intervals. In our  
previous
work the case when the support consists of 2 intervals was studied.  
This paper
complements our previous analysis and studied the case when the support
consists of a single interval. By using Riemann-Hilbert analysis, we  
have shown
that under proper rescaling of the eigenvalues, the limiting  
correlation kernel
is given by the sine kernel and the Airy kernel in the bulk and the  
edge of the
spectrum respectively. As a consequence, the behavior of the largest  
eigenvalue
in this model is described by the Tracy-Widom distribution.


  http://arxiv.org/abs/0812.2863

---------------------------------------------------------------

7894. SEQUENTIAL MULTIPLE HYPOTHESIS TESTING IN PRESENCE OF CONTROL  
VARIABLES

Andrey Novikov

Suppose that at any stage of a statistical experiment a control  
variable $X$
that affects the distribution of the observed data $Y$ at this stage  
can be
used. The distribution of $Y$ depends on some unknown parameter $\theta 
$, and
we consider the problem of testing multiple hypotheses $H_1: \theta= 
\theta_1$,
$H_2: \theta=\theta_2, ...$, $H_k: \theta=\theta_k$ allowing the data  
to be
controlled by $X$, in the following sequential context.
   The experiment starts with assigning a value $X_1$ to the control  
variable
and observing $Y_1$ as a response. After some analysis, another value  
$X_2$ for
the control variable is chosen, and $Y_2$ as a response is observed,  
etc. It is
supposed that the experiment eventually stops, and at that moment a  
final
decision in favor of one of the hypotheses $H_1,...$, $H_k$ is to be  
taken. In
this article, our aim is to characterize the structure of optimal  
sequential
testing procedures based on data obtained from an experiment of this  
type in
the case when the observations $Y_1, Y_2,..., Y_n$ are independent,  
given
controls $X_1,X_2,..., X_n$, $n=1,2,...$.


  http://arxiv.org/abs/0812.2712

---------------------------------------------------------------

7895. THE LARGEST EIGENVALUES OF SAMPLE COVARIANCE MATRICES FOR A  
SPIKED  POPULATION: DIAGONAL CASE

Delphine F\'eral (IMB) and  Sandrine P\'ech\'e (IF)

We consider large complex random sample covariance matrices obtained  
from
"spiked populations", that is when the true covariance matrix is  
diagonal with
all but finitely many eigenvalues equal to one. We investigate the  
limiting
behavior of the largest eigenvalues when the population and the sample  
sizes
both become large. Under some conditions on moments of the sample  
distribution,
we prove that the asymptotic fluctuations of the largest eigenvalues  
are the
same as for a complex Gaussian sample with the same true covariance.  
The real
setting is also considered.


  http://arxiv.org/abs/0812.2320

---------------------------------------------------------------

7896. THE STATISTICAL RESTRICTED ISOMETRY PROPERTY AND THE WIGNER  
SEMICIRCLE  DISTRIBUTION OF INCOHERENT DICTIONARIES

Shamgar Gurevich (University of California Berkeley) and Ronny Hadani   
(University of Chicago)

In this article we present a statistical version of the Candes-Tao  
restricted
isometry property (SRIP for short) which holds in general for any  
incoherent
dictionary which is a disjoint union of orthonormal bases. In  
addition, we show
that, under appropriate normalization, the eigenvalues of the  
associated Gram
matrix fluctuate around 1 according to the Wigner semicircle  
distribution. The
result is then applied to various dictionaries that arise naturally in  
the
setting of finite harmonic analysis, giving, in particular, a better
understanding on a remark of Applebaum-Howard-Searle-Calderbank  
concerning RIP
for the Heisenberg dictionary of chirp like functions.


  http://arxiv.org/abs/0812.2602

---------------------------------------------------------------

7897. SYNCHRONIZATION OF DISCRETE-TIME DYNAMICAL NETWORKS WITH TIME- 
VARYING  COUPLINGS

Wenlian Lu and  Fatihcan M. Atay and  J\"urgen Jost

We study the local complete synchronization of discrete-time dynamical
networks with time-varying couplings. Our conditions for the temporal  
variation
of the couplings are rather general and include both variations in the  
network
structure and in the reaction dynamics; the reactions could, for  
example, be
driven by a random dynamical system. A basic tool is the concept of  
Hajnal
diameter which we extend to infinite Jacobian matrix sequences. The  
Hajnal
diameter can be used to verify synchronization and we show that it is
equivalent to other quantities which have been extended to time- 
varying cases,
such as the projection radius, projection Lyapunov exponents, and  
transverse
Lyapunov exponents. Furthermore, these results are used to investigate  
the
synchronization problem in coupled map networks with time-varying  
topologies
and possibly directed and weighted edges. In this case, the Hajnal  
diameter of
the infinite coupling matrices can be used to measure the  
synchronizability of
the network process. As we show, the network is capable of  
synchronizing some
chaotic map if and only if there exists an integer T>0 such that for  
any time
interval of length T, there exists a vertex which can access other  
vertices by
directed paths in that time interval.


  http://arxiv.org/abs/0812.2706

---------------------------------------------------------------

7898. CONE STRUCTURE OF $L^2$-WASSERSTEIN SPACES

Asuka Takatsu

In this paper, we prove that if a base space has a cone structure,  
then so
does its $L^2$-Wasserstein space. Furthermore, we investigate  
relations between
the base spaces of the both cones. Conversely, we show when an
$L^2$-Wasserstein space has a cone structure satisfying certain  
conditions,
then its underlying space is also a cone.


  http://arxiv.org/abs/0812.2752

---------------------------------------------------------------

7899. ASYMPTOTICS FOR THE SIZE OF THE LARGEST COMPONENT SCALED TO "LOG  
N" IN  INHOMOGENEOUS RANDOM GRAPHS

Tatyana S. Turova

We study the inhomogeneous random graphs in the subcritical case. We  
derive
an exact formula for the size of the largest connected component  
scaled to
$\log n$ where $n$ is the size of the graph. This generalizes the  
recent result
for the "rank 1 case". Here we discover that the same well-known  
equation for
the survival probability, whose positive solution determines the  
asymptotics of
the size of the largest component in the supercritical case, plays the  
crucial
role in the subcritical case as well. But now these are the negative  
solutions
which come into play.


  http://arxiv.org/abs/0812.3007

---------------------------------------------------------------

7900. MATHEMATICAL MODEL FOR RESISTANCE AND OPTIMAL STRATEGY

Blandine Berard Bergery (IECN) and  Christophe Profeta (IECN) and   
Etienne  Tanr\'e (INRIA Sophia Antipolis / INRIA Lorraine / IECN)

We propose a mathematical model for one pattern of charts studied in
technical analysis: in a phase of consolidation, the price of a risky  
asset
goes down $\xi$ times after hitting a resistance level. We construct a
mathematical strategy and we calculate the expectation of the wealth  
for the
logaritmic utility function. Via simulations, we compare the strategy  
with the
standard one.


  http://arxiv.org/abs/0812.3027

---------------------------------------------------------------

7901. NORMAL APPROXIMATION FOR COVERAGE MODELS OVER BINOMIAL POINT  
PROCESSES

Larry Goldstein and  Mathew D. Penrose

We give error bounds which demonstrate optimal rates of convergence in  
the
CLT for the total covered volume and the number of isolated shapes, for
germ-grain models with fixed grain radius over a binomial point  
process of $n$
points in a toroidal spatial region of volume $n$. The proof is based on
Stein's method via size-biased couplings.


  http://arxiv.org/abs/0812.3084

---------------------------------------------------------------

7902. PARAMETER ESTIMATION FOR ROUGH DIFFERENTIAL EQUATIONS

Anastasia Papavasiliou and  Christophe Ladroue

We construct an estimator based on "signature matching" for differential
equations driven by rough paths and we prove its consistency and  
asymptotic
normality. Note that the the Moment Matching estimator is a special  
case of
this estimator.


  http://arxiv.org/abs/0812.3102

---------------------------------------------------------------

7903. CRITICAL VALUE OF THE QUANTUM ISING MODEL ON STAR-LIKE GRAPHS

Jakob E. Bj\"ornberg

We present a rigorous determination of the critical value of the  
ground-state
quantum Ising model in a transverse field, on a class of planar graphs  
which we
call star-like. These include the star graph, which is a junction of  
several
copies of Z at a single point. Our approach is to use the graphical,  
or FK-,
representation of the model, and the probabilistic and geometric tools
associated with it.


  http://arxiv.org/abs/0812.3113

---------------------------------------------------------------

7904. NON-COLLIDING JACOBI PROCESSES AS LIMITS OF MARKOV CHAINS ON   
GELFAND-TSETLIN GRAPH

Vadim Gorin

We introduce a stochastic dynamics related to the measures that arise in
harmonic analysis on the infinite-dimensional unitary group. Our  
dynamics is
obtained as a limit of a sequence of natural Markov chains on Gelfand- 
Tsetlin
graph. We compute finite-dimensional distributions of the limit Markov  
process,
the generator and eigenfunctions of the semigroup related to this  
process. The
limit process can be identified with Doob h-transform of a family of
independent diffusions. Space-time correlation functions of the limit  
process
have a determinantal form.


  http://arxiv.org/abs/0812.3146

---------------------------------------------------------------

7905. FROM SCHOENBERG TO PICK-NEVANLINNA: TOWARDS A COMPLETE PICTURE  
OF THE  VARIOGRAM CLASS

Emilio Porcu and  Rene L. Schilling

We show that a large subclass of variograms is closed under Schur  
products
and that some desirable stability properties, like the Schur product of
\emph{ad hoc} compositions, can be obtained under the proposed  
setting. We
introduce new classes of kernels of Schoenberg-L\'{e}vy type and show  
some
important properties of eventually constant, radially symmetric  
variograms. In
particular, we characterize eventually constant variograms in terms of  
their
permissibility in Euclidean spaces of arbitrary high dimension.


  http://arxiv.org/abs/0812.2936

---------------------------------------------------------------

7906. RUBINSTEIN DISTANCES ON CONFIGURATION SPACES

Laurent Decreusefond (LTCI) and  Ald\'eric Joulin and  Nicolas Savy

In this paper, we provide upper bounds on several Rubinstein-type  
distances
on the configuration space equipped with the Poisson measure. Our  
inequalities
involve the two well-known gradients, in the sense of Malliavin  
calculus, which
can be defined on this space. Actually, we show that depending on the  
distance
between configurations which is considered, it is one gradient or the  
other
which is the most effective. Some applications to distance estimates  
between
Poisson and other more sophisticated processes are also provided, and an
investigation of our results to functional inequalities completes this  
work.


  http://arxiv.org/abs/0812.3221

---------------------------------------------------------------

7907. INTERMITTENCY ON CATALYSTS: THREE-DIMENSIONAL SIMPLE SYMMETRIC  
EXCLUSION

J. Gaertner and  F. den Hollander and  G. Maillard

We continue our study of intermittency for the parabolic Anderson model
$\partial u/\partial t = \kappa\Delta u + \xi u$ in a space-time  
random medium
$\xi$, where $\kappa$ is a positive diffusion constant, $\Delta$ is  
the lattice
Laplacian on $\Z^d$, $d \geq 1$, and $\xi$ is a simple symmetric  
exclusion
process on $\Z^d$ in Bernoulli equilibrium. This model describes the  
evolution
of a \emph{reactant} $u$ under the influence of a \emph{catalyst} $\xi$.
   In G\"artner, den Hollander and Maillard (2007) we investigated the  
behavior
of the annealed Lyapunov exponents, i.e., the exponential growth rates  
as
$t\to\infty$ of the successive moments of the solution $u$. This led  
to an
almost complete picture of intermittency as a function of $d$ and $ 
\kappa$. In
the present paper we finish our study by focussing on the asymptotics  
of the
Lyaponov exponents as $\kappa\to\infty$ in the \emph{critical}  
dimension $d=3$,
which was left open in G\"artner, den Hollander and Maillard (2007)  
and which
is the most challenging. We show that, interestingly, this asymptotics  
is
characterized not only by a \emph{Green} term, as in $d\geq 4$, but  
also by a
\emph{polaron} term. The presence of the latter implies intermittency of
\emph{all} orders above a finite threshold for $\kappa$.


  http://arxiv.org/abs/0812.3311

---------------------------------------------------------------

7908. DISTANCES BETWEEN PAIRS OF VERTICES AND VERTICAL PROFILE IN  
CONDITIONED  GALTON--WATSON TREES

Luc Devroye and Svante Janson

We consider a conditioned Galton-Watson tree and prove an estimate of  
the
number of pairs of vertices with a given distance, or, equivalently,  
the number
of paths of a given length.
   We give two proofs of this result, one probabilistic and the other  
using
generating functions and singularity analysis.
   Moreover, the second proof yields a more general estimate for  
generating
functions, which is used to prove a conjecture by Bousquet-Melou and  
Janson
saying that the vertical profile of a randomly labelled conditioned
Galton-Watson tree converges in distribution, after suitable  
normalization, to
the density of ISE (Integrated Superbrownian Excursion).


  http://arxiv.org/abs/0812.3326

---------------------------------------------------------------

7909. COMPUTATION OF VAR AND CVAR USING STOCHASTIC APPROXIMATIONS AND   
UNCONSTRAINED IMPORTANCE SAMPLING

Olivier Aj Bardou (PMA and  GDF-RDD) and  Noufel Frikha (PMA and  GDF- 
RDD) and  G.  Pag\`es (PMA)

Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are two risk
measures which are widely used in the practice of risk management.  
This paper
deals with the problem of computing both VaR and CVaR using stochastic
approximation (with decreasing steps): we propose a first Robbins-Monro
procedure based on Rockaffelar-Uryasev's identity for the CVaR. The  
convergence
rate of this algorithm to its target satisfies a Gaussian Central Limit
Theorem. As a second step, in order to speed up the initial procedure,  
we
propose a recursive importance sampling (I.S.) procedure which induces a
significant variance reduction of both VaR and CVaR procedures. This  
idea,
which goes back to the seminal paper of B. Arouna, follows a new  
approach
introduced by V. Lemaire and G. Pag\`es. Finally, we consider a  
deterministic
moving risk level to speed up the initialization phase of the  
algorithm. We
prove that the convergence rate of the resulting procedure is ruled by a
Central Limit Theorem with minimal variance and its efficiency is  
illustrated
by considering several typical energy portfolios.


  http://arxiv.org/abs/0812.3381

---------------------------------------------------------------

7910. A NUMERICAL ALGORITHM FOR ZERO COUNTING II: RANDOMIZATION AND  
CONDITION

Felipe Cucker and  Teresa Krick and  Gregorio Malajovich and Mario  
Wschebor

In a recent paper [A numerical algorithm for zero counting I:  
complexity and
accuracy . J. of Complexity 24, 5-6, pp 582-605 (Oct-Dec 2008)] we  
analyzed a
numerical algorithm for computing the number of real zeros of a  
polynomial
system. The analysis relied on a condition number k(f) for the input  
system f.
In this paper we continue this analysis by looking at k(f) as a random  
variable
derived from imposing a probability measure on the space of polynomial  
systems.
We give bounds for both the tail P{k(f) > a} and the expected value  
E(log
k(f)).


  http://arxiv.org/abs/0812.3281

---------------------------------------------------------------

7911. EVOLUTION BY MEAN CURVATURE IN SUB-RIEMANNIAN GEOMETRIES: A  
STOCHASTIC  APPROACH

Nicolas Dirr and  Federica Dragoni and Max von Renesse

We study the phenomenon of evolution by horizontal mean curvature flow  
in
sub-Riemannian geometries. We use a stochastic approach to prove the  
existence
of a generalized evolution in these spaces. In particular we show that  
the
value function of suitable family of stochastic control problems  
solves in the
viscosity sense the level set equation for the evolution by horizontal  
mean
curvature flow.


  http://arxiv.org/abs/0812.3288

---------------------------------------------------------------

7912. EPIDEMIC MODELLING: ASPECTS WHERE STOCHASTICITY MATTERS

Tom Britton and  David Lindenstrand

Epidemic models are always simplifications of real world epidemics.  
Which
real world features to include, and which simplifications to make,  
depend both
on the disease of interest and on the purpose of the modelling. In the  
present
paper we discuss some such purposes for which a \emph{stochastic}  
model is
preferable to a \emph{deterministic} counterpart. The two main examples
illustrate the importance of allowing the infectious and latent  
periods to be
random when focus lies on the \emph{probability} of a large epidemic  
outbreak
and/or on the initial \emph{speed}, or growth rate, of the epidemic. A
consequence of the latter is that estimation of the basic reproduction  
number
$R_0$ is sensitive to assumptions about the distributions of the  
infectious and
latent periods when using the data from the early stages of an  
outbreak, which
we illustrate with data from the SARS outbreak. Some further examples  
are also
discussed as are some practical consequences related to these stochastic
aspects.


  http://arxiv.org/abs/0812.3505

---------------------------------------------------------------

7913. ON THE ALMOST SURE CENTRAL LIMIT THEOREM FOR VECTOR  
MARTINGALES:  CONVERGENCE OF MOMENTS AND STATISTICAL APPLICATIONS

Bernard Bercu (IMB and  INRIA Bordeaux - Sud-Ouest) and  Peggy C\'enac  
(IMB) and   Guy Fayolle (INRIA Rocquencourt)

We investigate the almost sure asymptotic properties of vector  
martingale
transforms. Assuming some appropriate regularity conditions both on the
increasing process and on the moments of the martingale, we prove that
normalized moments of any even order converge in the almost sure  
cental limit
theorem for martingales. A conjecture about almost sure upper bounds  
under
wider hypotheses is formulated. The theoretical results are supported by
examples borrowed from statistical applications, including linear
autoregressive models and branching processes with immigration, for  
which new
asymptotic properties are established on estimation and prediction  
errors.


  http://arxiv.org/abs/0812.3528

---------------------------------------------------------------

7914. ESTIMATION OF THE INSTANTANEOUS VOLATILITY AND DETECTION OF  
VOLATILITY  JUMPS

A. Alvarez and  F. Panloup and  M. Pontier and  N. Savy

Concerning price processes, the fact that the volatility is not  
constant has
been observed for a long time. So we deal with models as
$dX_t=\mu_tdt+\sigma_tdW_t$ where $\sigma$ is a stochastic process.  
Recent
works on volatility modeling suggest that we should incorporate jumps  
in the
volatility process. Empirical observations suggest that simultaneous  
jumps on
the price \underline{and} the volatility \cite{BarShep1,ConTan} exist.  
The
hypothesis that jumps occur simultaneously makes the problem of  
volatility jump
detection reduced to the prices jump detection. But in case of this  
hypothesis
failure, we try to work in this direction.
   Among others, we use Jacod and Ait-Sahalia' recent work \cite{jac1}  
giving
estimators of cumulated volatility $\int_0^t|\sigma_s|^pds$ for any $p 
\geq 2.$
   This tool allows us to deliver an estimator of instantaneous  
volatility.
Moreover we prove a central limit theorem for it. Obviously, such a  
theorem
provides a confidence interval for the instantaneous volatility and  
leads us to
a test of the jump existence hypothesis. For instance, we consider a  
simplest
model having volatility jumps, when volatility is piecewise constant:
$\sigma_t=\sum_{i=0}^{N_t-1} \sigma_i \1_{[\tau_i,\tau_{i+1}[}(t).$  
The jump
times are $\tau_i, i\geq 1,$ and $\sigma_i$ is a $\F_{\tau_{i}}$- 
measurable
random variable. Another example is studied: $\sigma_t=|Y_t|$ where $ 
(Y_t)$ is
a solution to a L\'evy driven SDE, with suitable coefficients.


  http://arxiv.org/abs/0812.3538

---------------------------------------------------------------

7915. ON THE ANNEALED LARGE DEVIATION RATE FUNCTION FOR A MULTI- 
DIMENSIONAL  RANDOM WALK IN RANDOM ENVIRONMENT

Jonathon Peterson and Ofer Zeitouni

We derive properties of the rate function in Varadhan's (annealed) large
deviation principle for multidimensional, ballistic random walk in  
random
environment, in a certain neighborhood of the zero set of the rate  
function.
Our approach relates the LDP to that of regeneration times and  
distances. The
analysis of the latter is possible due to the i.i.d. structure of
regenerations.


  http://arxiv.org/abs/0812.3619

---------------------------------------------------------------

7916. OPTIMAL DETECTION OF HOMOGENEOUS SEGMENT OF OBSERVATIONS IN  
STOCHASTIC  SEQUENCE

Wojciech Sarnowski and  Krzysztof Szajowski

We register a Markov process. At random moment $\theta$ the  
distribution of
observed sequence changes. Using probability maximizing approach the  
optimal
stopping rule is identified. For the particular case of disorder the  
explicit
solution is obtained.


  http://arxiv.org/abs/0812.3632

---------------------------------------------------------------

7917. TIME MANAGEMENT IN A POISSON FISHING MODEL

Anna Karpowicz and  Krzysztof Szajowski

The aim of the paper is to extend the model of "fishing problem". The  
simple
formulation is following. The angler goes to fishing. He buys fishing  
ticket
for a fixed time. There are two places for fishing at the lake. The  
fishes are
caught according to renewal processes which are different at both  
places. The
fishes' weights and the inter-arrival times are given by the sequences  
of
i.i.d. random variables with known distribution functions. These  
distributions
are different for the first and second fishing place. The angler's  
satisfaction
measure is given by difference between the utility function dependent  
on size
of the caught fishes and the cost function connected with time. On  
each place
the angler has another utility functions and another cost functions.  
In this
way, the angler's relative opinion about these two places is modeled.  
For
example, on the one place better sort of fish can be caught with bigger
probability or one of the places is more comfortable. Obviously our  
angler
wants to have as much satisfaction as possible and additionally he  
have to
leave the lake before the fixed moment. Therefore his goal is to find  
two
optimal stopping times in order to maximize his satisfaction. The  
first time
corresponds to the moment, when he eventually should change the place  
and the
second time, when he should stop fishing. These stopping times have to  
be less
than the fixed time of fishing. The value of the problem and the optimal
stopping times are derived.


  http://arxiv.org/abs/0812.3651

---------------------------------------------------------------

7918. MAXIMUM EMPIRICAL LIKELIHOOD ESTIMATION OF THE SPECTRAL MEASURE  
OF AN  EXTREME VALUE DISTRIBUTION

John H. J. Einmahl and  Johan Segers

Consider a random sample from a bivariate distribution function $F$ in  
the
max-domain of attraction of an extreme value distribution function $G 
$. This
$G$ is characterized by two extreme value indices and a spectral  
measure, the
latter determining the tail dependence structure of $F$. A major issue  
in
multivariate extreme value theory is the estimation of the spectral  
measure
$\Phi_p$ with respect to the $L_p$ norm. For every $p \in [1, \infty] 
$, a
nonparametric maximum empirical likelihood estimator is proposed for $ 
\Phi_p$.
The main novelty is that these estimators are guaranteed to satisfy  
the moment
constraints by which spectral measures are characterized. Asymptotic  
normality
of the estimators is proved under conditions that allow for tail  
independence.
Moreover, the conditions are easily verifiable as we demonstrate  
through a
number of theoretical examples. A simulation study shows substantially  
improved
performance of the new estimators. Two case studies illustrate how to  
implement
the methods in practice.


  http://arxiv.org/abs/0812.3485

---------------------------------------------------------------

7919. A RANK-BASED SELECTION WITH CARDINAL PAYOFFS AND A COST OF CHOICE

Krzysztof Szajowski

A version of the secretary problem is considered. The ranks of items,  
whose
values are independent, identically distributed random variables
$X_1,X_2,...,X_n$ from a uniform distribution on $[0; 1]$, are observed
sequentially by the grader. He has to select exactly one item, when it  
appears,
and receives a payoff which is a function of the unobserved  
realization of
random variable assigned to the item diminished by some cost. The  
methods of
analysis are based on the existence of an embedded Markov chain and  
use the
technique of backward induction. The result is a generalization of the
selection model considered by Bearden(2006). The asymptotic behaviour  
of the
solution is also investigated.


  http://arxiv.org/abs/0812.3483

---------------------------------------------------------------

7920. ULTRASPHERICAL TYPE GENERATING FUNCTIONS FOR ORTHOGONAL  
POLYNOMIALS

Nizar Demni

We characterize probability distributions of all order finite moments  
gaving
ultraspherical type generating functions for orthogonal polynomials.


  http://arxiv.org/abs/0812.3666

---------------------------------------------------------------

7921. SIMULTANEOUS ASYMPTOTICS FOR THE SHAPE OF RANDOM YOUNG TABLEAUX  
WITH  GROWINGLY RESHUFFLED ALPHABETS

Jean-Christophe Breton and  Christian Houdr\'e

Given a random word of size n whose letters are drawn independently  
from an
ordered alphabet of size m, the fluctuations of the shape of the  
associated
random Young tableaux are investigated, when both n and m converge  
together to
infinity. If m does not grow too fast and if the draws are uniform, the
limiting shape is the same as the limiting spectrum of the GUE. In the
non-uniform case, a control of both highest probabilities will ensure  
the
convergence of the first row of the tableau towards the Tracy-Widom
distribution.


  http://arxiv.org/abs/0812.3672

---------------------------------------------------------------

7922. THE GAUSSIAN APPROXIMATION FOR MULTI-COLOR GENERALIZED  
FRIEDMAN'S URN  MODEL

Li-Xin Zhang and Feifang Hu

The Friedman's urn model is a popular urn model which is widely used  
in many
disciplines. In particular, it is extensively used in treatment  
allocation
schemes in clinical trials. In this paper, we prove that both the urn
composition process and the allocation proportion process can be  
approximated
by a multi-dimensional Gaussian process almost surely for a multi-color
generalized Friedman's urn model with non-homogeneous generating  
matrices. The
Gaussian process is a solution of a stochastic differential equation.  
This
Gaussian approximation together with the properties of the Gaussian  
process is
important for the understanding of the behavior of the urn process and  
is also
useful for statistical inferences. As an application, we obtain the  
asymptotic
properties including the asymptotic normality and the law of the  
iterated
logarithm for a multi-color generalized Friedman's urn model as well  
as the
randomized-play-the-winner rule as a special case.


  http://arxiv.org/abs/0812.3697

---------------------------------------------------------------

7923. THE DURATION PROBLEM WITH MULTIPLE EXCHANGES

Charles E.M. Pearce and  Krzysztof Szajowski and  Mitsushi Tamaki

We treat a version of the multiple-choice secretary problem called the
multiple-choice duration problem, in which the objective is to  
maximize the
time of possession of relatively best objects. It is shown that, for the
$m$--choice duration problem, there exists a sequence (s1,s2,...,sm) of
critical numbers such that, whenever there remain k choices yet to be  
made,
then the optimal strategy immediately selects a relatively best object  
if it
appears at or after time $s_k$ ($1\leq k\leq m$). We also exhibit an
equivalence between the duration problem and the classical best-choice
secretary problem. A simple recursive formula is given for calculating  
the
critical numbers when the number of objects tends to infinity.  
Extensions are
made to models involving an acquisition or replacement cost.


  http://arxiv.org/abs/0812.3765

---------------------------------------------------------------

7924. THE CRITICAL Z-INVARIANT ISING MODEL VIA DIMERS: THE PERIODIC CASE

C\'edric Boutillier and  B\'eatrice de Tili\`ere

We study a large class of critical two-dimensional Ising models namely
critical Z-invariant Ising models on periodic graphs, example of which  
are the
classical square, triangular and honeycomb lattice at the critical  
temperature.
Fisher introduced a correspondence between the Ising model and the  
dimer model
on a decorated graph, thus setting dimer techniques as a powerful tool  
for
understanding the Ising model. In this paper, we give a full  
description of the
dimer model corresponding to the critical Z-invariant Ising model. We  
prove
that the dimer characteristic polynomial is equal (up to a constant)  
to the
critical Laplacian characteristic polynomial, and defines a Harnack  
curve of
genus 0. We prove an explicit expression for the free energy, and for  
the Gibbs
measure obtained as weak limit of Boltzmann measures.


  http://arxiv.org/abs/0812.3848

---------------------------------------------------------------

7925. IMMIGRATED URN MODELS - ASYMPTOTIC PROPERTIES AND APPLICATIONS

Li-Xin Zhang and  Feifang Hu and  Siu Hung Cheung and Wei Sum Chan

Urn models have been widely studied and applied in both scientific and  
social
disciplines. In clinical studies, the adoption of urn models in  
treatment
allocation schemes has been proved to be beneficial to both  
researchers, by
providing more efficient clinical trials, and patients, by increasing  
the
probability of receiving the better treatment. In this paper, we  
endeavor to
derive a very general class of immigrated urn models that incorporates  
the
immigration mechanism into the urn process. Important asymptotic  
properties are
developed and illustrative examples are provided to demonstrate the
applicability of our proposed class of urn models. In general, the  
immigrated
urn model has smaller variability than the corresponding urn model.  
Therefore,
it is more powerful when used in clinical trials.


  http://arxiv.org/abs/0812.3698

---------------------------------------------------------------

7926. ARTIFICIAL INTELLIGENCE FOR BIDDING HEX

Sam Payne and Elina Robeva

We present a Monte Carlo algorithm for efficiently finding near  
optimal moves
and bids in the game of Bidding Hex. The algorithm is based on the  
recent
solution of Random-Turn Hex by Peres, Schramm, Sheffield, and Wilson  
together
with Richman's work connecting random-turn games to bidding games.


  http://arxiv.org/abs/0812.3677

---------------------------------------------------------------

7927. PREDICTABILITY IN SPATIALLY EXTENDED SYSTEMS WITH MODEL  
UNCERTAINTY

Jinqiao Duan

Macroscopic models for spatially extended systems under random  
influences are
often described by stochastic partial differential equations (SPDEs).
   Some techniques for understanding solutions of such equations, such  
as
estimating correlations, Liapunov exponents and impact of noises, are
discussed. They are relevant for understanding predictability in  
spatially
extended systems with model uncertainty, for example, in physics,  
geophysics
and biological sciences. The presentation is for a wide audience.


  http://arxiv.org/abs/0812.3679

---------------------------------------------------------------

7928. A NEW FAMILY OF COVARIATE-ADJUSTED RESPONSE ADAPTIVE DESIGNS AND  
THEIR  ASYMPTOTIC PROPERTIES

Li-Xin Zhang and Feifang Hu

It is often important to incorporating covariate information in the  
design of
clinical trials. In literature, there are many designs of using  
stratification
and covariate-adaptive randomization to balance on certain known  
covariate.
Recently Zhang, Hu, Cheung and Chan (2007) have proposed a family of
covariate-adjusted response-adaptive (CARA) designs and studied their
asymptotic properties. However, these CARA designs often have high
variabilities. In this paper, we propose a new family of covariate- 
adjusted
response-adaptive (CARA) designs. We show that the new designs have  
smaller
variabilities and therefore more efficient.


  http://arxiv.org/abs/0812.3691

---------------------------------------------------------------

7929. MULTI-COLOR RANDOMLY REINFORCED URN FOR ADAPTIVE DESIGNS

Li-Xin Zhang and  Feifang Hu and  Siu Hung Cheung and Wei Sum Chan

The response-adaptive design driven by randomly reinforced urn model is
optimal in the sense that it allocate patients to the best treatment  
with
probability converging to one. This paper illustrates asymptotic  
properties for
multi-color reinforced urn models. Results on the rate of convergence  
of the
number of patients assigned to each treatment are obtained under minimum
requirement of conditions and the distributions of the limits are found.
Asymptotic distributions of the Wald test statistic for testing mean
differences are obtained both under the null hypothesis and alternate
hypothesis. The asymptotic behavior for the non-homogenous is also  
studied.


  http://arxiv.org/abs/0812.3699

---------------------------------------------------------------

7930. A PROBABLISTIC ORIGIN FOR A NEW CLASS OF BIVARIATE POLYNOMIALS

Michael R. Hoare and Mizan Rahman

We present here a probabilistic approach to the generation of new  
polynomials
in two discrete variables. This extends our earlier work on the  
'classical'
orthogonal polynomials in a previously unexplored direction, resulting  
in the
discovery of an exactly soluble eigenvalue problem corresponding to a  
bivariate
Markov chain with a transition kernel formed by a convolution of simple
binomial and trinomial distributions. The solution of the relevant
eigenfunction problem, giving the spectral resolution of the kernel,  
leads to
what we believe to be a new class of orthogonal polynomials in two  
discrete
variables. Possibilities for the extension of this approach are  
discussed.


  http://arxiv.org/abs/0812.3879

---------------------------------------------------------------

7931. SCALING LIMITS FOR SYMMETRIC ITO-LEVY PROCESSES IN RANDOM MEDIUM

Remi Rhodes; Vincent Vargas

We are concerned with scaling limits of the solutions to stochastic
differential equations with stationary coefficients driven by Poisson  
random
measures and Brownian motions. We state an annealed convergence  
theorem, in
which the limit exhibits a diffusive or superdiffusive behavior,  
depending on
the integrability properties of the Poisson random measure


  http://arxiv.org/abs/0812.3904

---------------------------------------------------------------

7932. OPTIMAL STOPPING OF A RISK PROCESS WHEN CLAIMS ARE COVERED  
IMMEDIATELY

Bogdan K. Muciek and  Krzysztof J. Szajowski

The optimal stopping problem for the risk process with interests rates  
and
when claims are covered immediately is considered. An insurance company
receives premiums and pays out claims which have occured according to  
a renewal
process and which have been recognized by them. The capital of the  
company is
invested at interest rate $\alpha\in\Re^{+}$, the size of claims  
increase at
rate $\beta\in\Re^{+}$ according to inflation process. The immediate  
payment of
claims decreases the company investment by rate $\alpha_1$. The aim is  
to find
the stopping time which maximizes the capital of the company. The  
improvement
to the known models by taking into account different scheme of claims  
payment
and the possibility of rejection of the request by the insurance  
company is
made. It leads to essentially new risk process and the solution of  
optimal
stopping problem is different.


  http://arxiv.org/abs/0812.3925

---------------------------------------------------------------

7933. REFLECTED BACKWARD SDES WITH GENERAL JUMPS

S.Hamadene and Y.Ouknine

In the first part of this paper we give a solution for the one- 
dimensional
reflected backward stochastic differential equation (BSDE for short)  
when the
noise is driven by a Brownian motion and an independent Poisson point  
process.
The reflecting process is right continuous with left limits (rcll for  
short)
whose jumps are arbitrary. We first prove existence and uniqueness of  
the
solution for a specific coefficient in using a method based on a  
combination of
penalization and the Snell envelope theory. To show the result in the  
general
framework we use a fixed point argument in an appropriate space. The  
second
part of the paper is related to BSDEs with two reflecting barriers.  
Once more
we prove the existence and uniqueness of the solution of the BSDE.


  http://arxiv.org/abs/0812.3965

---------------------------------------------------------------

7934. RADIAL DUNKL PROCESSES ASSOCIATED WITH DIHEDRAL SYSTEMS

Nizar Demni

We stduy radial Dunkl processes associated with dihedral systems: we  
derive
the semi group, the generalized Bessel function, the Dunkl-Hermite  
polynomials.
Then we give a skew product decomposition by means of independent Bessel
processes and we compute the tail distribution of the first hitting  
time of the
boundary of Weyl chamber.


  http://arxiv.org/abs/0812.4002

---------------------------------------------------------------

7935. ISING (CONFORMAL) FIELDS AND CLUSTER AREA MEASURES

Federico Camia and  Charles M. Newman

We provide a representation for the scaling limit of the d=2 critical  
Ising
magnetization field as a (conformal) random field using SLE (Schramm- 
Loewner
Evolution) clusters and associated renormalized area measures. The  
renormalized
areas are from the scaling limit of the critical FK (Fortuin-Kasteleyn)
clusters and the random field is a convergent sum of the area measures  
with
random signs. Extensions to off-critical scaling limits, to d=3 and to  
Potts
models are also considered.


  http://arxiv.org/abs/0812.4030

---------------------------------------------------------------

7936. ON THE SUPREMUM OF CERTAIN FAMILIES OF STOCHASTIC PROCESSES

Wenbo V. Li and  Natesh S. Pillai and  Robert L. Wolpert

We consider a family of stochastic processes $\{X_t^\epsilon, t \in T\} 
$ on a
metric space $T$, with a parameter $\epsilon \downarrow 0$. We study the
conditions under which \lim_{\e \to 0} \P \Big(\sup_{t \in T} |X_t^\e|  
< \delta
\Big) =1 when one has the \textit{a priori} estimate on the modulus of
continuity and the value at one point. We compare our problem to the  
celebrated
Kolmogorov continuity criteria for stochastic processes, and finally  
give an
application of our main result for stochastic intergrals with respect to
compound Poisson random measures with infinite intensity measures.


  http://arxiv.org/abs/0812.4062

---------------------------------------------------------------

7937. DEFAULT TIMES, NON ARBITRAGE CONDITIONS AND CHANGE OF  
PROBABILITY  MEASURES

Delia Coculescu and  Monique Jeanblanc and Ashkan Nikeghbali

In this paper we give a financial justification, based on non arbitrage
conditions, of the $(H)$ hypothesis in default time modelling. We also  
show how
the $(H)$ hypothesis is affected by an equivalent change of probability
measure. The main technique used here is the theory of progressive  
enlargements
of filtrations.


  http://arxiv.org/abs/0812.4064

---------------------------------------------------------------

7938. FLUCTUATIONS OF THE EMPIRICAL QUANTILES OF INDEPENDENT BROWNIAN  
MOTIONS

Jason Swanson

We consider $n$ independent, identically distributed one-dimensional  
Brownian
motions, $B_j(t)$, where $B_j(0)$ has a rapidly decreasing, smooth  
density
function $f$. The empirical quantiles, or pointwise order statistics,  
are
denoted by $B_{j:n}(t)$, and we are interested in a sequence of  
quantiles
$Q_n(t) = B_{j(n):n}(t)$, where $j(n)/n \to \alpha \in (0,1)$. This  
sequence
converges in probability in $C[0,\infty)$ to $q(t)$, the $\alpha$- 
quantile of
the law of $B_j(t)$. Our main result establishes the convergence in  
law in
$C[0,\infty)$ of the fluctuation processes $F_n = n^{1/2}(Q_n - q)$.  
The limit
process $F$ is a centered Gaussian process and we derive an explicit  
formula
for its covariance function. We also show that $F$ has many of the  
same local
properties as $B^{1/4}$, the fractional Brownian motion with Hurst  
parameter $H
= 1/4$. For example, it is a quartic variation process, it has H\"older
continuous paths with any exponent $\gamma < 1/4$, and (at least  
locally) it
has increments whose correlation is negative and of the same order of  
magnitude
as those of $B^{1/4}$.


  http://arxiv.org/abs/0812.4102

---------------------------------------------------------------

7939. ASYMPTOTICS OF THE NORM OF ELLIPTICAL RANDOM VECTORS

Enkelejd Hashorva

In this paper we consider elliptical random vectors X in R^d,d>1 with
stochastic representation A R U where R is a positive random radius  
independent
of the random vector U which is uniformly distributed on the unit  
sphere of R^d
and A is a given matrix. The main result of this paper is an asymptotic
expansion of the tail probability of the norm of X derived under the  
assumption
that R has distribution function is in the Gumbel or the Weibull max- 
domain of
attraction.


  http://arxiv.org/abs/0812.4105

---------------------------------------------------------------

7940. NON-EQUILIBRIUM DYNAMICS OF DYSON'S MODEL WITH INFINITE PARTICLES

Makoto Katori and  Hideki Tanemura

Dyson's model is a one-dimensional system of Brownian motions with  
long-range
repulsive forces acting between any pair of particles with strength
proportional to the inverse of distances. We give sufficient  
conditions for
initial configurations so that Dyson's model with infinite number of  
particles
is well defined in the sense that any multitime correlation function  
is given
by a determinant with a locally integrable kernel. The class of
infinite-dimensional configurations satisfying our conditions is large  
enough
to study non-equilibrium dynamics. For example, a relaxation process  
starting
from a configuration, in which each lattice point of $\Z$ is occupied  
by one
particle, to the stationary state, which is the determinantal point  
process
with the sine kernel $\mu_{\sin}$, is determined. The invariant measure
$\mu_{\sin}$ also satisfies our conditions and Dyson's model starting  
from
$\mu_{\sin}$, which is a reversible process, is identified with the  
infinite
particle system, which is determinantal with the extended sine kernel  
studied
in the random matrix theory. We also show that this infinite-dimensional
reversible process is Markovian.


  http://arxiv.org/abs/0812.4108

---------------------------------------------------------------

7941. THRESHOLD BEHAVIOUR AND FINAL OUTCOME OF AN EPIDEMIC ON A RANDOM  
NETWORK  WITH HOUSEHOLD STRUCTURE

Frank Ball and  David Sirl and Pieter Trapman

This paper considers a stochastic SIR (susceptible$\to$infective$\to 
$removed)
epidemic model in which individuals may make infectious contacts in  
two ways,
both within `households' (which for ease of exposition are assumed to  
have
equal size) and along the edges of a random graph describing  
additional social
contacts. Heuristically-motivated branching process approximations are
described, which lead to a threshold parameter for the model and  
methods for
calculating the probability of a major outbreak, given few initial  
infectives,
and the expected proportion of the population who are ultimately  
infected by
such a major outbreak. These approximate results are shown to be exact  
as the
number of households tends to infinity by proving associated limit  
theorems.
Moreover, simulation studies indicate that these asymptotic results  
provide
good approximations for modestly sized finite populations. The  
extension to
unequal sized households is discussed briefly.


  http://arxiv.org/abs/0812.4110

---------------------------------------------------------------

7942. A USEFUL RELATIONSHIP BETWEEN EPIDEMIOLOGY AND QUEUEING THEORY

Pieter Trapman and Martin Bootsma

In this paper we establish a relation between the spread of infectious
diseases and the dynamics of so called M/G/1 queues with processor  
sharing. The
in epidemiology well known relation between the spread of epidemics and
branching processes and the in queueing theory well known relation  
between
M/G/1 queues and birth death processes will be combined to provide a  
framework
in which results from queueing theory can be used in epidemiology and  
vice
versa.
   In particular, we consider the number of infectious individuals in  
a standard
SIR epidemic model at the moment of the first detection of the  
epidemic, where
infectious individuals are detected at a constant per capita rate. We  
use a
result from the literature on queueing processes to show that this  
number of
infectious individuals is geometrically distributed.


  http://arxiv.org/abs/0812.4135

---------------------------------------------------------------

7943. NOTE ON RADIAL DUNKL PROCESSES

Nizar Demni

This note encloses relatively short proofs of the following known  
results:
the radial Dunkl process associated with a reduced system and a strictly
positive multiplicity function is the unique strong solution for all  
time t of
a stochastic differential equation of a singular drift (see [11] for the
original proof and [4] for a proof under additional restrictions), the  
first
hitting time of the Weyl chamber by a radial Dunkl process is finite  
almost
surely for small values of the multiplicity function. Our proof of the  
second
mentioned result gives more information than the original one.


  http://arxiv.org/abs/0812.4269

---------------------------------------------------------------

7944. STOCHASTICALLY STABLE GLOBALLY COUPLED MAPS WITH BISTABLE  
THERMODYNAMIC  LIMIT

Jean-Baptiste Bardet (IRMAR and  LMRS) and  Gerhard Keller and   
Roland  Zweim\"uller

We study systems of globally coupled interval maps, where the identical
individual maps have two expanding, fractional linear, onto branches,  
and where
the coupling is introduced via a parameter - common to all individual  
maps -
that depends in an analytic way on the mean field of the system. We  
show: 1)
For the range of coupling parameters we consider, finite-size coupled  
systems
always have a unique invariant probability density which is strictly  
positive
and analytic, and all finite-size systems exhibit exponential decay of
correlations. 2) For the same range of parameters, the self-consistent
Perron-Frobenius operator which captures essential aspects of the  
corresponding
infinite-size system (arising as the limit of the above when the  
system size
tends to infinity), undergoes a supercritical pitchfork bifurcation  
from a
unique stable equilibrium to the coexistence of two stable and one  
unstable
equilibrium.


  http://arxiv.org/abs/0812.4040

---------------------------------------------------------------

7945. BOUNDING BASIC CHARACTERISTICS OF SPATIAL EPIDEMICS WITH A NEW   
PERCOLATION MODEL

Ronald Meester and Pieter Trapman

We introduce a new percolation model to describe and analyze the  
spread of an
epidemic on a general directed and locally finite graph. We assign a
two-dimensional random weight vector to each vertex of the graph in  
such a way
that the weights of different vertices are i.i.d., but the two entries  
of the
vector assigned to a vertex need not be independent. The probability  
for an
edge to be open depends on the weights of its end vertices, but  
conditionally
on the weights, the states of the edges are independent of each other.  
In an
epidemiological setting, the vertices of a graph represent the  
individuals in a
(social) network and the edges represent the connections in the  
network. The
weights assigned to an individual denote its (random) infectivity and
susceptibility, respectively. We show that one can bound the percolation
probability and the expected size of the cluster of vertices that can be
reached by an open path starting at a given vertex from above and  
below by the
corresponding quantities for respectively independent bond and site  
percolation
with certain densities; this generalizes a result of Kuulasmaa. Many  
models in
the literature are special cases of our general model.


  http://arxiv.org/abs/0812.4353

---------------------------------------------------------------

7946. MARTINGALE-COBOUNDARY REPRESENTATION FOR A CLASS OF RANDOM FIELDS

Mikhail Gordin

A stationary random sequence admits under some assumptions a  
representation
as the sum of two others: one of them is a martingale difference  
sequence, and
another is a so-called coboundary. Such a representation can be used for
proving some limit theorems by means of the martingale approximation. A
multivariate version of such a decomposition is presented in the paper  
for a
class of random fields generated by several commuting non-invertible
probability preserving transformations. In this representation  
summands of
mixed type appear which behave with respect to some groupof directions  
of the
parameter space as reversed multiparameter martingale differences (in  
the sense
of one of several known definitions) while they look as coboundaries  
relative
to the other directions. Applications to limit theorems will be  
published
elsewhere.


  http://arxiv.org/abs/0812.4414

---------------------------------------------------------------

7947. CONSTRUCTION OF SIGNED MULTIPLICATIVE CASCADES

Julien Barral and  Xiong Jin and Benoit Mandelbrot

The theory of positive $T$-martingales was developed in order to set  
up a
general framework including the positive measure-valued martingales  
initially
considered for intermittent turbulence modelling. We consider the  
natural
extension consisting in allowing the martingale to take complex  
values. We
focus on martingales constructed on the line: $T$ is the interval  
$[0,1]$.
Then, random measures are replaced by random functions. We specify a  
large
class of such martingales, which contains the complex extension of $b$- 
adic
canonical cascades, compound Poisson cascades, and more generally  
infinitely
divisible cascades. For the elements of this class, we find a sufficient
condition for their almost sure uniform convergence to a non-trivial  
limit.
Such limit provide new examples of multifractal processes.


  http://arxiv.org/abs/0812.4556

---------------------------------------------------------------

7948. CONVERGENCE OF SIGNED MULTIPLICATIVE CASCADES

Julien Barral and  Xiong Jin and  Benoit Mandelbrot

This paper extends the familiar sequences of random measures obtained on
$[0,1]$ via $b$-adic independent cascades by allowing the random weights
invoked in the cascades to take real, or complex values. This yields  
sequences
of random functions. The asymptotic behavior of these sequences is
investigated. We obtain a sufficient condition for the almost sure  
convergence
of these signed cascades to non-trivial statistically self-similar  
limit. Under
suitable assumptions, the limit function can be represented almost  
surely as a
monofractal function in multifractal time. When the sufficient  
condition for
convergence does not hold, in most of the cases we show that either  
the limit
is 0 or the sequence diverges almost surely. In the later case, under  
some
condition we prove a functional central limit theorem, which claims  
that there
is a natural normalization making the sequence convergent in law to a  
standard
Brownian motion in multifractal time.


  http://arxiv.org/abs/0812.4557

---------------------------------------------------------------

7949. POLYNOMIAL PROCESSES AND THEIR APPLICATIONS TO MATHEMATICAL  
FINANCE

Christa Cuchiero and  Martin Keller-Ressel and  Josef Teichmann

We introduce a class of Markov stochastic processes called $m$- 
polynomial,
for which the calculation of (mixed) moments up to order $m$ only  
requires the
computation of matrix exponentials. This class contains affine  
processes,
Feller processes with quadratic squared diffusion coefficient, as well  
as
L\'evy-driven SDEs with affine vector fields. Thus, many popular  
models such as
the classical Black-Scholes, exponential L\'evy or affine models are  
covered by
this setting. The applications range from statistical GMM estimation  
to option
pricing. For instance, the efficient and easy computation of moments can
successfully be used for variance reduction techniques in Monte Carlo
simulations.


  http://arxiv.org/abs/0812.4740

---------------------------------------------------------------

7950. POLYNOMIAL BIRTH-DEATH DISTRIBUTION APPROXIMATION IN  
WASSERSTEIN  DISTANCE

Aihua Xia and Fuxi Zhang

The polynomial birth-death distribution (abbr. as PBD) on $\ci=\{0,1,2,
 >...\}$ or $\ci=\{0,1,2, ..., m\}$ for some finite $m$ introduced in  
Brown &
Xia (2001) is the equilibrium distribution of the birth-death process  
with
birth rates $\{\alpha_i\}$ and death rates $\{\beta_i\}$, where $\a_i 
\ge0$ and
$\b_i\ge0$ are polynomial functions of $i\in\ci$. The family includes  
Poisson,
negative binomial, binomial and hypergeometric distributions. In this  
paper, we
give probabilistic proofs of various Stein's factors for the PBD  
approximation
with $\a_i=a$ and $\b_i=i+bi(i-1)$ in terms of the Wasserstein  
distance. The
paper complements the work of Brown & Xia (2001) and generalizes the  
work of
Barbour & Xia (2006) where Poisson approximation ($b=0$) in the  
Wasserstein
distance is investigated. As an application, we establish an upper  
bound for
the Wasserstein distance between the PBD and Poisson binomial  
distribution and
show that the PBD approximation to the Poisson binomial distribution  
is much
more precise than the approximation by the Poisson or shifted Poisson
distributions.


  http://arxiv.org/abs/0812.4847

---------------------------------------------------------------

7951. ON THE BOSE-EINSTEIN DISTRIBUTION AND BOSE CONDENSATION

V. P. Maslov (1 and 2) and V. E. Nazaikinskii (2) ((1) Moscow State   
University, (2) Institute for Problems in Mechanics, RAS, Moscow)

For a system of identical Bose particles sitting on integer energy  
levels, we
give sharp estimates for the convergence of the sequence of occupation  
numbers
to the Bose-Einstein distribution and for the Bose condensation effect.


  http://arxiv.org/abs/0812.4885

---------------------------------------------------------------

7952. A NEW APPROACH OF POINT ESTIMATION FROM TRUNCATED OR GROUPED  
AND  CENSORED DATA

Ahmed Guellil (USTHB) and  Tewfik Kernane (USTHB)

We propose a new approach for estimating the parameters of a probability
distribution. It consists on combining two new methods of estimation.  
The first
is based on the definition of a new distance measuring the difference  
between
variations of two distributions on a finite number of points from  
their support
and on using this measure for estimation purposes by the method of  
minimum
distance. For the second method, given an empirical discrete  
distribution, we
build up an auxiliary discrete theoretical distribution having the  
same support
of the first and depending on the same parameters of the parent  
distribution of
the data from which the empirical distribution emanated. We estimate  
then the
parameters from the empirical distribution by the usual statistical  
methods. In
practice, we propose to compute the two estimations, the second based on
maximum likelihood principle of known theoretical properties, and the  
first
being as a control of the effectiveness of the obtained estimation,  
and for
which we prove the convergence in probability, so we have also a  
criterion on
the quality of the information contained in the observations. We apply  
the
approach to truncated or grouped and censored data situations to give  
the
flavour on the effectiveness of the approach. We give also some  
interesting
perspectives of the approach including model selection from truncated  
data,
estimation of the initial trial value in the celebrate EM algorithm in  
the case
of truncation and merged normal populations, a test of goodness of fit  
based on
the new distance, quality of estimations and data.


  http://arxiv.org/abs/0802.2155

---------------------------------------------------------------

7953. RANDOM COMPLEX DYNAMICS AND SEMIGROUPS OF HOLOMORPHIC MAPS

Hiroki Sumi

We investigate the random dynamics of rational maps and the dynamics of
semigroups of rational maps on the Riemann sphere. We see that the  
both fields
are related to each other very deeply. We investigate spectral  
properties of
transition operators and the dynamics of associated semigroups of  
rational
maps. We define several kinds of Julia sets of the associated Markov  
processes
and we study the properties and the dimension of them. Moreover, we  
investigate
"singular functions on the complex plane". In particular, we consider  
the
functions $T$ which represent the probability of tending to infinity  
with
respect to the random dynamics of polynomials. Under certain  
conditions these
functions $T$ are complex analogues of the devil's staircase and  
Lebesgue's
singular functions. More precisely, we show that these functions $T$ are
continuous on the Riemann sphere and vary only on the Julia sets of  
associated
semigroups. Furthermore, by using ergodic theory and potential theory,  
we
investigate the non-differentiability and regularity of these  
functions. We
find many phenomena which can hold in the random complex dynamics and  
the
dynamics of semigroups of rational maps, but cannot hold in the usual  
iteration
dynamics of a single holomorphic map. We carry out a systematic study  
of these
phenomena and their mechanisms.


  http://arxiv.org/abs/0812.4483



-----------------------------------
Stefano M. Iacus
Department of Economics,
Business and Statistics
University of Milan
Via Conservatorio, 7
I-20123 Milan - Italy
Ph.: +39 02 50321 461
Fax: +39 02 50321 505
http://www.economia.unimi.it/iacus
------------------------------------------------------------------------------------
Please don't send me Word or PowerPoint attachments if not
absolutely necessary. See:
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From pas at lists.imstat.org  Tue Mar  3 02:14:14 2009
From: pas at lists.imstat.org (Probability Abstract Service)
Date: Tue, 03 Mar 2009 09:14:14 +0100
Subject: [PAS] Probability Abstracts 108
Message-ID: <8D0D54CC-103B-48F7-82E7-B0F80F88C4F1@unimi.it>

Probability Abstracts 108
This document contains abstracts 7954-8212
from Jan-1-2009 to February-28-2009.
They have been mailed on Mar 3, 2009.


This letter can be also found on line at
http://pas.imstat.org/Letters/letter_108.shtml


-----------------------------------------------

7954. Regularity of Ornstein-Uhlenbeck processes driven by a L{\'e}vy  
white noise
Author(s): Zdzis{\l}aw Brze{\'z}niak and Jerzy Zabczyk

Abstract: The paper is concerned with spatial and time regularity of  
solutions to linear stochastic evolution equation perturbed by L\'evy  
white noise "obtained by subordination of a Gaussian white noise".  
Sufficient conditions for spatial continuity are derived. It is also  
shown that solutions do not have in general \cadlag modifications.  
General results are applied to equations with fractional Laplacian.  
Applications to Burgers stochastic equations are considered as well.

http://arxiv.org/abs/0901.0028


7955. Weak Solutions of the Stochastic Landau-Lifshitz-Gilbert Equation
Author(s): Z. Brzezniak and B. Goldys

Abstract: The Landau-Lifshitz-Gilbert equation perturbed by a  
multiplicative space-dependent noise is considered for a ferromagnet  
filling a bounded three-dimensional domain. We show the existence of  
weak martingale solutions taking values in a sphere $\mathbb S^2$. The  
regularity of weak solutions is also discussed. Some of the regularity  
results are new even for the deterministic Landau-Lifshitz-Gilbert  
equation.

http://arxiv.org/abs/0901.0039


7956. Conditions for certain ruin for the generalised Ornstein- 
Uhlenbeck process and the structure of the upper and lower bounds
Author(s): Damien Bankovsky

Abstract: For a bivariate \Levy process $(\xi_t,\eta_t)_{t\geq 0}$ the  
generalised Ornstein-Uhlenbeck (GOU) process is defined as \ 
[V_t:=e^{\xi_t}(z+\int_0^t e^{-\xi_{s-}}\ud \eta_s), t\ge0,\]where $z 
\in\mathbb{R}.$ We present conditions on the characteristic triplet of  
$(\xi,\eta)$ which ensure certain ruin for the GOU. We present a  
detailed analysis on the structure of the upper and lower bounds and  
the sets of values on which the GOU is almost surely increasing, or  
decreasing. This paper is the sequel to \cite{BankovskySly08}, which  
stated conditions for zero probability of ruin, and completes a  
significant aspect of the study of the GOU.

http://arxiv.org/abs/0901.0207


7957. Current and density fluctuations for interacting particle  
systems with anomalous diffusive behavior
Author(s): M. Jara

Abstract: We prove density and current fluctuations for two examples  
of symmetric, interacting particle systems with anomalous diffusive  
behavior: the zero-range process with long jumps and the zero-range  
process with degenerated bond disorder. As an application, we obtain  
subdiffusive behavior of a tagged particle in a simple exclusion  
process with variable diffusion coefficient.

http://arxiv.org/abs/0901.0229


7958. Order-invariant Measures on Causal Sets
Author(s): Graham Brightwell and Malwina Luczak

Abstract: A causal set is a partially ordered set on a countably  
infinite ground-set such that each element is above finitely many  
others. A natural extension of a causal set is an enumeration of its  
elements which respects the order. We bring together two different  
classes of random processes. In one class, we are given a fixed causal  
set, and we consider random natural extensions of this causal set: we  
think of the random enumeration as being generated one point at a  
time. In the other class of processes, we generate a random causal  
set, again working from the bottom up, adding one new maximal element  
at each stage. Processes of both types can exhibit a property called  
order-invariance: if we stop the process after some fixed number of  
steps, then, conditioned on the structure of the causal set, every  
possible order of generation of its elements is equally likely. We  
develop a framework for the study of order-invariance which includes  
both types of example: order-invariance is then a property of  
probability measures on a certain space. Our main result is a  
description of the extremal order-invariant measures.

http://arxiv.org/abs/0901.0240


7959. Spatial Epidemics and Local Times for Critical Branching Random  
Walks in Dimensions 2 and 3
Author(s): Steven P. Lalley and Xinghua Zheng

Abstract: The behavior at criticality of spatial SIR (susceptible/ 
infected/recovered) epidemic models in dimensions two and three is  
investigated. In these models, finite populations of size N are  
situated at the vertices of the integer lattice, and infectious  
contacts are limited to individuals at the same or at neighboring  
sites. Susceptible individuals, once infected, remain contagious for  
one unit of time and then recover, after which they are immune to  
further infection. It is shown that the measure-valued processes  
associated with these epidemics, suitably scaled, converge, in the  
large-N limit, either to a standard Dawson-Watanabe process (super- 
Brownian motion) or to a Dawson-Watanabe process with location- 
dependent killing, depending on the size of the the initially infected  
set. A key element of the argument is a proof of Adler's 1993  
conjecture that the local time processes associated with branching  
random walks converge to the local time density process associated  
with the limiting super-Brownian motion.

http://arxiv.org/abs/0901.0246


7960. Representation of gaussian small ball probabilities in $l_2$
Author(s): Andr\'e Mas (I3M)

Abstract: Let $z=\sum_{i=1}^{+\infty}x_{i}^{2}/a_{i}^{2}$ where the  
$x_{i}$'s are i.d.d centered with unit variance gaussian random  
variables and $(a_{i}) _{i\in\mathbb{N}}$ an increasing sequence such  
that $\sum _{i=1}^{+\infty}a_{i}^{-2}<+\infty$. We propose an  
exponential-integral representation theorem for the gaussian small  
ball probability $\mathbb{P}% (z<\varepsilon) $ when $\varepsilon 
\downarrow0$. We start from a result by Meyer-Wolf, Zeitouni (1993)  
and Dembo, Meyer-Wolf, Zeitouni (1995) who computed this probability  
by means of series. We prove that $\mathbb{P}% (z<\varepsilon) $  
belongs to a class of functions introduced by de Haan, well-known in  
extreme value theory, the class Gamma, for which an explicit  
exponential-integral representation is available. The converse  
implication holds under a mild additional assumption. Some  
applications are underlined in connection with statistical inference  
for random functions.

http://arxiv.org/abs/0901.0264


7961. Adjustment coefficient for risk processes in some dependent  
contexts
Author(s): H. Cossette and E. Marceau and V. Maume-Deschamps

Abstract: Following an article by Muller and Pflug, we study the  
adjustment coefficient of ruin theory in a context of temporal  
dependency. We provide a consistent estimator of this coefficient, and  
perform some simulations.

http://arxiv.org/abs/0901.0182


7962. Maximum Entropy on Compact Groups
Author(s): Peter Harremoes

Abstract: On a compact group the Haar probability measure plays the  
role as uniform distribution. The entropy and rate distortion theory  
for this uniform distribution is studied. New results and simplified  
proofs on convergence of convolutions on compact groups are presented  
and they can be formulated as entropy increases to its maximum.  
Information theoretic techniques and Markov chains play a crucial  
role. The rate of convergence is shown to be exponential. The results  
are also formulated via rate distortion functions.

http://arxiv.org/abs/0901.0015


7963. p-Adic Spherical Coordinates and Their Applications
Author(s): Anatoly N. Kochubei

Abstract: On the space $\mathbb Q_p^n$, where $p\ne 2$ and $p$ does  
not divide $n$, we construct a p-adic counterpart of spherical  
coordinates. As applications, a description of homogeneous  
distributions on $\mathbb Q_p^n$ and a skew product decomposition of p- 
adic L\'evy processes are given.

http://arxiv.org/abs/0901.0071


7964. Order-invariant Measures on Fixed Causal Sets
Author(s): Graham Brightwell and Malwina Luczak

Abstract: A causal set is a countably infinite poset in which every  
element is above finitely many others; causal sets are exactly the  
posets that have a linear extension with the order-type of the natural  
numbers -- we call such a linear extension a {\em natural extension}.  
We study probability measures on the set of natural extensions of a  
causal set, especially those measures having the property of {\em  
order-invariance}: if we condition on the set of the bottom $k$  
elements of the natural extension, each possible ordering among these  
$k$ elements is equally likely. We give sufficient conditions for the  
existence and uniqueness of an order-invariant measure on the set of  
natural extensions of a causal set.

http://arxiv.org/abs/0901.0242


7965. Beta Jacobi processes
Author(s): Nizar Demni

Abstract: We define and study a multidimensional process that  
generalizes the eigenvalues of matrix Jacobi processes on the one hand  
and whose stationary distribution is given by the beta Jacobi ensemble  
on the other hand.

http://arxiv.org/abs/0901.0324


7966. Stein's lemma, Malliavin calculus, and tail bounds, with  
application to polymer fluctuation exponent
Author(s): Frederi G. Viens

Abstract: We consider a random variable X satisfying almost-sure  
conditions involving G:= where DX is X's Malliavin derivative and  
L^{-1} is the inverse Ornstein-Uhlenbeck operator. A lower- (resp.  
upper-) bound condition on G is proved to imply a Gaussian-type lower  
(resp. upper) bound on the tail P[X>z]. Bounds of other natures are  
also given. A key ingredient is the use of Stein's lemma, including  
the explicit form of the solution of Stein's equation relative to the  
function 1_{x>z}, and its relation to G. Another set of comparable  
results is established, without the use of Stein's lemma, using  
instead a formula for the density of a random variable based on G,  
recently devised by the author and Ivan Nourdin. As an application,  
via a Mehler-type formula for G, we show that the Brownian polymer in  
a Gaussian environment which is white-noise in time and positively  
correlated in space has deviations of Gaussian type and a fluctuation  
exponent \chi=1/2. We also show this exponent remains 1/2 after a non- 
linear transformation of the polymer's Hamiltonian.

http://arxiv.org/abs/0901.0383


7967. General discrete random walk with variable absorbing probabilities
Author(s): Theo van Uem

Abstract: We obtain expected number of arrivals, probability of  
arrival, absorption probabilities and expected time before absorption  
for a general discrete random walk with variable absorbing  
probabilities on a finite interval using Fibonacci numbers

http://arxiv.org/abs/0901.0469


7968. Random Current Representation for Transverse Field Ising Models
Author(s): Nicholas Crawford and Dmitry Ioffe

Abstract: Recently, a random current representation for transverse  
field Ising models has been introduced in \cite{ILN}. This  
representation is a space-time version of the classical random current  
representation exploited by Aizenman et. al. %It is a space-time  
version of the classical random current representation \cite{Ai82,  
ABF, AF}. In this paper we formulate and prove corresponding space- 
time versions of the classical switching lemma and show how they  
generate various correlation inequalities. In particular we prove  
exponential decay of truncated two-point functions at positive  
magnetic fields in $\sfz$-direction and address the issue of the  
sharpness of phase transition.

http://arxiv.org/abs/0812.4834


7969. Invariant manifolds for random and stochastic partial  
differential equations
Author(s): Tomas Caraballo and Jinqiao Duan and Kening Lu and Bjorn  
Schmalfuss

Abstract: Random invariant manifolds are geometric objects useful for  
understanding complex dynamics under stochastic influences. Under a  
nonuniform hyperbolicity or a nonuniform exponential dichotomy  
condition, the existence of random pseudo-stable and pseudo-unstable  
manifolds for a class of \emph{random} partial differential equations  
and \emph{stochastic} partial differential equations is shown. Unlike  
the invariant manifold theory for stochastic \emph{ordinary}  
differential equations, random norms are not used. The result is then  
applied to a nonlinear stochastic partial differential equation with  
linear multiplicative noise.

http://arxiv.org/abs/0901.0382


7970. An upper bound for front propagation velocities inside moving  
populations
Author(s): A. Gaudilliere and F.R. Nardi

Abstract: We consider a two type (red and blue or $R$ and $B$)  
particle population that evolves on the $d$-dimensional lattice  
according to some reaction-diffusion process $R+B\to 2R$ and starts  
with a single red particle and a density $\rho$ of blue particles. For  
two classes of models we give an upper bound on the propagation  
velocity of the red particles front with explicit dependence on $\rho 
$. In the first class of models red and blue particles respectively  
evolve with a diffusion constant $D_R=1$ and a possibly time dependent  
jump rate $D_B \geq 0$ -- more generally blue particles follow some  
independent bistochastic process and this also includes long range  
random walks with drift and various deterministic processes. We then  
get in all dimensions an upper bound of order $\max(\rho,\sqrt\rho)$  
that depends only on $\rho$ and $d$ and not on the specific process  
followed by blue particles, in particular that does not depend on $D_B 
$. We argue that for $d \geq 2$ or $\rho \geq 1$ this bound can be  
optimal (in $\rho$), while for the simplest case with $d=1$ and $\rho  
< 1$ known as the frog model, we give a better bound of order $\rho$.  
In the second class of models particles evolve with exclusion and  
possibly attraction inside a large two-dimensional box with periodic  
boundary conditions according to Kawasaki dynamics (that turns into  
simple exclusion when the attraction is set to zero.) In a low density  
regime we then get an upper bound of order $\sqrt\rho$. This proves a  
long-range decorrelation of dynamical events in this low density regime.

http://arxiv.org/abs/0901.0586


7971. A unifying formulation of the Fokker-Planck-Kolmogorov equation  
for general stochastic hybrid systems (extended version)
Author(s): Julien Bect

Abstract: A general formulation of the Fokker-Planck-Kolmogorov (FPK)  
equation for stochastic hybrid systems is presented, within the  
framework of Generalized Stochastic Hybrid Systems (GSHS). The FPK  
equation describes the time evolution of the probability law of the  
hybrid state. Our derivation is based on the concept of mean jump  
intensity, which is related to both the usual stochastic intensity (in  
the case of spontaneous jumps) and the notion of probability current  
(in the case of forced jumps). This work unifies all previously known  
instances of the FPK equation for stochastic hybrid systems, and  
provides GSHS practitioners with a tool to derive the correct  
evolution equation for the probability law of the state in any given  
example.

http://arxiv.org/abs/0901.0615


7972. Infinite rate mutually catalytic branching in infinitely many  
colonies. Construction, characterization and convergence
Author(s): Achim Klenke and Leonid Mytnik

Abstract: We construct a mutually catalytic branching process on a  
countable site space with infinite "branching rate". The finite rate  
mutually catalytic model, in which the rate of branching of one  
population at a site is proportional to the mass of the other  
population at that site, was introduced by Dawson and Perkins in  
[DP98]. We show that our model is the limit for a class of models and  
in particular for the Dawson- Perkins model as the rate of branching  
goes to infinity. Our process is characterized as the unique solution  
to a martingale problem. We also give a characterization of the  
process as a weak solution of an infinite system of stochastic  
integral equations driven by a Poisson noise.

http://arxiv.org/abs/0901.0623


7973. On the growth of the supercritical long-range percolation  
cluster on $\mathbb{Z}^d$ and an application for spatial epidemics
Author(s): Pieter Trapman

Abstract: We consider long-range percolation on $\mathbb{Z}^d$ in  
which the measure on the configuration of edges is a product measure  
and the probability that two vertices at distance $r$ share an edge is  
given by $p(r) = 1-\exp[-\lambda(r)] \in (0,1)$. Here $\lambda(r)$ is  
a strictly positive, non-increasing regularly varying function. We  
investigate the asymptotic growth of the size of the $k$-ball around  
the origin, $|\mathcal{B}_k|$, i.e. the number of vertices that are  
within graph-distance $k$ of the origin, for $k \to \infty$ for  
different $\lambda(r)$. We show that conditioned on the origin being  
in the infinite component, non-empty classes of non-increasing  
regularly varying $\lambda(r)$ exist for which respectively $| 
\mathcal{B}_k|^{1/k} \to \infty$ almost surely, there exist $1 < a_2 <  
\infty$ such that $\lim_{k\to \infty} \mathbb{P}(a_1<|\mathcal{B}_k| 
^{1/k}< a_2) = 1$, $|\mathcal{B}_k|^{1/k} \to 1$ almost surely. This  
result can be applied to spatial $SIR$ epidemics. In particular, we  
show that it is possible to construct a distribution of long-range  
contacts between individuals only depending on their distance, such  
that the number of infectious individuals in the $k$-th infection  
generation stochastically dominates an exponentially growing function.

http://arxiv.org/abs/0901.0661


7974. Isomorphism and Symmetries in Random Phylogenetic Trees
Author(s): Philippe Flajolet and Miklos Bona

Abstract: The probability that two randomly selected phylogenetic  
trees of the same size are isomorphic is found to be asymptotic to a  
decreasing exponential modulated by a polynomial factor. The number of  
symmetrical nodes in a random phylogenetic tree of large size obeys a  
limiting Gaussian distribution, in the sense of both central and local  
limits. The probability that two random phylogenetic trees have the  
same number of symmetries asymptotically obeys an inverse square-root  
law. Precise estimates for these problems are obtained by methods of  
analytic combinatorics, involving bivariate generating functions,  
singularity analysis, and quasi-powers approximations.

http://arxiv.org/abs/0901.0696


7975. Convolution symmetries of integrable hierarchies, matrix models  
and $\tau$-functions
Author(s): J. Harnad and A. Yu. Orlov

Abstract: Generalized convolution symmetries of integrable hierarchies  
of KP-Toda and 2KP-Toda type have the effect of multiplying the  
Fourier coefficients of the Baker-Akhiezer function by a specified  
sequence of constants. The induced action on the associated fermionic  
Fock space is diagonal in the standard orthonormal base determined by  
occupation sites and labeled by partitions. The coefficients in the  
single and double Schur function expansions of the associated $\tau$- 
functions, which are the Pl\"ucker coordinates of a decomposable  
element, are multiplied by the corresponding diagonal factors.  
Applying such transformations to matrix integrals, we obtain new  
matrix models of externally coupled type which are also KP-Toda or 2KP- 
Toda $\tau$-functions. More general multiple integral representations  
of tau functions are similarly obtained, as well as finite  
determinantal expressions for them.

http://arxiv.org/abs/0901.0323


7976. New bounds for the free energy of directed polymer in dimension  
1+1 and 1+2
Author(s): Hubert Lacoin

Abstract: We study the free energy of the directed polymer in random  
environment in dimension 1+1 and 1+2. For dimension 1, we improve the  
statement of Comets and Vargas concerning very strong disorder by  
giving sharp estimates on the free energy at high temperature. In  
dimension 2, we prove that very strong disorder holds at all  
temperatures, thus solving a long standing conjecture.

http://arxiv.org/abs/0901.0699


7977. Phantom Probability
Author(s): Yehuda Izhakian and Zur Izhakian

Abstract: The classical probability theory supports probability  
measures assigning each event with a fixed positive real value; aiming  
to formulate occurrences in real life, these measures are far from  
being satisfactory. The main innovation of this paper is the  
introduction of a new probability measure, enabling the assignment of  
events with varying probabilities that are recorded by ring elements;  
this measure still provides a Bayesian model, resembling the classical  
probability model. By introducing two principles for the possible  
variation of a probability (also known as uncertainty, ambiguity, or  
imprecise probability), together with the ``correct'' algebraic  
structure allowing the framing of these principles, we present the  
foundations for the theory of phantom probability, generalizing the  
classical probability theory in a natural way. This generalization  
preserves much of the well known properties, as well as familiar  
distribution functions, of the classical probability theory: moments,  
covariance, moment generating functions, the low of large numbers, and  
the central limit theorem are a few instances demonstrating the  
concept of the phantom probability theory.

http://arxiv.org/abs/0901.0902


7978. A new approach to mutual information. II
Author(s): Fumio Hiai and Takuho Miyamoto

Abstract: A new concept of mutual pressure is introduced for potential  
functions on both continuous and discrete compound spaces via discrete  
micro-states of permutations, and its relations with the usual  
pressure and the mutual information are established. This paper is a  
continuation of the paper of Hiai and Petz in Banach Center  
Publications, Vol. 78.

http://arxiv.org/abs/0901.1072


7979. When do nonlinear filters achieve maximal accuracy?
Author(s): Ramon van Handel

Abstract: The nonlinear filter for an ergodic signal observed in white  
noise is said to achieve maximal accuracy if the stationary filtering  
error vanishes as the signal to noise ratio diverges. We give a  
general characterization of the maximal accuracy property in terms of  
various systems theoretic notions. When the signal state space is a  
finite set explicit necessary and sufficient conditions are obtained,  
while the linear Gaussian case reduces to a classic result of  
Kwakernaak and Sivan (1972).

http://arxiv.org/abs/0901.1084


7980. A CLT for the L^{2} modulus of continuity of Brownian local time
Author(s): Xia Chen and Wenbo Li and Michael B. Marcus and Jay Rosen

Abstract: Let $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote  
the local time of Brownian motion and \[ \alpha_{t}:=\int_{- 
\infty}^{\infty} (L^{x}_{t})^{2} dx . \] Let $\eta=N(0,1)$ be  
independent of $\alpha_{t}$. For each fixed $t$ \[ {\int_{- 
\infty}^{\infty} (L^{x+h}_{t}- L^{x}_{t})^{2} dx- 4ht\over h^{3/2}}  
\stackrel{\mathcal{L}}{\to}({64 \over 3})^{1/2}\sqrt{\alpha_{t}} \eta,  
\] as $h\rar 0$. Equivalently \[ {\int_{-\infty}^{\infty} (L^{x 
+1}_{t}- L^{x}_{t})^{2} dx- 4t\over t^{3/4}} \stackrel{\mathcal{L}} 
{\to}({64 \over 3} )^{1/2}\sqrt{\alpha_{1}} \eta, \] as $t\rar\infty$.

http://arxiv.org/abs/0901.1102


7981. Asymptotic behaviour of a general reversible chemical reaction- 
diffusion equation
Author(s): Ivan Gentil (CEREMADE) and Boguslaw Zegarlinski

Abstract: In this work, we prove the existence and the exponential  
decay to equilibrium of a general reversible chemical reaction- 
diffusion equation with same but general diffusion. Moreover, we prove  
the optimal asymptotic behaviour in the "two-by-two" case.

http://arxiv.org/abs/0901.1241


7982. Projecting the Fokker-Planck Equation onto a finite dimensional  
exponential family
Author(s): Damiano Brigo and Giovanni Pistone

Abstract: In the present paper we discuss problems concerning  
evolutions of densities related to Ito diffusions in the framework of  
the statistical exponential manifold. We develop a rigorous approach  
to the problem, and we particularize it to the orthogonal projection  
of the evolution of the density of a diffusion process onto a finite  
dimensional exponential manifold. It has been shown by D. Brigo (1996)  
that the projected evolution can always be interpreted as the  
evolution of the density of a different diffusion process. We give  
also a compactness result when the dimension of the exponential family  
increases, as a first step towards a convergence result to be  
investigated in the future. The infinite dimensional exponential  
manifold structure introduced by G. Pistone and C. Sempi is used and  
some examples are given.

http://arxiv.org/abs/0901.1308


7983. Collisions and Spirals of Loewner Traces
Author(s): Joan Lind and Donald E. Marshall and and Steffen Rohde

Abstract: We analyze Loewner traces driven by functions asymptotic to K 
\sqrt{1-t}. We prove a stability result when K is not 4 and show that  
K=4 can lead to non locally connected hulls. As a consequence, we  
obtain a driving term \lambda(t) so that the hulls driven by K 
\lambda(t) are generated by a continuous curve for all K > 0 with K  
not equal to 4 but not when K = 4, so that the space of driving terms  
with continuous traces is not convex. As a byproduct, we obtain an  
explicit construction of the traces driven by K\sqrt{1-t} and a  
conceptual proof of the corresponding results of Kager, Nienhuis and  
Kadanoff, math-ph/0309006

http://arxiv.org/abs/0901.1157


7984. A Better Way to Deal the Cards
Author(s): Mark Conger and Jason Howald

Abstract: This thesis considers the effect of riffle shuffling on  
decks of cards, allowing for some cards to be indistinguishable from  
other cards. The dual problem of dealing a game with hands, such as  
bridge or poker, is also considered. The Gilbert-Shannon-Reeds model  
of card shuffling is used, along with variation distance for measuring  
how close to uniform a deck has become. The surprising results are  
that for a deck with only two types of cards (such as red and black),  
the shuffler can greatly improve the randomness of the deck by  
insuring that the top and bottom cards are the same before shuffling.  
And in the case of dealing cards for a game with "hands", such as  
bridge or poker, the normal method of dealing cyclically around the  
table is very far from optimal. In the case of a well-shuffled bridge  
deck, changing to another dealing method is as good as doing 3.7 extra  
shuffles. How the deck is cut in poker affects its randomness as well.

http://arxiv.org/abs/0901.1324


7985. Semi-infinite TASEP with a Complex Boundary Mechanism
Author(s): Nicky Sonigo (UMPA-Ensl)

Abstract: We consider a totally asymmetric exclusion process on the  
positive half-line. When particles enter in the system according to a  
Poisson source, Liggett has computed all the limit distributions when  
the initial distribution has an asymptotic density. In this paper we  
consider systems for which particles enter at the boundary according  
to a complex mechanism depending on the current configuration in a  
finite neighborhood of the origin. For this kind of models, we prove a  
strong law of large numbers for the number of particles entered in the  
system at a given time. Our main tool is a new representation of the  
model as a multi-type particle system with infinitely many particle  
types.

http://arxiv.org/abs/0901.1364


7986. Two kinds of conditionings for stable L\'evy processes
Author(s): Kouji Yano

Abstract: Two kinds of conditionings for one-dimensional stable L\'evy  
processes are discussed via $ h $-transforms of excursion measures:  
One is to stay positive, and the other is to avoid the origin.

http://arxiv.org/abs/0901.1374


7987. Mixture of the Riesz distribution with respect to the  
multivariate Poisson
Author(s): Abdelhamid Hassairi and Mahdi Louati

Abstract: The aim of this paper is to study the mixture of the Riesz  
distribution on symmetric matrices with respect to the multivariate  
Poisson distribution. We show, in particular, that this distribution  
is related to the modified Bessel function of the first kind. We also  
study the generated natural exponential family. We determine the  
domain of the means and the variance function of this family.

http://arxiv.org/abs/0901.1390


7988. Tails of multivariate Archimedean copulas
Author(s): Arthur Charpentier and Johan Segers

Abstract: A complete and user-friendly directory of tails of  
Archimedean copulas is presented which can be used in the selection  
and construction of appropriate models with desired properties. The  
results are synthesized in the form of a decision tree: Given the  
values of some readily computable characteristics of the Archimedean  
generator, the upper and lower tails of the copula are classified into  
one of three classes each, one corresponding to asymptotic dependence  
and the other two to asymptotic independence. For a long list of  
single-parameter families, the relevant tail quantities are computed  
so that the corresponding classes in the decision tree can easily be  
determined. In addition, new models with tailor-made upper and lower  
tails can be constructed via a number of transformation methods. The  
frequently occurring category of asymptotic independence turns out to  
conceal a surprisingly rich variety of tail dependence structures.

http://arxiv.org/abs/0901.1521


7989. The phase transition of the quantum Ising model is sharp
Author(s): J. E. Bj\"ornberg and G. R. Grimmett

Abstract: An analysis is presented of the phase transition of the  
quantum Ising model with transverse field on the d-dimensional  
hypercubic lattice. It is shown that there is a unique sharp  
transition. The value of the critical point is calculated rigorously  
in one dimension. The first step is to express the quantum Ising model  
in terms of a (continuous) classical Ising model in d+1 dimensions. A  
so-called `random-parity' representation is developed for the latter  
model, similar to the random-current representation for the classical  
Ising model on a discrete lattice. Certain differential inequalities  
are proved. Integration of these inequalities yields the sharpness of  
the phase transition, and also a number of other facts concerning the  
critical and near-critical behaviour of the model under study.

http://arxiv.org/abs/0901.0328


7990. A cautionary tale on the efficiency of some adaptive Monte Carlo  
Schemes
Author(s): Yves F. Atchade

Abstract: There is a growing interest in the literature for adaptive  
Markov Chain Monte Carlo methods based on sequences of random  
transition kernels $\{P_n\}$ where the kernel $P_n$ is allowed to have  
an invariant distribution $\pi_n$ not necessarily equal to the  
distribution of interest $\pi$ (target distribution). These algorithms  
are designed such that as $n\to\infty$, $P_n$ converges to $P$, a  
kernel that has the correct invariant distribution $\pi$. Typically, $P 
$ is a kernel with good convergence properties, but one that cannot be  
directly implemented. It is then expected that the algorithm will  
inherit the good convergence properties of $P$. The equi-energy  
sampler of \cite{kzw06} is an example of this type of adaptive MCMC.  
We show in this paper, that the asymptotic variance of this type of  
adaptive MCMC is always at least as large as the asymptotic variance  
of the Markov chain with transition kernel $P$. We also show by  
simulation that the difference can be substantial.

http://arxiv.org/abs/0901.1378


7991. The Logarithmic Sobolev Inequality in Infinite dimensions for  
Unbounded Spin Systems on the Lattice with non Quadratic Interactions
Author(s): Ioannis Papageorgiou (Imperial College London)

Abstract: We are interested in the Logarithmic Sobolev Inequality for  
the infinite volume Gibbs measure with no quadratic interactions. We  
consider unbounded spin systems on the one dimensional Lattice with  
interactions that go beyond the usual strict convexity and without  
uniform bound on the second derivative. We assume that the one  
dimensional single-site measure with boundaries satisfies the Log- 
Sobolev inequality uniformly on the boundary conditions and we  
determine conditions under which the Log-Sobolev Inequality can be  
extended to the infinite volume Gibbs measure.

http://arxiv.org/abs/0901.1403


7992. Degree-distribution Stability of Evolving Networks
Author(s): Zhenting Hou and Xiangxing Kong and Dinghua Shi and  
Guanrong Chen and Qinggui Zhao

Abstract: In this paper, we abstract a kind of stochastic processes  
from evolving processes of evolving networks, this process is called  
evolving network Markov chains. Thus the degree distribution of  
evolving network is transformed to the corresponding problem of  
evolving network Markov chains. First we investigate the evolving  
network Markov chains, and get its exact formulas and obtain a  
criteria to judge whether the steady degree distribution is power-law  
or not. Then we apply it to evolving networks. With this method, we  
get a rigorous, exact and unified solution of the steady degree  
distribution for evolving networks.

http://arxiv.org/abs/0901.1418


7993. Perturbing the Logarithmic Sobolev Inequality for Unbounded Spin  
Systems on the Lattice with non Quadratic Interactions
Author(s): Ioannis Papageorgiou (Imperial College London)

Abstract: We consider unbounded spin systems on the one dimensional  
Lattice with interactions that go beyond the usual strict convexity  
and without uniform bound on the second derivative. We assume that the  
one dimensional without interactions (boundary-free) measure satisfies  
the Logarithmic Sobolev inequality and we determine conditions under  
which the Log-Sobolev Inequality can be extended to the infinite  
volume Gibbs measure.

http://arxiv.org/abs/0901.1482


7994. Correlation inequalities of GKS type for the Potts model
Author(s): Geoffrey Grimmett

Abstract: Correlation inequalities are presented for functionals of a  
ferromagnetic Potts model with external field, using the random- 
cluster representation. These results extend earlier inequalities of  
Ganikhodjaev--Razak and Schonmann, and yield also GKS-type  
inequalities when the spin-space is taken as the set of qth roots of  
unity.

http://arxiv.org/abs/0901.1625


7995. Entropic Measure on Multidimensional Spaces
Author(s): Karl-Theodor Sturm

Abstract: We construct the entropic measure $\mathbb{P}^\beta$ on  
compact manifolds of any dimension. It is defined as the push forward  
of the Dirichlet process (another random probability measure, well- 
known to exist on spaces of any dimension) under the {\em conjugation  
map} $$\Conj:\mathcal{P}(M)\to\mathcal{P}(M).$$ This conjugation map  
is a continuous involution. It can be regarded as the canonical  
extension to higher dimensional spaces of a map between probability  
measures on 1-dimensional spaces characterized by the fact that the  
distribution functions of $\mu$ and $\Conj(\mu)$ are inverse to each  
other. We also present an heuristic interpretation of the entropic  
measure as $$d\mathbb{P}^\beta(\mu)=\frac{1}{Z}\exp(-\beta\cdot {Ent}  
(\mu|m))\cdot d\mathbb{P}^0(\mu).$$

http://arxiv.org/abs/0901.1815


7996. Approximation of target problems in Blackwell spaces
Author(s): Giacomo Aletti and Diane Saada

Abstract: On a weakly Blackwell space we show how to define a Markov  
chain approximating problem, for the target problem. The approximating  
problem is proved to converge to the optimal reduced problem under  
different pseudometrics. A computational example of compression of  
information is discussed.

http://arxiv.org/abs/0901.1871


7997. Distribution of Random Variables on the Symmetric Group
Author(s): Vytas Zacharovas

Abstract: The well known Erdos-Turan law states that the logarithm of  
an order of a random permutation is asymptotically normally  
distributed. The aim of this work is to estimate convergence rate in  
this theorem and also to prove analogous result for distribution of  
the logarithm of an order of a random permutation on a certain class  
of subsets of the symmetric group. We also study the asymptotic  
behavior of the mean values of multiplicative functions on the  
symmetric group and the results we obtain are of independent interest  
besides their application to the investigation of the remainder term  
in the Erdos-Turan law. We also study a related problem of  
distribution of the degree of a splitting field of a random polynomial  
and obtain sharp estimates for its convergence rate to normal law. In  
research we apply both probabilistic and analytic methods. Some  
analytic methods used here have their origins in the probabilistic  
number theory, and some have their roots in the theory of summation of  
divergent series. One of the approaches we use is to apply Tauberian  
type estimates for Voronoi summability of divergent series to analyze  
the generating functions of the mean values of multiplicative functions.

http://arxiv.org/abs/0901.1733


7998. Classification of E_0--Semigroups by Product Systems
Author(s): Michael Skeide

Abstract: In these notes we tie up some loose ends in the theory of  
E_0-semigroups and their classification by product systems of Hilbert  
modules. We explain how the notion of cocycle conjugacy must be  
modified in order to see how product systems classify E_0-semigroups.  
Actually, we will find two notions of cocycle conjugacy (which for  
Hilbert spaces coincide) that lead to classification up to isomorphism  
of product systems and up to Morita equivalence of product systems,  
respectively. (In between there is also a classification up to  
generalized isomorphism of product systems.) Apart from these new  
results, we provide also general versions of results known for Hilbert  
modules with unit vectors. In this context it is also indispensable to  
review the notions of Morita equivalent product systems and Morita  
equivalent Hilbert modules, adding some generalities that have not yet  
been mentioned. In any case, we underline the outstanding role played  
by Morita equivalence in the relation between E_0-semigroups and  
product systems. As usual with Morita equivalence, the most satisfying  
form of the results we find for von Neumann algebras. Some of the C*- 
versions of the results will depend on countability assumptions.  
Altogether, we have now a complete the theory of the classification of  
normal E_0-semigroups on B^a(E) by product systems of von Neumann  
correspondences. We have the same theory for the classification of  
strict E_0-semigroups by product systems of C*-correspondences under  
countability hypotheses. In both cases, we apply our theory to prove  
that a Markov semigroup admits a Hudson-Parthasarathy dilation if and  
only if it is spatial.

http://arxiv.org/abs/0901.1798


7999. A finite dimensional filter with exponential conditional density
Author(s): Damiano Brigo

Abstract: In this paper we consider the continuous--time nonlinear  
filtering problem, which has an infinite--dimensional solution in  
general, as proved by Chaleyat--Maurel and Michel. There are few  
examples of nonlinear systems for which the optimal filter is finite  
dimensional, in particular Kalman's, Benes', and Daum's filters. In  
the present paper, we construct new classes of scalar nonlinear  
filtering problems admitting finite--dimensional filters. We consider  
a given (nonlinear) diffusion coefficient for the state equation, a  
given (nonlinear) observation function, and a given finite-- 
dimensional exponential family of probability densities. We construct  
a drift for the state equation such that the resulting nonlinear  
filtering problem admits a finite--dimensional filter evolving in the  
prescribed exponential family augmented by the observaton function and  
its square.

http://arxiv.org/abs/0901.1952


8000. Brownian motion with respect to time-changing Riemannian  
metrics, applications to Ricci flow
Author(s): Kol\'eh\'e Abdoulaye Coulibaly-Pasquier (LMA)

Abstract: We generalize Brownian motion on a Riemannian manifold to  
the case of a family of metrics which depends on time. Such questions  
are natural for equations like the heat equation with respect to time  
dependent Laplacians (inhomogeneous diffusions). In this paper we are  
in particular interested in the Ricci flow which provides an intrinsic  
family of time dependent metrics. We give a notion of parallel  
transport along this Brownian motion, and establish a generalization  
of the Dohrn-Guerra or damped parallel transport, Bismut integration  
by part formulas, and gradient estimate formulas. One of our main  
results is a characterization of the Ricci flow in terms of the damped  
parallel transport. At the end of the paper we give an intrinsic  
definition of the damped parallel transport in terms of stochastic  
flows, and derive an intrinsic martingale which may provide  
information about singularities of the flow.

http://arxiv.org/abs/0901.1999


8001. Some differential systems driven by a fBm with Hurst parameter  
greater than 1/4
Author(s): Samy Tindel (IECN) and Iv\'an Torrecilla (UB)

Abstract: This note is devoted to show how to push forward the  
algebraic integration setting in order to treat differential systems  
driven by a noisy input with H\"older regularity greater than 1/4.  
After recalling how to treat the case of ordinary stochastic  
differential equations, we mainly focus on the case of delay  
equations. A careful analysis is then performed in order to show that  
a fractional Brownian motion with Hurst parameter H>1/4 fulfills the  
assumptions of our abstract theorems.

http://arxiv.org/abs/0901.2010


8002. The cut metric, random graphs, and branching processes
Author(s): Bela Bollobas and Svante Janson and Oliver Riordan

Abstract: In this paper we study the component structure of random  
graphs with independence between the edges. Under mild assumptions, we  
determine whether there is a giant component, and find its asymptotic  
size when it exists. We assume that the sequence of matrices of edge  
probabilities converges to an appropriate limit object (a kernel), but  
only in a very weak sense, namely in the cut metric. Our results thus  
generalize previous results on the phase transition in the already  
very general inhomogeneous random graph model we introduced recently,  
as well as related results of Bollob\'as, Borgs, Chayes and Riordan,  
all of which involve considerably stronger assumptions. We also prove  
corresponding results for random hypergraphs; these generalize our  
results on the phase transition in inhomogeneous random graphs with  
clustering.

http://arxiv.org/abs/0901.2091


8003. Hankel determinants of Dirichlet series
Author(s): H. Monien

Abstract: We derive a general expression for the Hankel determinants  
of a Dirichlet series F(s) and derive the asymptotic behavior for the  
special case that F(s) is the Riemann zeta function. In this case the  
Hankel determinant is a discrete analogue of the Selberg integral and  
can be viewed as a matrix integral with discrete measure. We briefly  
comment on its relation to Plancherel measures.

http://arxiv.org/abs/0901.1883


8004. A Lower Bound on the Capacity of Wireless Erasure Networks with  
Random Node Locations
Author(s): Rayyan G. Jaber and Jeffrey G. Andrews

Abstract: In this paper, a lower bound on the capacity of wireless ad  
hoc erasure networks is derived in closed form in the canonical case  
where $n$ nodes are uniformly and independently distributed in the  
unit area square. The bound holds almost surely and is asymptotically  
tight. We assume all nodes have fixed transmit power and hence two  
nodes should be within a specified distance $r_n$ of each other to  
overcome noise. In this context, interference determines outages, so  
we model each transmitter-receiver pair as an erasure channel with a  
broadcast constraint, i.e. each node can transmit only one signal  
across all its outgoing links. A lower bound of $\Theta(n r_n)$ for  
the capacity of this class of networks is derived. If the broadcast  
constraint is relaxed and each node can send distinct signals on  
distinct outgoing links, we show that the gain is a function of $r_n$  
and the link erasure probabilities, and is at most a constant if the  
link erasure probabilities grow sufficiently large with $n$. Finally,  
the case where the erasure probabilities are themselves random  
variables, for example due to randomness in geometry or channels, is  
analyzed. We prove somewhat surprisingly that in this setting,  
variability in erasure probabilities increases network capacity.

http://arxiv.org/abs/0901.1936


8005. Soliton dynamics for the Korteweg-de Vries equation with  
multiplicative homogeneous noise
Author(s): Anne De Bouard (CMAP) and Arnaud Debussche (IRMAR)

Abstract: We consider a randomly perturbed Korteweg-de Vries equation.  
The perturbation is a random potential depending both on space and  
time, with a white noise behavior in time, and a regular, but  
stationary behavior in space. We investigate the dynamics of the  
soliton of the KdV equation in the presence of this random  
perturbation, assuming that the amplitude of the perturbation is  
small. We estimate precisely the exit time of the perturbed solution  
from a neighborhood of the modulated soliton, and we obtain the  
modulation equations for the soliton parameters. We moreover prove a  
central limit theorem for the dispersive part of the solution, and  
investigate the asymptotic behavior in time of the limit process.

http://arxiv.org/abs/0901.1965


8006. H"older index for density states of (alpha,1,beta)- 
superprocesses at a given point
Author(s): Klaus Fleischmann and Leonid Mytnik and Vitali Wachtel

Abstract: A H"older regularity index at given points for density  
states of (alpha,1,beta)-superprocesses with alpha>1+beta is  
determined. It is shown that this index is strictly greater than the  
optimal index of local H"older continuity for those density states.

http://arxiv.org/abs/0901.2315


8007. On weak approximation of U-statistics
Author(s): Masoud M. Nasari

Abstract: This paper investigates weak convergence of U-statistics via  
approximation in probability. The classical condition that the second  
moment of the kernel of the underlying U-statistic exists is relaxed  
to having 4/3 moments only (modulo a logarithmic term). Furthermore,  
the conditional expectation of the kernel is only assumed to be in the  
domain of attraction of the normal law (instead of the classical two- 
moment condition).

http://arxiv.org/abs/0901.2343


8008. An Excursion-Theoretic Approach to Stability of Discrete-Time  
Stochastic Hybrid Systems
Author(s): Debasish Chatterjee and Soumik Pal

Abstract: We address stability of a class of Markovian discrete-time  
stochastic hybrid systems. This class of systems is characterized by  
the state-space of the system being partitioned into a safe or target  
set and its exterior, and the dynamics of the system being different  
in each domain. We give conditions for $L_1$-boundedness of Lyapunov  
functions based on certain negative drift conditions outside the  
target set, together with some more minor assumptions. We then apply  
our results to a wide class of randomly switched systems (or iterated  
function systems), for which we give conditions for global asymptotic  
stability almost surely and in $L_1$. The systems need not be time- 
homogeneous, and our results apply to certain systems for which  
functional-analytic or martingale-based estimates are difficult or  
impossible to get.

http://arxiv.org/abs/0901.2269


8009. Counterexamples in the theory of fair division
Author(s): Theodore P. Hill and Kent E. Morrison

Abstract: The formal mathematical theory of fair division has a rich  
history dating back at least to Steinhaus in the 1940's. In recent  
work in this area, several general classes of errors have appeared  
along with confusion about the necessity and sufficiency of certain  
hypotheses. It is the purpose of this article to correct the  
scientific record and to point out with concrete examples some of the  
pitfalls that have led to these mistakes. These examples may serve as  
guideposts for future work.

http://arxiv.org/abs/0901.2360


8010. Pricing and trading credit default swaps in a hazard process model
Author(s): Tomasz R. Bielecki and Monique Jeanblanc and Marek Rutkowski

Abstract: In the paper we study dynamics of the arbitrage prices of  
credit default swaps within a hazard process model of credit risk. We  
derive these dynamics without postulating that the immersion property  
is satisfied between some relevant filtrations. These results are then  
applied so to study the problem of replication of general defaultable  
claims, including some basket claims, by means of dynamic trading of  
credit default swaps.

http://arxiv.org/abs/0901.2390


8011. Poisson process approximation for dependent superposition of  
point processes
Author(s): Louis H. Y. Chen and Aihua Xia

Abstract: Although the study of weak convergence of superposition of  
point processes to the Poisson process dates back to the work of  
Grigelionis in 1963, it was only recently that Schuhmacher (2005a)  
obtained error bounds for the weak convergence. Schuhmacher considered  
dependent supposition, truncated the individual point processes to  
0--1 point processes and then applied Stein's method to the latter. In  
this paper we take a different approach to the problem by using Palm  
theory and Stein's method, thereby expressing the error bounds in  
terms of the mean measures of the individual point processes, which is  
not possible by Schuhmacher's approach. We consider locally dependent  
supposition as a generalization of the locally dependent point process  
introduced in Chen and Xia (2004) and apply the main theorem to the  
superposition of thinned point processes and of renewal processes.

http://arxiv.org/abs/0901.2445


8012. Busemann functions and equilibrium measures in last passage  
percolation
Author(s): Eric Cator and Leandro P.R. Pimentel

Abstract: The interplay between two-dimensional percolation growth  
models and one-dimensional particle processes has always been a  
fruitful source of interesting mathematical phenomena. In this paper  
we develop a connection between the construction of Busemann functions  
in the Hammersley last-passage percolation model with i.i.d. random  
weights, and the existence, ergodicity and uniqueness of equilibrium  
measures for the related (multi-class) interacting particle process.  
As we shall see, in the classical Hammersley model where each point  
has weight one, this approach brings a new and rather geometrical  
solution of the longest increasing subsequence problem, as well as a  
detailed description of the scaling behavior of the Busemann function  
along different directions.

http://arxiv.org/abs/0901.2450


8013. Asymptotic optimality of maximum pressure policies in stochastic  
processing networks
Author(s): J. G. Dai and Wuqin Lin

Abstract: We consider a class of stochastic processing networks.  
Assume that the networks satisfy a complete resource pooling  
condition. We prove that each maximum pressure policy asymptotically  
minimizes the workload process in a stochastic processing network in  
heavy traffic. We also show that, under each quadratic holding cost  
structure, there is a maximum pressure policy that asymptotically  
minimizes the holding cost. A key to the optimality proofs is to prove  
a state space collapse result and a heavy traffic limit theorem for  
the network processes under a maximum pressure policy. We extend a  
framework of Bramson [Queueing Systems Theory Appl. 30 (1998) 89--148]  
and Williams [Queueing Systems Theory Appl. 30 (1998b) 5--25] from the  
multiclass queueing network setting to the stochastic processing  
network setting to prove the state space collapse result and the heavy  
traffic limit theorem. The extension can be adapted to other studies  
of stochastic processing networks.

http://arxiv.org/abs/0901.2451


8014. State-dependent Foster-Lyapunov criteria for subgeometric  
convergence of Markov chains
Author(s): Stephen B. Connor and Gersende Fort

Abstract: We consider a form of state-dependent drift condition for a  
general Markov chain, whereby the chain subsampled at some  
deterministic time satisfies a geometric Foster-Lyapunov condition. We  
present sufficient criteria for such a drift condition to exist, and  
use these to partially answer a question posed by Connor & Kendall  
(2007) concerning the existence of so-called 'tame' Markov chains.  
Furthermore, we show that our 'subsampled drift condition' implies the  
existence of finite moments for the return time to a small set.

http://arxiv.org/abs/0901.2453


8015. Central limit theorem for the solution of the Kac equation
Author(s): Ester Gabetta and Eugenio Regazzini

Abstract: We prove that the solution of the Kac analogue of  
Boltzmann's equation can be viewed as a probability distribution of a  
sum of a random number of random variables. This fact allows us to  
study convergence to equilibrium by means of a few classical  
statements pertaining to the central limit theorem. In particular, a  
new proof of the convergence to the Maxwellian distribution is  
provided, with a rate information both under the sole hypothesis that  
the initial energy is finite and under the additional condition that  
the initial distribution has finite moment of order $2+\delta$ for  
some $\delta$ in $(0,1]$. Moreover, it is proved that finiteness of  
initial energy is necessary in order that the solution of Kac's  
equation can converge weakly. While this statement may seem to be  
intuitively clear, to our knowledge there is no proof of it as yet.

http://arxiv.org/abs/0901.2464


8016. The asymptotic distribution and Berry--Esseen bound of a new  
test for independence in high dimension with an application to  
stochastic optimization
Author(s): Wei-Dong Liu and Zhengyan Lin and Qi-Man Shao

Abstract: Let $\mathbf{X}_1,...,\mathbf{X}_n$ be a random sample from  
a $p$-dimensional population distribution. Assume that $c_1n^{\alpha} 
\leq p\leq c_2n^{\alpha}$ for some positive constants $c_1,c_2$ and $ 
\alpha$. In this paper we introduce a new statistic for testing  
independence of the $p$-variates of the population and prove that the  
limiting distribution is the extreme distribution of type I with a  
rate of convergence $O((\log n)^{5/2}/\sqrt{n})$. This is much faster  
than $O(1/\log n)$, a typical convergence rate for this type of  
extreme distribution. A simulation study and application to stochastic  
optimization are discussed.

http://arxiv.org/abs/0901.2468


8017. Optimal stopping and free boundary characterizations for some  
Brownian control problems
Author(s): Amarjit Budhiraja and Kevin Ross

Abstract: A singular stochastic control problem with state constraints  
in two-dimensions is studied. We show that the value function is $C^1$  
and its directional derivatives are the value functions of certain  
optimal stopping problems. Guided by the optimal stopping problem, we  
then introduce the associated no-action region and the free boundary  
and show that, under appropriate conditions, an optimally controlled  
process is a Brownian motion in the no-action region with reflection  
at the free boundary. This proves a conjecture of Martins, Shreve and  
Soner [SIAM J. Control Optim. 34 (1996) 2133--2171] on the form of an  
optimal control for this class of singular control problems. An  
important issue in our analysis is that the running cost is Lipschitz  
but not $C^1$. This lack of smoothness is one of the key obstacles in  
establishing regularity of the free boundary and of the value  
function. We show that the free boundary is Lipschitz and that the  
value function is $C^2$ in the interior of the no-action region. We  
then use a verification argument applied to a suitable $C^2$  
approximation of the value function to establish optimality of the  
conjectured control.

http://arxiv.org/abs/0901.2474


8018. The contact process in a dynamic random environment
Author(s): Daniel Remenik

Abstract: We study a contact process running in a random environment  
in $\mathbb {Z}^d$ where sites flip, independently of each other,  
between blocking and nonblocking states, and the contact process is  
restricted to live in the space given by nonblocked sites. We give a  
partial description of the phase diagram of the process, showing in  
particular that, depending on the flip rates of the environment,  
survival of the contact process may or may not be possible for large  
values of the birth rate. We prove block conditions for the process  
that parallel the ones for the ordinary contact process and use these  
to conclude that the critical process dies out and that the complete  
convergence theorem holds in the supercritical case.

http://arxiv.org/abs/0901.2480


8019. A von Neumann theorem for uniformly distributed sequences of  
partitions
Author(s): Ingrid Carbone and Aljosa Volcic (University of Calabria -  
Italy)

Abstract: In this paper we consider permutations of sequences of  
partitions, obtaining a result which parallels von Neumann's theorem  
on permutations of dense sequences and uniformly distributed sequences  
of points.

http://arxiv.org/abs/0901.2531


8020. Fermionic construction of partition functions for two-matrix  
models and perturbative Schur function expansions
Author(s): J. Harnad and A.Yu. Orlov

Abstract: A new representation of the 2N fold integrals appearing in  
various two-matrix models that admit reductions to integrals over  
their eigenvalues is given in terms of vacuum state expectation values  
of operator products formed from two-component free fermions. This is  
used to derive the perturbation series for these integrals under  
deformations induced by exponential weight factors in the measure,  
expressed as double and quadruple Schur function expansions,  
generalizing results obtained earlier for certain two-matrix models.  
Links with the coupled two-component KP hierarchy and the two- 
component Toda lattice hierarchy are also derived.

http://arxiv.org/abs/math-ph/0512056


8021. Synchronization of dissipative dynamical systems driven by non- 
Gaussian Levy noises
Author(s): Xianming Liu and Jinqiao Duan and Jicheng Liu and Peter E.  
Kloeden

Abstract: Dynamical systems driven by Gaussian noises have been  
considered extensively in modeling, simulation and theory. However,  
complex systems in engineering and science are often subject to non- 
Gaussian fluctuations or uncertainties. A coupled dynamical system  
under non- Gaussian Levy noises is considered. After discussing  
cocycle prop- erty, stationary orbits and random attractors, a  
synchronization phe- nomenon is shown to occur, when the drift terms  
of the coupled system satisfy certain dissipativity and integrability  
conditions. The synchro- nization result implies that coupled  
dynamical systems share a dy- namical feature in some asymptotic sense.

http://arxiv.org/abs/0901.2446


8022. Exact Asymptotic for the Tail of Maximum of Smooth Random Field  
Distribution
Author(s): E. Ostrovsky

Abstract: We obtain in this paper using the saddle point method the  
expression for the exact asymptotic for the tail of maximum of smooth  
(twice continuous differentiable) random field (process) distribution.

http://arxiv.org/abs/0901.2714


8023. Averaging of Hamiltonian flows with an ergodic component
Author(s): Dmitry Dolgopyat and Leonid Koralov

Abstract: We consider a process on $\mathbb{T}^2$, which consists of  
fast motion along the stream lines of an incompressible periodic  
vector field perturbed by white noise. It gives rise to a process on  
the graph naturally associated to the structure of the stream lines of  
the unperturbed flow. It has been shown by Freidlin and Wentzell  
[Random Perturbations of Dynamical Systems, 2nd ed. Springer, New York  
(1998)] and [Mem. Amer. Math. Soc. 109 (1994)] that if the stream  
function of the flow is periodic, then the corresponding process on  
the graph weakly converges to a Markov process. We consider the  
situation where the stream function is not periodic, and the flow  
(when considered on the torus) has an ergodic component of positive  
measure. We show that if the rotation number is Diophantine, then the  
process on the graph still converges to a Markov process, which spends  
a positive proportion of time in the vertex corresponding to the  
ergodic component of the flow.

http://arxiv.org/abs/0901.2776


8024. Optimal approximation rate of certain stochastic integrals
Author(s): Heikki Sepp\"al\"a

Abstract: Given an increasing function $H:[0,1)\to [0,\infty)$ and $$  
A_n(H):=\inf_{\tau\in \mathcal{T}_n}(\sum_{i=1}^n \int_{t_{i-1}}^{t_i}  
(t_i-t)H^2(t)dt)^{{1/2}}, $$ where $\mathcal{T}_n:=\ 
{\tau=(t_i)_{i=0}^n: 0=t_0<...

http://arxiv.org/abs/0901.2777


8025. Weak solutions for forward--backward SDEs--a martingale problem  
approach
Author(s): Jin Ma and Jianfeng Zhang and Ziyu Zheng

Abstract: In this paper, we propose a new notion of Forward--Backward  
Martingale Problem (FBMP), and study its relationship with the weak  
solution to the forward--backward stochastic differential equations  
(FBSDEs). The FBMP extends the idea of the well-known (forward)  
martingale problem of Stroock and Varadhan, but it is structured  
specifically to fit the nature of an FBSDE. We first prove a general  
sufficient condition for the existence of the solution to the FBMP. In  
the Markovian case with uniformly continuous coefficients, we show  
that the weak solution to the FBSDE (or equivalently, the solution to  
the FBMP) does exist. Moreover, we prove that the uniqueness of the  
FBMP (whence the uniqueness of the weak solution) is determined by the  
uniqueness of the viscosity solution of the corresponding quasilinear  
PDE.

http://arxiv.org/abs/0901.2790


8026. Some local approximations of Dawson--Watanabe superprocesses
Author(s): Olav Kallenberg

Abstract: Let $\xi$ be a Dawson--Watanabe superprocess in $\mathbb{R}^d 
$ such that $\xi_t$ is a.s. locally finite for every $t\geq 0$. Then  
for $d\geq2$ and fixed $t>0$, the singular random measure $\xi_t$ can  
be a.s. approximated by suitably normalized restrictions of Lebesgue  
measure to the $\varepsilon$-neighborhoods of $\operatorname {supp} 
\xi_t$. When $d\geq3$, the local distributions of $\xi_t$ near a  
hitting point can be approximated in total variation by those of a  
stationary and self-similar pseudo-random measure $\tilde{\xi}$. By  
contrast, the corresponding distributions for $d=2$ are locally  
invariant. Further results include improvements of some classical  
extinction criteria and some limiting properties of hitting  
probabilities. Our main proofs are based on a detailed analysis of the  
historical structure of $\xi$.

http://arxiv.org/abs/0901.2840


8027. Trivial intersection of $\sigma$-fields and Gibbs sampling
Author(s): Patrizia Berti and Luca Pratelli and Pietro Rigo

Abstract: Let $(\Omega,\mathcal{F},P)$ be a probability space and $ 
\mathcal{N}$ the class of those $F\in\mathcal{F}$ satisfying $P(F)\in 
\{0,1\}$. For each $\mathcal{G}\subset\mathcal{F}$, define $ 
\overline{\mathcal{G}}=\sigma(\mathcal{G}\cup\mathcal {N})$. Necessary  
and sufficient conditions for $\overline{\mathcal{A}}\cap 
\overline{\mathcal{B}}=\overline {\mathcal {A}\cap\mathcal{B}}$, where  
$\mathcal{A},\mathcal{B}\subset\mathcal{F}$ are sub-$\sigma$-fields,  
are given. These conditions are then applied to the (two-component)  
Gibbs sampler. Suppose $X$ and $Y$ are the coordinate projections on $ 
(\Omega,\mathcal{F})=(\mathcal{X}\times\mathcal{Y},\mathcal {U}\otimes  
\mathcal{V})$ where $(\mathcal{X},\mathcal{U})$ and $(\mathcal{Y}, 
\mathcal{V})$ are measurable spaces. Let $(X_n,Y_n)_{n\geq0}$ be the  
Gibbs chain for $P$. Then, the SLLN holds for $(X_n,Y_n)$ if and only  
if $\overline{\sigma(X)}\cap\overline{\sigma(Y)}=\mathcal{N}$, or  
equivalently if and only if $P(X\in U)P(Y\in V)=0$ whenever $U\in 
\mathcal{U}$, $V\in\mathcal{V}$ and $P(U\times V)=P(U^c\times V^c)=0$.  
The latter condition is also equivalent to ergodicity of $(X_n,Y_n)$,  
on a certain subset $S_0\subset\Omega$, in case $\mathcal{F}= 
\mathcal{U}\otimes\mathcal{V}$ is countably generated and $P$  
absolutely continuous with respect to a product measure.

http://arxiv.org/abs/0901.2851


8028. Ornstein-Uhlenbeck Equations with time-dependent coefficients  
and Levy Noise in finite and infinite dimensions
Author(s): F. Kn\"able

Abstract: We solve a time-dependent linear SPDE with additive Levy  
noise in the mild and weak sense. Existence of a generalized invariant  
measure for the associated transition semigroup is established and the  
generator is characterized on the corresponding L^2-space. The square  
field operator is calculated, allowing to derive a Poincare and a  
Harnack inequality.

http://arxiv.org/abs/0901.2887


8029. Evolution Systems of Measures for Non-autonomous Ornstein- 
Uhlenbeck Processes with Levy noise
Author(s): Robert Wooster

Abstract: We examine the question of existence and uniqueness of  
evolution systems of measures for non-autonomous Ornstein-Uhlenbeck- 
type processes with jumps. In particular, we give examples where we  
explicitly compute the densities of such families of measures.

http://arxiv.org/abs/0901.2899


8030. Depinning of a polymer in a multi-interface medium
Author(s): Francesco Caravenna and Nicolas P\'etr\'elis

Abstract: In this paper we consider a model which describes a polymer  
chain interacting with an infinity of equi-spaced linear interfaces.  
The distance between two consecutive interfaces is denoted by T = T_N  
and is allowed to grow with the size N of the polymer. When the  
polymer receives a positive reward for touching the interfaces, its  
asymptotic behavior has been derived in a previous paper, showing that  
a transition occurs when T_N \approx log(N). In the present paper, we  
deal with the so-called depinning case, i.e., the polymer is repelled  
rather than attracted by the interfaces. Using techniques from renewal  
theory, we determine the scaling behavior of the model for large N as  
a function of T_N, showing that two transitions occur, when T_N  
\approx N^{1/3} and when T_N \approx N^{1/2} respectively.

http://arxiv.org/abs/0901.2902


8031. A martingale approach to continuous time marginal structural  
models
Author(s): Kjetil Roysland

Abstract: Marginal structural models were introduced in order to  
provide estimates of causal effects from interventions based on  
observational studies in epidemiological research. We present a  
variant of the marginal structural strategy in continuous time using  
martingale theory and marked point processes. This offers a  
mathematical interpretation of marginal structural models that has not  
been available before. Our approach starts with a characterization of  
reasonable models of randomized trials in terms of local independence.  
Such a model gives a martingale measure that is equivalent to the  
observational measure. The continuous time likelihood ratio process  
with respect to these two probability measures corresponds to the  
weights in a discrete time marginal structural model. In order to do  
inference for the new measure, we can simulate sampling using the  
observed data weighted by this likelihood ratio.

http://arxiv.org/abs/0901.2593


8032. The compositional construction of Markov processes
Author(s): L. de Francesco Albasini and N. Sabadini and R.F.C. Walters

Abstract: We describe an algebra for composing automata in which the  
actions have probabilities. We illustrate by showing how to calculate  
the probability of reaching deadlock in k steps in a model of the  
classical Dining Philosopher problem, and show, using the Perron- 
Frobenius Theorem, that this probability tends to 1 as k tends to  
infinity.

http://arxiv.org/abs/0901.2434


8033. Ham Sandwich with Mayo: A Stronger Conclusion to the Classical  
Ham Sandwich Theorem
Author(s): John H. Elton and Theodore P. Hill

Abstract: The conclusion of the classical ham sandwich theorem of  
Banach and Steinhaus may be strengthened: there always exists a common  
bisecting hyperplane that touches each of the sets, that is,  
intersects the closure of each set. Hence, if the knife is smeared  
with mayonnaise, a cut can always be made so that it will not only  
simultaneously bisect each of the ingredients, but it will also spread  
mayonnaise on each. A discrete analog of this theorem says that n  
finite nonempty sets in n-dimensional Euclidean space can always be  
simultaneously bisected by a single hyperplane that contains at least  
one point in each set. More generally, for n compactly-supported  
positive finite Borel measures in Euclidean n-space, there is always a  
hyperplane that bisects each of the measures and intersects the  
support of each measure.

http://arxiv.org/abs/0901.2589


8034. A Trotter type approach to infinite rate mutually catalytic  
branching
Author(s): Achim Klenke and Mario Oeler

Abstract: Dawson and Perkins (1998) constructed a stochastic model of  
an interacting two-type population indexed by a countable site space  
which locally undergoes a mutually catalytic branching mechanism.  
Klenke and Mytnik (2009) showed that as the branching rate approaches  
infinity the process converges to a process that is called the  
infinite rate mutually catalytic branching process. It is most  
conveniently characterised as the solution to a certain martingale  
problem. While Klenke and Mytnik used a noise equation approach in  
order to construct a solution to this martingale problem, the aim of  
this paper is to provide a Trotter type construction.

http://arxiv.org/abs/0901.2993


8035. Condenser physics applied to Markov chains - A brief  
introduction to potential theory
Author(s): A. Gaudilliere

Abstract: These notes constitute the introduction to potential theory  
I exposed at the XIIth brazilian school of probability inside  
Elisabetta Scoppola's Introduction to Metastability.

http://arxiv.org/abs/0901.3053


8036. Simulation and approximation of Levy-driven stochastic  
differential equations
Author(s): Nicolas Fournier

Abstract: We consider the problem of the simulation of Levy-driven  
stochastic differential equations. It is generally impossible to  
simulate the increments of a Levy-process. Thus in addition to an  
Euler scheme, we have to simulate approximately these increments. We  
use a method in which the large jumps are simulated exactly, while the  
small jumps are approximated by Gaussian variables. Using some recent  
results of Rio about the central limit theorem, in the spirit of the  
famous paper by Komlos-Major-Tsunady, we derive an estimate for the  
strong error of this numerical scheme. This error remains reasonnable  
when the Levy measure is very singular near 0, which is not the case  
when neglecting the small jumps. In the same spirit, we study the  
problem of the approximation of a Levy-driven S.D.E. by a Brownian  
S.D.E. when the Levy process has no large jumps.

http://arxiv.org/abs/0901.3082


8037. On the Convergence of the Ensemble Kalman Filter
Author(s): Jan Mandel and Loren Cobb and and Jonathan D. Beezley

Abstract: Convergence of the ensemble Kalman filter in the limit for  
large ensembles to the Kalman filter is proved. In each step of the  
filter, convergence of the ensemble sample covariance follows from a  
weak law of large numbers for exchangeable random variables, Slutsky's  
theorem gives weak convergence of ensemble members, and $L^p$ bounds  
on the ensemble then give $L^p$ convergence.

http://arxiv.org/abs/0901.2951


8038. A process very similar to multifractional Brownian motion
Author(s): Antoine Ayache (LPP) and Pierre R. Bertrand (INRIA Saclay -  
Ile de France)

Abstract: In Ayache and Taqqu (2005), the multifractional Brownian  
(mBm) motion is obtained by replacing the constant parameter $H$ of  
the fractional Brownian motion (fBm) by a smooth enough functional  
parameter $H(.)$ depending on the time $t$. Here, we consider the  
process $Z$ obtained by replacing in the wavelet expansion of the fBm  
the index $H$ by a function $H(.)$ depending on the dyadic point $k/2^j 
$. This process was introduced in Benassi et al (2000) to model fBm  
with piece-wise constant Hurst index and continuous paths. In this  
work, we investigate the case where the functional parameter satisfies  
an uniform H\"older condition of order $\beta>\sup_{t\in \rit} H(t)$  
and ones shows that, in this case, the process $Z$ is very similar to  
the mBm in the following senses: i) the difference between $Z$ and a  
mBm satisfies an uniform H\"older condition of order $d>\sup_{t\in \R}  
H(t)$; ii) as a by product, one deduces that at each point $t\in \R$  
the pointwise H\"older exponent of $Z$ is $H(t)$ and that $Z$ is  
tangent to a fBm with Hurst parameter $H(t)$.

http://arxiv.org/abs/0901.2808


8039. Max-plus Stochastic Control and Risk-sensitivity
Author(s): Wendell H. Fleming and Hidehiro Kaise and Shuenn-Jyi Sheu

Abstract: In the Maslov idempotent probability calculus, expectations  
of random variables are defined so as to be linear with respect to max- 
plus addition and scalar multiplication. This paper considers control  
problems in which the objective is to minimize the max-plus  
expectation of some max-plus additive running cost. Such problems  
arise naturally as limits of some types of risk sensitive stochastic  
control problems. The value function is a viscosity solution to a  
quasivariational inequality (QVI) of dynamic programming. Equivalence  
of this QVI to a nonlinear parabolic PDE with discontinuous  
Hamiltonian is used to prove a comparison theorem for viscosity sub-  
and super-solutions. An example from math finance is given, and an  
application in nonlinear H-infinity control is sketched.

http://arxiv.org/abs/0901.3007


8040. Factorization of Joint Probability Mass Functions into Parity  
Check Interactions
Author(s): M. F. Bayramoglu and A. \"Ozg\"ur Y{\i}lmaz

Abstract: We show that any joint probability mass function (PMF) can  
be expressed as a product of parity check factors and factors of  
degree one, if the alphabet size is appropriate for defining a parity  
check equation. In other words, marginalization or maximization of a  
joint PMF is equivalent to a decoding task as long as a finite field  
can be constructed over the alphabet of the PMF. In factor graph  
terminology this claim means that a factor graph representing such a  
joint PMF always has an equivalent Tanner graph. We provide a  
systematic method based on the Hilbert space of PMFs and orthogonal  
projections for obtaining this factorization.

http://arxiv.org/abs/0901.3056


8041. Moderate deviations in random graphs and Bernoulli random matrices
Author(s): Hanna D\"oring and Peter Eichelsbacher

Abstract: We prove a moderate deviation principle for subgraph count  
statistics of Erdos-Renyi random graphs. This is equivalent in showing  
a moderate deviation principle for the trace of a power of a Bernoulli  
random matrix. It is done via an estimation of the log-Laplace  
transform and the Gaertner-Ellis theorem. We obtain upper bounds on  
the upper tail probabilities of the number of occurrences of small  
subgraphs. The method of proof is used to show supplemental moderate  
deviation principles for a class of symmetric statistics, including  
non-degenerate U-statistics with independent or Markovian entries.

http://arxiv.org/abs/0901.3246


8042. From the long jump random walk to the fractional Laplacian
Author(s): Enrico Valdinoci

Abstract: This note illustrates how a simple random walk with possibly  
long jumps is related to fractional powers of the Laplace operator.  
The exposition is elementary and self-contained.

http://arxiv.org/abs/0901.3261


8043. Limit theorems for random spatial drainage networks
Author(s): Mathew D. Penrose and Andrew R. Wade

Abstract: Suppose that under the action of gravity, liquid drains  
through the unit $d$-cube via a minimal-length network of channels  
constrained to pass through random sites and to flow with nonnegative  
component in one of the canonical orthogonal basis directions of $\R^d 
$, $d \geq 2$. The resulting network is a version of the so-called  
minimal directed spanning tree. We give laws of large numbers and  
convergence in distribution results on the large-sample asymptotic  
behaviour of the total power-weighted edge-length of the network on  
uniform random points in $(0,1)^d$. The distributional results exhibit  
a weight-dependent phase transition between Gaussian and boundary- 
effect-derived distributions. These boundary contributions are  
characterized in terms of limits of the so-called on-line nearest- 
neighbour graph, a natural model of spatial network evolution, for  
which we also present some new results. Also, we give a convergence in  
distribution result for the length of the longest edge in the drainage  
network; when $d=2$, the limit is expressed in terms of Dickman-type  
variables.

http://arxiv.org/abs/0901.3297


8044. The algebraic difference of two random Cantor sets: the Larsson  
family
Author(s): F.Michel Dekking and Karoly Simon and and Balazs Szekely

Abstract: In this paper we consider a family of random Cantor sets on  
the line and consider the question whether the condition that the sum  
of the Hausdorff dimensions is larger than one implies the existence  
of interior points in the difference set of two independent copies. We  
give a new and complete proof that this is the case for the random  
Cantor sets introduced by Per Larsson.

http://arxiv.org/abs/0901.3304


8045. A Stochastic Approach for Parameterizing Unresolved Scales in a  
System with Memory
Author(s): Aijun Du and Jinqiao Duan

Abstract: Complex systems display variability over a broad range of  
spatial and temporal scales. Some scales are unresolved due to  
computational limitations. The impact of these unresolved scales on  
the resolved scales needs to be parameterized or taken into account.  
One stochastic parameterization scheme is devised to take the effects  
of unresolved scales into account, in the context of solving a  
nonlinear partial differential equation with memory (a time-integral  
term), via large eddy simulations. The obtained large eddy simulation  
model is a stochastic partial differential equation. Numerical  
experiments are performed to compare the solutions of the original  
system and of the stochastic large eddy simulation model.

http://arxiv.org/abs/0901.3312


8046. The mean width of circumscribed random polytopes
Author(s): K\'aroly J. B\"or\"oczky and Rolf Schneider

Abstract: For a given convex body K in $R^d$, a random polytope  
$K^{(n)}$ is defined (essentially) as the intersection of $n$  
independent closed halfspaces containing $K$ and having an isotropic  
and (in a specified sense) uniform distribution. We prove upper and  
lower bounds, of optimal orders, for the difference of the mean widths  
of $K^{(n)}$ and K, as n tends to infinity. For a simplicial polytope  
P, a precise asymptotic formula for the difference of the mean widths  
of $P^{(n)}$ and P is obtained.

http://arxiv.org/abs/0901.3343


8047. The Asymptotic Shape Theorem for Generalized First Passage  
Percolation
Author(s): Michael Bj\"orklund

Abstract: We generalize the asymptotic shape theorem in first passage  
percolation on $\Z^d$ to cover the case of general semimetrics. We  
prove a structure theorem for equivariant semimetrics on topological  
groups and an extended version of the maximal inequality for $\Z^d$- 
cocycles by D. Boivin and Y. Derriennic in the vector-valued case.  
This inequality will imply a very general form of Kingman's  
subadditive ergodic theorem. For certain classes of generalized first  
passage percolation we prove further structure theorems and provide  
rates of convergence in the asymptotic shape theorem. We also  
establish a general form of the multiplicative ergodic theorem by A.  
Karlsson and F. Ledrappier for cocycles with values in separable  
Banach spaces with the Radon-Nikod\'ym property.

http://arxiv.org/abs/0901.3449


8048. Excursions of the integral of the Brownian motion
Author(s): Emmanuel Jacob (PMA)

Abstract: The integrated Brownian motion is sometimes known as the  
Langevin process. Lachal studied several excursion laws induced by the  
latter. Here we follow a different point of view developed by Pitman  
for general stationary processes. We first construct a stationary  
Langevin process and then determine explicitly its stationary  
excursion measure. This is then used to provide new descriptions of It 
\^o's excursion measure of the Langevin process reflected at a  
completely inelastic boundary, which has been introduced recently by  
Bertoin.

http://arxiv.org/abs/0901.3464


8049. Expansion of the propagation of chaos for Bird and Nanbu systems
Author(s): Sylvain Rubenthaler (JAD)

Abstract: The Bird and Nanbu systems are particle systems used to  
approximate the solution of Boltzmann mollified equation. In  
particular, they have the propagation of chaos property. Following  
[GM94], we use coupling techniques and resultson branching processes  
to write an expansion of the error in the propagation of chaos in  
terms of the number of particles, for slightly more general systems  
than the ones cited above. As explained in [DMPR] and [DMPR09], this  
result will lead to the proof of the convergence of U-statistics for  
these systems.

http://arxiv.org/abs/0901.3476


8050. Normal approximation for isolated balls in an urn allocation model
Author(s): Mathew D. Penrose

Abstract: Consider throwing $n$ balls at random into $m$ urns, each  
ball landing in urn $i$ with probability $p_i$. Let $S$ be the  
resulting number of singletons, i.e., urns containing just one ball.  
We give an error bound for the Kolmogorov distance from $S$ to the  
normal, and estimates on its variance. These show that if $n$, $m$ and  
$(p_i, 1 \leq i \leq m)$ vary in such a way that $\sup_i p_i =  
O(n^{-1})$, then $S$ satisfies a CLT if and only if $n^2 \sum_i p_i^2$  
tends to infinity, and demonstrate an optimal rate of convergence in  
the CLT in this case. In the uniform case $(p_i \equiv m^{-1}) with $m 
$ and $n$ growing proportionately, we provide bounds with better  
asymptotic constants. The proof of the error bounds are based on  
Stein's method via size-biased couplings.

http://arxiv.org/abs/0901.3493


8051. Zonal polynomials and hypergeometric functions of quaternion  
matrix argument
Author(s): Fei Li and Yifeng Xue

Abstract: We define zonal polynomials of quaternion matrix argument  
and deduce some important formulae of zonal polynomials and  
hypergeometric functions of quaternion matrix argument. As an  
application, we give the distributions of the largest and smallest  
eigenvalues of a quaternion central Wishart matrix $W\sim\mathbb{Q}W(n, 
\Sigma)$, respectively.

http://arxiv.org/abs/0901.3379


8052. The mean width of random polytopes circumscribed around a convex  
body
Author(s): K\'aroly J. B\"or\"oczky and Ferenc Fodor and Daniel Hug

Abstract: Let K be a d-dimensional convex body, and let $K^{(n)}$ be  
the intersection of n halfspaces containing $K$ whose bounding  
hyperplanes are independent and identically distributed. Under  
suitable distributional assumptions, we prove an asymptotic formula  
for the expectation of the difference of the mean widths of $K^{(n)}$  
and K, and another asymptotic formula for the expectation of the  
number of facets of $K^{(n)}$. These results are achieved by  
establishing an asymptotic result on weighted volume approximation of  
$K$ and by "dualizing" it using polarity.

http://arxiv.org/abs/0901.3419


8053. Generalized Whittle-Mat$\acute{\text{E}}$rn random field as a  
model of correlated fluctuations
Author(s): S.C. Lim and L.P. Teo

Abstract: This paper considers a generalization of Gaussian random  
field with covariance function of Whittle-Mat$\acute{\text{e}}$rn  
family. Such a random field can be obtained as the solution to the  
fractional stochastic differential equation with two fractional  
orders. Asymptotic properties of the covariance functions belonging to  
this generalized Whittle-Mat$\acute{\text{e}}$rn family are studied,  
which are used to deduce the sample path properties of the random  
field. The Whittle-Mat$\acute{\text{e}}$rn field has been widely used  
in modeling geostatistical data such as sea beam data, wind speed,  
field temperature and soil data. In this article we show that  
generalized Whittle-Mat$\acute{\text{e}}$rn field provides a more  
flexible model for wind speed data.

http://arxiv.org/abs/0901.3581


8054. Logconcave Random Graphs
Author(s): Alan Frieze and Santosh Vempala and Juan Vera

Abstract: We propose the following model of a random graph on n  
vertices. Let F be a distribution in R_+^{n(n-1)/2} with a coordinate  
for every pair i$ with 1 \le i,j \le n. Then G_{F,p} is the  
distribution on graphs with n vertices obtained by picking a random  
point X from F and defining a graph on n vertices whose edges are  
pairs ij for which X_{ij} \le p. The standard Erd\H{o}s-R\'{e}nyi  
model is the special case when F is uniform on the 0-1 unit cube. We  
examine basic properties such as the connectivity threshold for quite  
general distributions. We also consider cases where the X_{ij} are the  
edge weights in some random instance of a combinatorial optimization  
problem. By choosing suitable distributions, we can capture random  
graphs with interesting properties such as triangle-free random graphs  
and weighted random graphs with bounded total weight.

http://arxiv.org/abs/0901.3697


8055. On a random number of disorders
Author(s): Krzysztof Szajowski

Abstract: We register a random sequence constructed based on Markov  
processes by switching between them. At two random moments $\theta_1$,  
$\theta_2$, where $0\leq \theta_1 \leq \theta_2$, the source of  
observations is changed. In effect the number of homogeneous segments  
is random. The transition probabilities of each process are known and  
\emph{a priori} distribution of the disorder moments is given. The  
various questions are formulated concerning the distribution changes  
in the model in the former research. The random number of  
distributional segments creates new problems in solutions of the  
problems formulated for model with deterministic number of segments.  
Two cases are presented in details. In the first one the objectives is  
to stop on between the disorder moments and in the second one our  
objective is to find the strategy which immediately detects the  
distribution changes. Both problems are reformulated to optimal  
stopping of the observed sequences. The detailed analysis of the  
problem is presented to show the form of optimal decision function.

http://arxiv.org/abs/0901.3795


8056. On the global maximum of the solution to a stochastic heat  
equation with compact-support initial data
Author(s): Mohammud Foondun and Davar Khoshnevisan

Abstract: Consider a stochastic heat equation $\partial_t u = \kappa  
\partial^2_{xx}u+\sigma(u)\dot{w}$ for a space-time white noise $ 
\dot{w}$ and a constant $\kappa>0$. Under some suitable conditions on  
the the initial function $u_0$ and $\sigma$, we show that the quantity  
\limsup_{t\to\infty}t^{-1}\ln\E(\sup_{x\in\R} |u_t(x)|^2) is bounded  
away from zero and infinity by explicit multiples of $1/\kappa$. Our  
proof works by demonstrating quantitatively that the peaks of the  
stochastic process $x\mapsto u_t(x)$ are highly concentrated for  
infinitely-many large values of $t$. In the special case of the  
parabolic Anderson model--where $\sigma(u)= \lambda u$ for some $ 
\lambda>0$--this "peaking" is a way to make precise the notion of  
physical intermittency.

http://arxiv.org/abs/0901.3814


8057. A phase diagram for a stochastic reaction diffusion system
Author(s): Carl Mueller and Roger Tribe

Abstract: In this paper a stochastic reaction diffusion system is  
considered, which models the spread of a finite population reacting  
with a non-renewable resource in the presence of individual based  
noise. A two-parameter phase diagram is established to describe the  
large time evolution, distinguishing between certain death or possible  
life of the population.

http://arxiv.org/abs/0901.3859


8058. New Classes of Infinitely Divisible Distributions Related to the  
Goldie-Steutel-Bondesson Class
Author(s): Takahiro Aoyama and Alexander Lindner and Makoto Maejima

Abstract: Recently, many classes of infinitely divisible distributions  
on R^d have been characterized in several ways. Among others, the  
first way is to use Levy measures, the second one is to use  
transformations of Levy measures, and the third one is to use mappings  
of infinitely divisible distributions defined by stochastic integrals  
with respect to Levy processes. In this paper, we are concerned with a  
class of mappings, by which we construct new classes of infinitely  
divisible distributions on R^d. Then we study a special case in R^1,  
which is the class of infinitely divisible distributions without  
Gaussian parts generated by stochastic integrals with respect to a  
fixed compound Poisson processes on R^1. This is closely related to  
the Goldie-Steutel-Bondesson class.

http://arxiv.org/abs/0901.3874


8059. Affine Diffusion Processes: Theory and Applications
Author(s): Damir Filipovic and Eberhard Mayerhofer

Abstract: We revisit affine diffusion processes on general and on the  
canonical state space in particular. A detailed study of theoretic and  
applied aspects of this class of Markov processes is given. In  
particular, we derive admissibility conditions and provide a full  
proof of existence and uniqueness through stochastic invariance of the  
canonical state space. Existence of exponential moments and the full  
range of validity of the affine transform formula are established.  
This is applied to the pricing of bond and stock options, which is  
illustrated for the Vasicek, Cox-Ingersoll-Ross and Heston models.

http://arxiv.org/abs/0901.4003


8060. Limiting behaviors of the Brownian motions on hyperbolic spaces
Author(s): Hiroyuki Matsumoto

Abstract: By adopting the upper half space realizations of the real,  
complex and quaternionic hyperbolic spaces and solving the  
corresponding stochastic differential equations, we can represent the  
Brownian motions on these classical families of the hyperbolic spaces  
as explicit Wiener functionals. Using the representations, we show  
that the almost sure convergence of the Brownian motions and the  
central limit theorems for the radial components as time tends to  
infinity are easily obtained. We also give a straightforward strategy  
to obtain the explicit expressions for the Poisson kernels by  
combining the representations with some results on the distributions  
of the random variables which are defined by the perpetual (infinite)  
integrals of the usual geometric Brownian motions with negative drifts.

http://arxiv.org/abs/0901.4028


8061. Growth Rates and Explosions in Sandpiles
Author(s): Anne Fey and Lionel Levine and Yuval Peres

Abstract: We study the abelian sandpile growth model, where n  
particles are added at the origin on a stable background configuration  
in Z^d. Any site with at least 2d particles then topples by sending  
one particle to each neighbor. We find that with constant background  
height h <= 2d-2, the diameter of the set of sites that topple has  
order n^{1/d}. This was previously known only for h

http://arxiv.org/abs/0901.3805


8062. Generalized kinetic Maxwell type models of granular gases
Author(s): A.V. Bobylev and C. Cercignani and I.M. Gamba

Abstract: We consider generalizations of kinetic granular gas models  
given by Boltzmann equations of Maxwell type. These type of models for  
non-linear elastic or inelastic interactions, have many applications  
in physics, dynamics of granular gases, economy, etc. We present the  
problem and develop its form in the space of characteristic functions,  
i.e. Fourier transforms of probability measures, from a very general  
point of view, including those with arbitrary polynomial non- 
linearities and in any dimension space. We find a whole class of  
generalized Maxwell models that satisfy properties that characterize  
the existence and asymptotic of dynamically scaled or self-similar  
solutions, often referred as {\em homogeneous cooling states}. Of  
particular interest is a concept interpreted as an operator  
generalization of usual Lipschitz conditions which allows to describe  
the behavior of solutions to the corresponding initial value problem.  
In particular, we present, in the most general case, existence of self  
similar solutions and study, in the sense of probability measures, the  
convergence of dynamically scaled solutions associated with the Cauchy  
problem to those self-similar solutions, as time goes to infinity. In  
addition we show that the properties of these self-similar solutions  
lead to non classical equilibrium stable states exhibiting power  
tails. These results apply to different specific problems related to  
the Boltzmann equation (with elastic and inelastic interactions) and  
show that all physically relevant properties of solutions follow  
directly from the general theory developed in this presentation.

http://arxiv.org/abs/0901.3864


8063. Choice-memory tradeoff in allocations
Author(s): Noga Alon and Ori Gurel-Gurevich and Eyal Lubetzky

Abstract: In the classical balls-and-bins paradigm, where n balls are  
placed independently and uniformly in n bins, typically the number of  
bins with at least two balls in them has order n and the maximum  
number of balls in a bin has order (log n)/(log log n). It is well  
known that when each round offers k independent uniform options for  
bins, it is possible to typically achieve a constant maximal load if  
and only if k is at least of order (log n). Moreover, it is possible  
whp to avoid any collisions between (n/2) balls if (k> log_2 n). In  
this work, we extend this into the setting where only m bits of memory  
are available. We establish a tradeoff between the number of choices k  
and the memory m, dictated by the quantity km/n. Roughly put, we show  
that for (k m) larger than n, one can achieve a constant maximal load,  
while for (k m) smaller than n no substantial improvement can be  
gained over the case k=1 (i.e., a random allocation). For any (k =  
\Omega(log n)) and (m = \Omega(log^2 n)), one can achieve a constant  
load whp if (k m = \Omega(n)), yet the load is unbounded if (k m  
=o(n)). Similarly, if (k m > C n) then (n/2) balls can be allocated  
without any collisions whp, whereas for (k m < \epsilon n) there are  
typically order n collisions. Furthermore, we show that the load is  
whp at least log(n/m)/[log k + log log(n/m)]. In particular, for  
k=polylog(n), if m = n^{1-\delta} the optimal maximal load is of order  
(log n)/(log log n) (the same as in the case k=1), while m=2n suffices  
to ensure a constant load. Finally, we analyze non-adaptive allocation  
algorithms and give tight upper and lower bounds for their performance.

http://arxiv.org/abs/0901.4056


8064. Heat kernel estimates and Harnack inequalities for some  
Dirichlet forms with non-local part
Author(s): Mohammud Foondun

Abstract: We consider the Dirichlet form given by  
\sE(f,f)&=&{1/2}\int_{\bR^d}\sum_{i,j=1}^d a_{ij}(x)\frac{\partial  
f(x)}{\partial x_i} \frac{\partial f(x)}{\partial x_j} dx &+& 
\int_{\bR^d\times \bR^d} (f(y)-f(x))^2J(x,y)dxdy. Under the assumption  
that the $\{a_{ij}\}$ are symmetric and uniformly elliptic and with  
suitable conditions on $J$, the nonlocal part, we obtain upper and  
lower bounds on the heat kernel of the Dirichlet form. We also prove a  
Harnack inequality and a regularity theorem for functions that are  
harmonic with respect to $\sE$.

http://arxiv.org/abs/0901.4127


8065. On the Limiting Shape of Random Young Tableaux Associated to  
Inhomogeneous Words
Author(s): Christian Houdr\'e and Hua Xu

Abstract: The limiting shape of the random Young tableaux associated  
to the inhomogeneous word problem is identified as a multidimensional  
Brownian functional. This functional is thus identical in law to the  
spectrum of a certain matrix ensemble. The Poissonized word problem is  
also studied, and the asymptotic behavior of the shape analyzed.

http://arxiv.org/abs/0901.4138


8066. Mixing time of critical Ising model on trees is polynomial in  
the height
Author(s): Jian Ding and Eyal Lubetzky and Yuval Peres

Abstract: In the heat-bath Glauber dynamics for the Ising model on the  
lattice, physicists believe that the spectral gap of the continuous- 
time chain exhibits the following behavior. For some critical inverse- 
temperature $\beta_c$, the inverse-gap is bounded for $\beta < \beta_c 
$, polynomial in the surface area for $\beta = \beta_c$ and  
exponential in it for $\beta > \beta_c$. This has been proved for $ 
\Z^2$ except at criticality. So far, the only underlying geometry  
where the critical behavior has been confirmed is the complete graph.  
Recently, the dynamics for the Ising model on a regular tree, also  
known as the Bethe lattice, has been intensively studied. The facts  
that the inverse-gap is bounded for $\beta < \beta_c$ and exponential  
for $\beta > \beta_c$ were established, where $\beta_c$ is the  
critical spin-glass parameter, and the tree-height $h$ plays the role  
of the surface area. In this work, we complete the picture for the  
inverse-gap of the Ising model on the $b$-ary tree, by showing that it  
is indeed polynomial in $h$ at criticality. The degree of our  
polynomial bound does not depend on $b$, and furthermore, this result  
holds under any boundary condition. We also obtain analogous bounds  
for the mixing-time of the chain. In addition, we study the near  
critical behavior, and show that for $\beta > \beta_c$, the inverse- 
gap and mixing-time are both $\exp[\Theta((\beta-\beta_c) h)]$.

http://arxiv.org/abs/0901.4152


8067. Discretization-invariant Bayesian inversion and Besov space priors
Author(s): Matti Lassas. Eero Saksman and Samuli Siltanen

Abstract: Bayesian solution of an inverse problem for indirect  
measurement $M = AU + {\mathcal{E}}$ is considered, where $U$ is a  
function on a domain of $R^d$. Here $A$ is a smoothing linear operator  
and $ {\mathcal{E}}$ is Gaussian white noise. The data is a  
realization $m_k$ of the random variable $M_k = P_kA U+P_k  
{\mathcal{E}}$, where $P_k$ is a linear, finite dimensional operator  
related to measurement device. To allow computerized inversion, the  
unknown is discretized as $U_n=T_nU$, where $T_n$ is a finite  
dimensional projection, leading to the computational measurement model  
$M_{kn}=P_k A U_n + P_k {\mathcal{E}}$. Bayes formula gives then the  
posterior distribution $\pi_{kn}(u_n | m_{kn})\sim\pi_n(u_n) \exp(- 
{1/2}\|m_{kn} - P_kA u_n\|_2^2)$ in $R^d$, and the mean $U^{CM}_{kn}:= 
\int u_n \pi_{kn}(u_n | m_k) du_n$ is considered as the reconstruction  
of $U$. We discuss a systematic way of choosing prior distributions $ 
\prior_n$ for all $n\geq n_0>0$ by achieving them as projections of a  
distribution in a infinite-dimensional limit case. Such choice of  
prior distributions is {\em discretization-invariant} in the sense  
that $\prior_n$ represent the same {\em a priori} information for all  
$n$ and that the mean $U^{CM}_{kn}$ converges to a limit estimate as  
$k,n\to\infty$. Gaussian smoothness priors and wavelet-based Besov  
space priors are shown to be discretization invariant. In particular,  
Bayesian inversion in dimension two with $B^1_{11}$ prior is related  
to penalizing the $\ell^1$ norm of the wavelet coefficients of $U$.

http://arxiv.org/abs/0901.4220


8068. Note: Random-to-front shuffles on trees
Author(s): Anders Bj\"orner

Abstract: A Markov chain is considered whose states are orderings of  
an underlying fixed tree and whose transitions are local "random-to- 
front" reorderings, driven by a probability distribution on subsets of  
the leaves. The eigenvalues of the transition matrix are determined  
using Brown's theory of random walk on semigroups.

http://arxiv.org/abs/0901.4278


8069. Excited against the tide: A random walk with competing drifts
Author(s): Mark Holmes

Abstract: We study a random walk that has a drift $\frac{\beta}{d}$ to  
the right when located at a previously unvisited vertex and a drift $ 
\frac{\mu}{d}$ to the left otherwise. We prove that in high  
dimensions, for every $\mu$, the drift to the right is a strictly  
increasing and continuous function of $\beta$, and that there is  
precisely one value $\beta_0(\mu,d)$ for which the resulting speed is  
zero.

http://arxiv.org/abs/0901.4393


8070. Uniform shrinking and expansion under isotropic Brownian flows
Author(s): Peter Baxendale and Georgi Dimitroff

Abstract: We study some finite time transport properties of isotropic  
Brownian flows. Under a certain nondegeneracy condition on the  
potential spectral measure, we prove that uniform shrinking or  
expansion of balls under the flow over some bounded time interval can  
happen with positive probability. We also provide a control theorem  
for isotropic Brownian flows with drift. Finally, we apply the above  
results to show that under the nondegeneracy condition the length of a  
rectifiable curve evolving in an isotropic Brownian flow with strictly  
negative top Lyapunov exponent converges to zero as $t\to \infty$ with  
positive probability.

http://arxiv.org/abs/0901.4414


8071. Regeneration in Random Combinatorial Structures
Author(s): Alexander V. Gnedin

Abstract: Theory of Kingman's partition structures has two culminating  
points: the general paintbox representation, relating finite  
partitions to hypothetical infinite populations via a natural sampling  
procedure, known as Kingman's paintbox; a central example of the  
theory - the Ewens-Pitman two-parameter family of partitions. In these  
notes we further develop the theory by passing to structures enriched  
by the order on the collection of categories; extending the class of  
tractable models by exploring the idea of regeneration; analysing  
regenerative properties of the Ewens-Pitman partitions; studying  
asymptotic features of the regenerative compositions.

http://arxiv.org/abs/0901.4444


8072. Exact confidence intervals for the Hurst parameter of a  
fractional Brownian motion
Author(s): Jean-Christophe Breton (MIA) and Ivan Nourdin (PMA) and  
Giovanni Peccati (MODAL'X)

Abstract: In this short note, we show how to use concentration  
inequalities in order to build exact confidence intervals for the  
Hurst parameter associated with a one-dimensional fractional Brownian  
motion

http://arxiv.org/abs/0901.4456


8073. Universality of the Pearcey process
Author(s): Mark Adler and Nicolas Orantin and Pierre van Moerbeke

Abstract: Consider non-intersecting Brownian motions on the line  
leaving from the origin and forced to two arbitrary points. Letting  
the number of Brownian particles tend to infinity, and upon rescaling,  
there is a point of bifurcation, where the support of the density of  
particles goes from one interval to two intervals. In this paper, we  
show that at that very point of bifurcation a cusp appears, near which  
the Brownian paths fluctuate like the Pearcey process. This is a  
universality result within this class of problems. Tracy and Widom  
obtained such a result in the symmetric case, when the two target  
points are symmetric with regard to the origin. This asymmetry enabled  
us to improve considerably a result concerning the non-linear partial  
differential equations governing the transition probabilities for the  
Pearcey process, obtained by Adler and van Moerbeke.

http://arxiv.org/abs/0901.4520


8074. Is the critical percolation probability local?
Author(s): Itai Benjamini and Asaf Nachmias and Yuval Peres

Abstract: We show that the critical probability for percolation on a d- 
regular non-amenable graph of large girth is close to the critical  
probability for percolation on an infinite d-regular tree. This is a  
special case of a conjecture due to O. Schramm on the locality of p_c.  
We also prove a finite analogue of the conjecture for expander graphs.

http://arxiv.org/abs/0901.4616


8075. Nagaev method via Keller-Liverani theorem
Author(s): Lo\"ic Herv\'e (IRMAR) and Fran\c{c}oise P\`ene (LM)

Abstract: Nagaev's method, via the perturbation operator theorem of  
Keller and Liverani, has been exploited in recent papers to establish  
local limit and Berry-Essen type theorems for unbounded functionals of  
strongly ergodic Markov chains. The main difficulty of this approach  
is to prove Taylor expansions for the dominating eigenvalue of the  
Fourier kernels. This paper outlines this method and extends it by  
proving a multi-dimensional local limit theorem, a first-order  
Edgeworth expansion, and a multi-dimensional Berry-Esseen type theorem  
in the sense of Prohorov metric. When applied to uniformly or  
geometrically ergodic chains and to iterative Lipschitz models, the  
above cited limit theorems hold under moment conditions similar, or  
close, to those of the i.i.d. case.

http://arxiv.org/abs/0901.4617


8076. A survey on dynamical percolation
Author(s): Jeffrey E. Steif

Abstract: Percolation is one of the simplest and nicest models in  
probability theory/statistical mechanics which exhibits critical  
phenomena. Dynamical percolation is a model where a simple time  
dynamics is added to the (ordinary) percolation model. This dynamical  
model exhibits very interesting behavior. Our goal in thissurvey is to  
give an overview of the work in dynamical percolation that has been  
done (and some of which is in the process of being written up).

http://arxiv.org/abs/0901.4760


8077. A stochastic calculus for multidimensional fractional Brownian  
motion with arbitrary Hurst index
Author(s): Jeremie Unterberger (IECN)

Abstract: We construct in this article an explicit rough path over a  
multi-dimensional fractional Brownian motion $B$ with arbitrary Hurst  
index $H$ (in particular, for $H<1/4$) by regularizing an associated  
random Fourier series defined in \cite{Unt08}. The regularization  
procedure is applied to 'Fourier normal ordered' iterated integrals  
obtained by permuting the order of integration so that innermost  
integrals have highest Fourier modes. The algebraic properties of this  
rough path are best understood using the Hopf algebra structure of the  
algebra of decorated rooted trees. Rough path theory gives then a  
general procedure to define a stochastic calculus and solve stochastic  
differential equations driven by this very irregular process. A  
variant of our regularization scheme is also expected to apply to  
arbitrary deterministic H\"older paths. The last section is also  
dedicated to the definition of a related two-dimensional Gaussian  
process, called {\em antisymmetric two-dimensional fractional Brownian  
motion}, with the same regularity as $B$ but with dependent  
components, to which the above construction extends naturally.

http://arxiv.org/abs/0901.4771


8078. Weak KAM methods and ergodic optimal problems for countable  
Markov shifts
Author(s): Rodrigo Bissacot and Eduardo Garibaldi

Abstract: Let $\sigma$: S -> S be the left shift acting on S, a one- 
sided Markov subshift on a countable alphabet. Our intention is to  
guarantee the existence of $\sigma$-invariant Borel probabilities that  
maximize the integral of a given locally H\"older continuous potential  
A:S -> R. Under certain conditions, we are able to show not only that  
A-maximizing probabilities do exist, but also that they are  
characterized by the fact their support lies actually in a particular  
Markov subshift on a finite alphabet. To that end, we make use of  
objects dual to maximizing measures, the so-called sub-actions  
(concept analogous to subsolutions of the Hamilton-Jacobi equation),  
and specially the calibrated sub-actions (notion similar to weak KAM  
solutions).

http://arxiv.org/abs/0901.4640


8079. Ergodicity of multiplicative statistics
Author(s): Yuri Yakubovich

Abstract: For a subfamily of multiplicative measures on integer  
partitions we give conditions for properly rescaled associated Young  
diagrams to converge in probability to a certain deterministic curve  
named the limit shape of partitions. We provide explicit formulas for  
the scaling function and the limit shape covering some known and some  
new examples.

http://arxiv.org/abs/0901.4655


8080. Scaled limit and rate of convergence for the largest eigenvalue  
from the generalized Cauchy random matrix ensemble
Author(s): Joseph Najnudel and Ashkan Nikeghbali and Felix Rubin

Abstract: In this paper, we are interested in the asymptotic  
properties for the largest eigenvalue of the Hermitian random matrix  
ensemble, called the Generalized Cauchy ensemble $GCy$, whose  
eigenvalues PDF is given by $$\textrm{const}\cdot\prod_{1\leq j-1/2$  
and where $N$ is the size of the matrix ensemble. Using results by  
Borodin and Olshanski \cite{Borodin-Olshanski}, we first prove that  
for this ensemble, the largest eigenvalue divided by $N$ converges in  
law to some probability distribution for all $s$ such that $ 
\Re(s)>-1/2$. Using results by Forrester and Witte \cite{Forrester- 
Witte2} on the distribution of the largest eigenvalue for fixed $N$,  
we also express the limiting probability distribution in terms of some  
non-linear second order differential equation. Eventually, we show  
that the convergence of the probability distribution function of the  
re-scaled largest eigenvalue to the limiting one is at least of order $ 
(1/N)$.

http://arxiv.org/abs/0901.4800


8081. Wick Calculus For Nonlinear Gaussian Functionals
Author(s): Yaozhong Hu and Jia-an Yan

Abstract: This paper surveys some results on Wick product and Wick  
renormalization. The framework is the abstract Wiener space. Some  
known results on Wick product and Wick renormalization in the white  
noise analysis framework are presented for classical random variables.  
Some conditions are described for random variables whose Wick product  
or whose renormalization are integrable random variables. Relevant  
results on multiple Wiener integrals, second quantization operator,  
Malliavin calculus and their relations with the Wick product and Wick  
renormalization are also briefly presented. A useful tool for Wick  
product is the $S$-transform which is also described without the  
introduction of generalized random variables.

http://arxiv.org/abs/0901.4911


8082. Parameter estimation for fractional Ornstein-Uhlenbeck processes
Author(s): Yaozhong Hu and David Nualart

Abstract: We study a least squares estimator $\hat {\theta}_T$ for the  
Ornstein-Uhlenbeck process, $dX_t=\theta X_t dt+\sigma dB^H_t$, driven  
by fractional Brownian motion $B^H$ with Hurst parameter $H\ge  
\frac12$. We prove the strong consistence of $\hat {\theta}_T$ (the  
almost surely convergence of $\hat {\theta}_T$ to the true parameter $ 
{% \theta}$). We also obtain the rate of this convergence when $1/2\le  
H<3/4$, applying a central limit theorem for multiple Wiener  
integrals. This least squares estimator can be used to study other  
more simulation friendly estimators such as the estimator $\tilde  
\theta_T$ defined by (4.1).

http://arxiv.org/abs/0901.4925


8083. A note on adiabatic theorem for Markov chains and adiabatic  
quantum computation
Author(s): Yevgeniy Kovchegov

Abstract: We derive an adiabatic theorem for Markov chains using well  
known facts about mixing and relaxation times. We discuss the results  
in the context of the recent developments in adiabatic quantum  
computation.

http://arxiv.org/abs/0901.4954


8084. On generalized Cauchy-Stieltjes transforms of some Beta  
distributions
Author(s): Nizar Demni

Abstract: We express generalized Cauchy-Stieltjes transforms of some  
particular Beta distributions (of ultraspherical type generating  
functions for orthogonal polynomials) as a powered Cauchy-Stieltjes  
transform of some measure. For suitable values of the power parameter,  
the latter measure turns out to be a probability measure and its  
density is written down using Markov transforms. The discarded values  
give a negative answer to a deformed free probability unless a  
restriction on the power parameter is made. A particular symmetric  
distribution interpolating between Wigner and arcsine distributions is  
obtained. Its moments are expressed through a terminating  
hypergeometric series interpolating between Catalan and shifed Catalan  
numbers. for small values of the power parameter, the free cumulants  
are computed. Interesting opne problems related to a deformed  
representation theory of the infinite symmetric group and to a  
deformed Bozejko's convolution are discussed.

http://arxiv.org/abs/0902.0054


8085. On Brownian motion on the plane with membranes on rays with a  
common endpoint
Author(s): Olga V. Aryasova and Andrey Yu. Pilipenko

Abstract: We consider a Brownian motion on the plane with  
semipermeable membranes on n rays that have a common endpoint in the  
origin. We obtain the necessary and sufficient conditions for the  
process to reach the origin and we show that the probability of  
hitting the origin is equal to zero or one.

http://arxiv.org/abs/0902.0067


8086. Palm pairs and the general mass-transport principle
Author(s): Daniel Gentner and G\"unter Last

Abstract: We consider a lcsc group G acting properly on a Borel space  
S and measurably on an underlying sigma-finite measure space. Our  
first main result is a transport formula connecting the Palm pairs of  
jointly stationary random measures on S. A key (and new) technical  
result is a measurable disintegration of the Haar measure on G along  
the orbits. The second main result is an intrinsic characterization of  
the Palm pairs of a G-invariant random measure. We then proceed with  
deriving a general version of the mass-transport principle for  
possibly non-transitive and non-unimodular group operations first in a  
deterministic and then in its full probabilistic form.

http://arxiv.org/abs/0902.0068


8087. Cutpoints and resistance of random walk paths
Author(s): Itai Benjamini and Ori Gurel-Gurevich and Oded Schramm

Abstract: We construct a bounded degree graph G, such that a simple  
random walk on it is transient but the random walk path (i.e., the  
subgraph of all the edges the random walk has crossed) has only  
finitely many cutpoints, almost surely. We also prove that the  
expected number of cutpoints of any transient Markov chain is  
infinite. This answers two questions of James, Lyons and Peres.  
Additionally, we consider a simple random walk on a finite connected  
graph G that starts at some fixed vertex x and is stopped when it  
first visits some other fixed vertex y. We provide a lower bound on  
the expected effective resistance between x and y in the path of the  
walk, giving a partial answer to a question raised in http://arxiv.org/abs/math/0603060

http://arxiv.org/abs/0902.0115


8088. A passage to the Poisson-Dirichlet through the Bessel square  
processes
Author(s): Soumik Pal

Abstract: This principal result in this article is that every Poisson- 
Dirichlet distribution PD(0,a) is an asymptotically invariant  
distribution for a growing collection of independent Bessel square  
processes of dimension zero divided by their total sum, under the  
condition that the sum total of their initial values grows to infinity  
in probability. Implications in several areas of Probability theory  
have been discussed, including Brownian local time, Fernholz &  
Karatzas's Volatility Stabilized Market models of Mathematical  
Finance, Watterson's Infinitely Many Neutral Alleles model in  
Statistical Genetics, branching Bessel diffusions, and the Poisson- 
Dirichlet cascades. A key step involves generalization of a polar  
decomposition result involving squared Bessel processes that was  
observed by Warren & Yor in their study of the Brownian burglar.

http://arxiv.org/abs/0902.0116


8089. Variance decay for functionals of the environment viewed by the  
particle
Author(s): Jean-Christophe Mourrat

Abstract: For the random walk among random conductances, we prove an  
algebraic decay of the variance of a large class of functionals of the  
environment viewed by the particle, our main hypothesis being that the  
conductances are bounded away from zero. The basis of our method is  
the establishment of a Nash inequality, followed either by a  
comparison with the simple random walk or by a more direct analysis  
based on a martingale decomposition. As an example of application, we  
show that under certain conditions, our results imply an estimate of  
the speed of convergence of the mean square displacement of the walk  
towards its limit.

http://arxiv.org/abs/0902.0204


8090. Critical behavior in inhomogeneous random graphs
Author(s): Remco van der Hofstad

Abstract: We study the critical behavior of inhomogeneous random  
graphs where edges are present independently but with unequal edge  
occupation probabilities. We show that the critical behavior depends  
sensitively on the properties of the asymptotic degrees. Indeed, when  
the proportion of vertices with degree at least $k$ is bounded above  
by $k^{-\tau+1}$ for some $\tau>4$, the largest critical connected  
component is of order $n^{2/3}$, where $n$ denotes the size of the  
graph, as on the Erd\H{o}s-R\'enyi random graph. The restriction $ 
\tau>4$ corresponds to finite {\it third} moment of the degrees. When,  
the proportion of vertices with degree at least $k$ is asymptotically  
equal to $ck^{-\tau+1}$ for some $\tau\in (3,4),$ the largest critical  
connected component is of order $n^{(\tau-2)/(\tau-1)},$ instead. Our  
results show that, for inhomogeneous random graphs with a power-law  
degree sequence, the critical behavior admits a transition when the  
third moment of the degrees turns from finite to infinite. Similar  
phase transitions have been shown to occur for typical distances in  
such random graphs when the variance of the degrees turns from finite  
to infinite. We present further results related to the size of the  
critical or scaling window, and state conjectures for this and related  
random graph models.

http://arxiv.org/abs/0902.0216


8091. Shelf Life of Candidates in the Generalized Secretary Problem
Author(s): Krzysztof Szajowski and Mitsushi Tamaki

Abstract: A version of the secretary problem called the duration  
problem, in which the objective is to maximize the time of possession  
of relatively best objects or the second best, is treated. It is shown  
that in this duration problem there are threshold numbers $(k_1^ 
\star,k_2^\star)$ such that the optimal strategy immediately selects a  
relatively best object if it appears after time $k_1^\star$ and a  
relatively second best object if it appears after moment $k_2^\star$.  
When number of objects tends to infinity the thresholds values are $ 
\lfloor 0.417188N\rfloor$ and $\rfloor 0.120381N\rfloor$,  
respectively. The asymptotic mean time of shelf life of the object is  
$0.403827N$.

http://arxiv.org/abs/0902.0232


8092. On Stein's method for multivariate normal approximation
Author(s): Elizabeth S. Meckes

Abstract: The purpose of this paper is to synthesize the approaches  
taken by Chatterjee-Meckes and Reinert-R\"ollin in adapting Stein's  
method of exchangeable pairs for multivariate normal approximation.  
The more general linear regression condition of Reinert-R\"ollin  
allows for wider applicability of the method, while the method of  
bounding the solution of the Stein equation due to Chatterjee-Meckes  
allows for improved convergence rates. Two abstract normal  
approximation theorems are proved, one for use when the underlying  
symmetries of the random variables are discrete, and one for use in  
contexts in which continuous symmetry groups are present. The  
application to runs on the line from Reinert-R\"ollin is reworked to  
demonstrate the improvement in convergence rates, and a new  
application to joint value distributions of eigenfunctions of the  
Laplace-Beltrami operator on a compact Riemannian manifold is presented.

http://arxiv.org/abs/0902.0333


8093. Fermionic construction of tau functions and random processes
Author(s): John Harnad and Alexander Yu. Orlov

Abstract: Tau functions expressed as fermionic expectation values are  
shown to provide a natural and straightforward description of a number  
of random processes and statistical models involving hard core  
configurations of identical particles on the integer lattice, like a  
discrete version simple exclusion processes (ASEP), nonintersecting  
random walkers, lattice Coulomb gas models and others, as well as  
providing a powerful tool for combinatorial calculations involving  
paths between pairs of partitions. We study the decay of the initial  
step function within the discrete ASEP (d-ASEP) model as an example.

http://arxiv.org/abs/0704.1157


8094. Clustering Bounds on N-Point Correlations for Unbounded Spin  
Systems
Author(s): Abdelmalek Abdesselam and Aldo Procacci and Benedetto  
Scoppola

Abstract: We prove clustering estimates for the truncated  
correlations, i.e., cumulants of an unbounded spin system on the  
lattice. We provide a unified treatment, based on cluster expansion  
techniques, of four different regimes: large mass, small interaction  
between sites, large self-interaction, as well as the more delicate  
small self-interaction or `low temperature' regime. A clustering  
estimate in the latter regime is needed for the Bosonic case of the  
recent result obtained by Lukkarinen and Spohn on the rigorous control  
on kinetic scales of quantum fluids.

http://arxiv.org/abs/0901.4756


8095. Very large graphs
Author(s): Laszlo Lovasz

Abstract: In the last decade it became apparent that a large number of  
the most interesting structures and phenomena of the world can be  
described by networks: separable elements, with connections (or  
interactions) between certain pairs of them. These huge networks pose  
exciting challenges for the mathematician. Graph Theory (the  
mathematical theory of networks) faces novel, unconventional problems:  
these very large networks (like the Internet) are never completely  
known, in most cases they are not even well defined. Data about them  
can be collected only by indirect means like random local sampling.  
Dense networks (in which a node is adjacent to a positive percent of  
others nodes) and sparse networks (in which a node has a bounded  
number of neighbors) show very different behavior. From a practical  
point of view, sparse networks are more important, but at present we  
have more complete theoretical results for dense networks. The paper  
surveys relations with probability, algebra, extrema graph theory, and  
analysis.

http://arxiv.org/abs/0902.0132


8096. Carries, shuffling, and symmetric functions
Author(s): Persi Diaconis and Jason Fulman

Abstract: The "carries" when n random numbers are added base b form a  
Markov chain with an "amazing" transition matrix determined by Holte.  
This same Markov chain occurs in following the number of descents or  
rising sequences when n cards are repeatedly riffle shuffled. We give  
generating and symmetric function proofs and determine the rate of  
convergence of this Markov chain to stationarity. Similar results are  
given for type B shuffles. We also develop connections with Gaussian  
autoregressive processes and the Veronese mapping of commutative  
algebra.

http://arxiv.org/abs/0902.0179


8097. Poset limits and exchangeable random posets
Author(s): Svante Janson

Abstract: We develop a theory of limits of finite posets in close  
analogy to the recent theory of graph limits. In particular, we study  
representations of the limits by functions of two variables on a  
probability space, and connections to exchangeable random infinite  
posets.

http://arxiv.org/abs/0902.0306


8098. Random symmetrizations of measurable sets
Author(s): Aljosa Volcic

Abstract: In this paper we prove almost sure convergence to the ball,  
in the Nikodym metric, of sequences of random Steiner symmetrizations  
of bounded Caccioppoli and bounded measurable sets, paralleling a  
result due to Mani-Levitska concerning convex bodies.

http://arxiv.org/abs/0902.0462


8099. A L\'{e}vy input model with additional state-dependent services
Author(s): Zbigniew Palmowski and Maria Vlasiou

Abstract: We consider a queuing model with the workload evolving  
between consecutive i.i.d. exponential timers $\{e_q^{(i)} 
\}_{i=1,2,...}$ according to a spectrally positive L\'{e}vy process  
$Y(t)$ which is reflected at 0. When the exponential clock $e_q^{(i)}$  
ends, the additional state-dependent service requirement modifies the  
workload so that the latter is equal to $F_i(Y(e_q^{(i)}))$ at epoch  
$e^{(1)}_q+...+e^{(i)}_q$ for some random nonnegative i.i.d.  
functionals $F_i$. In particular, we focus on the case when  
$F_i(y)=(B_i-y)^+$, where $\{B_i\}_{i=1,2,...}$ are i.i.d. nonnegative  
random variables. We analyse the steady-state workload distribution  
for this model.

http://arxiv.org/abs/0902.0485


8100. Discretizing the fractional Levy area
Author(s): Andreas Neuenkirch and Samy Tindel (IECN) and J\'er\'emie  
Unterberger (IECN)

Abstract: In this article, we give sharp bounds for the Euler- and  
trapezoidal discretization of the Levy area associated to a d- 
dimensional fractional Brownian motion. We show that there are three  
different regimes for the exact root mean-square convergence rate of  
the Euler scheme. For H<3/4 the exact convergence rate is n^{-2H+1/2},  
where n denotes the number of the discretization subintervals, while  
for H=3/4 it is n^{-1} (log(n))^{1/2} and for H>3/4 the exact rate is  
n^{-1}. Moreover, the trapezoidal scheme has exact convergence rate  
n^{-2H+1/2} for H>1/2. Finally, we also derive the asymptotic error  
distribution of the Euler scheme. For H lesser than 3/4 one obtains a  
Gaussian limit, while for H>3/4 the limit distribution is of  
Rosenblatt type.

http://arxiv.org/abs/0902.0497


8101. Convergence of multi-class systems of fixed possibly infinite  
sizes
Author(s): Carl Graham (CMAP)

Abstract: Multi-class systems having possibly both finite and infinite  
classes are investigated under a natural partial exchangeability  
assumption. It is proved that the conditional law of such a system,  
given the vector of the empirical measures of its finite classes and  
directing measures of its infinite ones (given by the de Finetti  
Theorem), corresponds to sampling independently from each class,  
without replacement from the finite classes and i.i.d. from the  
directing measure for the infinite ones. The equivalence between the  
convergence of multi-exchangeable systems with fixed class sizes and  
the convergence of the corresponding vectors of measures is then  
established.

http://arxiv.org/abs/0902.0539


8102. A Bernstein type inequality and moderate deviations for weakly  
dependent sequences
Author(s): Florence Merlev\`ede (LAMA) and Magda Peligrad and Emmanuel  
Rio (LM-Versailles, INRIA Bordeaux - Sud-Ouest)

Abstract: In this paper we present a tail inequality for the maximum  
of partial sums of a weakly dependent sequence of random variables  
that are not necessarily bounded. The class considered includes  
geometrically and subgeometrically strongly mixing sequences. The  
result is then used to derive asymptotic moderate deviations results.  
Applications include classes of Markov chains, functions of linear  
processes with absolutely regular innovations and ARCH models

http://arxiv.org/abs/0902.0582


8103. Variance limite d'une marche al\'eatoire r\'eversible en milieu  
al\'eatoire sur Z (Limit of the Variance of a Reversible Random Walk  
in Random Medium on Z)
Author(s): J\'er\^ome Depauw (LMPT and FRDP) and Jean-Marc Derrien (LM- 
Brest)

Abstract: The Central Limit Theorem for the random walk on a  
stationary random network of conductances has been studied by several  
authors. In one dimension, when conductances and resistances are  
integrable, and following a method of martingale introduced by S.  
Kozlov (1985), we can prove the Quenched Central Limit Theorem. In  
that case the variance of the limit law is not null. When resistances  
are not integrable, the Annealed Central Limit Theorem with null  
variance was established by Y. Derriennic and M. Lin (personal  
communication). The quenched version of this last theorem is proved  
here, by using a very simple method. The similar problem for the  
continuous diffusion is then considered. Finally our method allows us  
to prove an inequality for the quadratic mean of a diffusion (X_t)_t  
at all time t.

http://arxiv.org/abs/0902.0584


8104. Belief propagation : an asymptotically optimal algorithm for the  
random assignment problem
Author(s): Justin Salez (INRIA Rocquencourt) and Devavrat Shah (MIT)

Abstract: The random assignment problem asks for the minimum-cost  
perfect matching in the complete $n\times n$ bipartite graph $\Knn$  
with i.i.d. edge weights, say uniform on $[0,1]$. In a remarkable work  
by Aldous (2001), the optimal cost was shown to converge to $\zeta(2)$  
as $n\to\infty$, as conjectured by M\'ezard and Parisi (1987) through  
the so-called cavity method. The latter also suggested a non-rigorous  
decentralized strategy for finding the optimum, which turned out to be  
an instance of the Belief Propagation (BP) heuristic discussed by  
Pearl (1987). In this paper we use the objective method to analyze the  
performance of BP as the size of the underlying graph becomes large.  
Specifically, we establish that the dynamic of BP on $\Knn$ converges  
in distribution as $n\to\infty$ to an appropriately defined dynamic on  
the Poisson Weighted Infinite Tree, and we then prove correlation  
decay for this limiting dynamic. As a consequence, we obtain that BP  
finds an asymptotically correct assignment in $O(n^2)$ time only. This  
contrasts with both the worst-case upper bound for convergence of BP  
derived by Bayati, Shah and Sharma (2005) and the best-known  
computational cost of $\Theta(n^3)$ achieved by Edmonds and Karp's  
algorithm (1972).

http://arxiv.org/abs/0902.0585


8105. Heat Conduction Networks: Disposition of Heat Baths and  
Invariant Measure
Author(s): Alain Camanes (LMJL)

Abstract: We consider a model of heat conduction networks consisting  
of oscillators in contact with heat baths at different temperatures.  
Our aim is to generalize the results concerning the existence and  
uniqueness of the stationnary state already obtained when the network  
is reduced to a chain of particles. Using Lasalle's principle, we  
establish a condition on the disposition of the heat baths among the  
network that ensures the uniqueness of the invariant measure. We will  
show that this condition is sharp when the oscillators are linear.  
Moreover, when the interaction between the particles is stronger than  
the pinning, we prove that this condition implies the existence of the  
invariant measure.

http://arxiv.org/abs/0902.0586


8106. On Small Perturbations of a Spin Glass System
Author(s): Louis-Pierre Arguin and Nicola Kistler

Abstract: We show through a simple example that perturbations of the  
Hamiltonian of a spin glass which cannot be detected at the level of  
the free energy can completely alter the behavior of the overlap. In  
particular, perturbations of order O(log N), with N the size of the  
system, suffice to have ultrametricity emerge in the thermodynamical  
limit.

http://arxiv.org/abs/0902.0294


8107. Some Rigorous Results on Semiflexible Polymers. I. Free and  
confined polymers
Author(s): Ostap Hryniv and Yvan Velenik

Abstract: We introduce a class of models of semiflexible polymers. The  
latter are characterized by a strong rigidity, the correlation length  
associated to the gradient-gradient correlations, called the  
persistence length, being of the same order as the polymer length. We  
determine the macroscopic scaling limit, from which we deduce bounds  
on the free energy of a polymer confined inside a narrow tube.

http://arxiv.org/abs/0902.0694


8108. A Finitization of the Bead Process
Author(s): Benjamin J. Fleming and Peter J. Forrester and Eric  
Nordenstam

Abstract: The bead process is the particle system defined on parallel  
lines, with underlying measure giving constant weight to all  
configurations in which particles on neighbouring lines interlace, and  
zero weight otherwise. Motivated by the statistical mechanical model  
of the tiling of an $abc$-hexagon by three species of rhombi, a  
finitized version of the bead process is defined. The corresponding  
joint distribution can be realized as an eigenvalue probability  
density function for a sequence of random matrices. The finitized bead  
process is determinantal, and we give the correlation kernel in terms  
of Jacobi polynomials. Two scaling limits are considered: a global  
limit in which the spacing between lines goes to zero, and a certain  
bulk scaling limit. In the global limit the shape of the support of  
the particles is determined, while in the bulk scaling limit the bead  
process kernel of Boutillier is reclaimed, after approriate  
identification of the anisotropy parameter therein.

http://arxiv.org/abs/0902.0709


8109. A few ideas about quantitative convergence of collison models to  
the mean field limit
Author(s): Remi Peyre

Abstract: We consider a stochastic N-particle model for the spatially  
homogeneous Boltzmann evolution and show its convergence to the  
associated Boltzmann equation when N tends to infinity. More  
precisely, for any time T>0 we bound over the distance between the  
empirical measure of the particle system and the measure given by  
Boltzmann evolution. That distance is computed in some homogeneous  
Sobolev space. The control we get is Gaussian, i.e. we prove that the  
distance is bigger than $x N^{-1/2}$ with a probability of type $e^{- 
x^2}$ at most. The two ingredients needed are first a control of  
fluctuations due to the discrete nature of collisions, secondly a kind  
of Lipschitz continuity for the Boltzmann collision kernel. The latter  
condition, in our present setting, is only satisfied for Maxwellian  
models. We also have to control the initial situation of the particle  
evolution, which we do by a kind of Chernoff inequality for the i.i.d.  
case. Numerical applications tend to show that our results are useful  
in practice.

http://arxiv.org/abs/0902.0721


8110. Isoperimetry for spherically symmetric log-concave probability  
measures
Author(s): Nolwen Huet (IMT)

Abstract: We prove an isoperimetric inequality for probability  
measures $\mu$ on $\mathbb{R}^n$ with density proportional to $\exp(- 
\phi(\lambda | x|))$, where $|x|$ is the euclidean norm on $ 
\mathbb{R}^n$ and $\phi$ is a non-decreasing convex function. It  
applies in particular when $\phi(x)=x^\alpha$ with $\alpha\ge1$. Under  
mild assumptions on $\phi$, the inequality is dimension-free if $ 
\lambda$ is chosen such that the covariance of $\mu$ is the identity.

http://arxiv.org/abs/0902.0743


8111. Correlated Drainage Model
Author(s): Siva Athreya and Sreekar Vadlamani

Abstract: In this article we present an example of a random oriented  
tree model on d-dimensional lattice, that is a forest in d=3 with  
positive probability. This is in contrast with the other random tree  
models in the literature which are a forest only when d strictly  
greater than 3.

http://arxiv.org/abs/0902.0762


8112. Total Current Fluctuations in ASEP
Author(s): Craig A. Tracy and Harold Widom

Abstract: A limit theorem for the total current in the asymmetric  
simple exclusion process (ASEP) with step initial condition is proved.  
This extends the result of Johansson on TASEP to ASEP.

http://arxiv.org/abs/0902.0821


8113. Univariate approximations in the infinite occupancy scheme
Author(s): A. D. Barbour

Abstract: The paper concerns the classical occupancy scheme with  
infinitely many boxes. We establish approximations to the  
distributions of the number of occupied boxes, and of the number of  
boxes containing exactly r balls, within the family of translated  
Poisson distributions. These are shown to be of ideal asymptotic  
order, with respect both to total variation distance and to the  
approximation of point probabilities. The proof is probabilistic,  
making use of a translated Poisson approximation theorem of R\"ollin  
(2005).

http://arxiv.org/abs/0902.0879


8114. Translated Poisson approximation to equilibrium distributions of  
Markov population processes
Author(s): Sanda N. Socoll and A. D. Barbour

Abstract: The paper is concerned with the equilibrium distributions of  
continuous-time density dependent Markov processes on the integers.  
These distributions are known typically to be approximately normal,  
and the approximation error, as measured in Kolmogorov distance, is of  
the smallest order that is compatible with their having integer  
support. Here, an approximation in the much stronger total variation  
norm is established, without any loss in the asymptotic order of  
accuracy; the approximating distribution is a translated Poisson  
distribution having the same variance and (almost) the same mean. Our  
arguments are based on the Stein-Chen method and Dynkin's formula.

http://arxiv.org/abs/0902.0884


8115. Local limit approximations for Markov population processes
Author(s): Sanda N. Socoll and A. D. Barbour

Abstract: The paper is concerned with the equilibrium distribution $ 
\Pi_n$ of the $n$-th element in a sequence of continuous-time density  
dependent Markov processes on the integers. Under a $(2+\a)$-th moment  
condition on the jump distributions, we establish a bound of order  
$O(n^{-(\a+1)/2}\sqrt{\log n})$ on the difference between the point  
probabilities of $\Pi_n$ and those of a translated Poisson  
distribution with the same variance. Except for the factor $\sqrt{\log  
n}$, the result is as good as could be obtained in the simpler setting  
of sums of independent integer-valued random variables. Our arguments  
are based on the Stein-Chen method and coupling.

http://arxiv.org/abs/0902.0886


8116. Random Walks on Directed Covers of Graphs
Author(s): Lorenz A. Gilch and Sebastian M\"uller

Abstract: Directed covers of finite graphs are also known as periodic  
trees or trees with finitely many cone types. We expand the existing  
theory of directed covers of finite graphs to those of infinite  
graphs. While the lower growth rate still equals the branching number,  
upper and lower growth rate do not longer coincide in general.  
Furthermore, the behaviour of random walks on directed covers of  
infinite graphs is more subtile. We provide a classification in  
recurrence and transience and point out that the critical random walk  
may be recurrent or transient. Our proof is based on the observation  
that recurrence of the random walk is equivalent to the almost sure  
extinction of an appropriate branching process. Two examples in random  
environment are provided: homesick random walk on infinite percolation  
clusters and random walk in random environment on directed covers.  
Furthermore, we calculate, under reasonable assumptions, the rate of  
escape with respect to suitable length functions and prove the  
existence of the asymptotic entropy including an explicit formula  
which is also a new result for directed covers of finite graphs. In  
particular, the asymptotic entropy of random walks on directed covers  
of finite graphs is positive if and only if the random walk is  
transient.

http://arxiv.org/abs/0902.0908


8117. About Gaussian filtering problems with general exponential  
quadratic criteria
Author(s): M.L.Keptsyna and A.Le Breton and M.Viot

Abstract: Filtering problems with general exponential quadratic  
criteria are investigated for Gauss-Markov processes. In this setting,  
the Linear Exponential Gaussian and Risk-Sensitive filtering problems  
are solved and it is shown that they may have different solutions.

http://arxiv.org/abs/0902.0940


8118. Randomized Kaczmarz solver for noisy linear systems
Author(s): Deanna Needell

Abstract: The Kaczmarz method is an iterative algorithm for solving  
systems of linear equations Ax=b. Theoretical convergence rates for  
this algorithm were largely unknown until recently when work was done  
on a randomized version of the algorithm. It was proved that for  
overdetermined systems, the randomized Kaczmarz method converges with  
expected exponential rate, independent of the number of equations in  
the system. Here we analyze the case where the system Ax=b is  
corrupted by noise, so we consider the system where Ax is  
approximately b + r where r is an arbitrary error vector. We prove  
that in this noisy version, the randomized method reaches an error  
threshold dependent on the matrix A with the same rate as in the error- 
free case. We provide examples showing our results are sharp in the  
general context.

http://arxiv.org/abs/0902.0958


8119. Transience/Recurrence and the speed of a one-dimensional random  
walk in a "have your cookie and eat it" environment
Author(s): Ross Pinsky

Abstract: Consider a simple random walk on the integers with the  
following transition mechanism. At each site $x$, the probability of  
jumping to the right is $\omega(x)\in[\frac12,1)$, until the first  
time the process jumps to the left from site $x$, from which time  
onward the probability of jumping to the right is $\frac12$. We  
investigate the transience/recurrence properties of this process in  
both deterministic and stationary, ergodic environments $\{\omega(x) 
\}_{x\in Z}$. In deterministic environments, we also study the speed  
of the process.

http://arxiv.org/abs/0902.1026


8120. Multiple orthogonal polynomial ensembles
Author(s): Arno B.J. Kuijlaars

Abstract: Multiple orthogonal polynomials are traditionally studied  
because of their connections to number theory and approximation  
theory. In recent years they were found to be connected to certain  
models in random matrix theory. In this paper we introduce the notion  
of a multiple orthogonal polynomial ensemble (MOP ensemble) and derive  
some of their basic properties. It is shown that Angelesco and  
Nikishin systems give rise to MOP ensembles and that the equilibrium  
problems that are associated with these systems have a natural  
interpretation in the context of MOP ensembles.

http://arxiv.org/abs/0902.1058


8121. Extremes of Levy processes with light tails
Author(s): Michael Braverman

Abstract: We give conditions under which the tail probability of the  
supremum over unit interval of a Levy process with light tail is  
equivalent to the tail of the value of the process at the right  
endpoint.

http://arxiv.org/abs/0902.1075


8122. Asymptotic directions in random walks in random environment  
revisited
Author(s): Alexander Drewitz and Alejandro F. Ram\'irez

Abstract: Recently Simenhaus proved that for any elliptic random walk  
in random environment, transience in the neighborhood of a given  
direction is equivalent to the a.s. existence of a deterministic  
asymptotic direction and to transience in any direction in the open  
half space defined by this asymptotic direction. Here we prove an  
improved version of this result and review some open problems.

http://arxiv.org/abs/0902.1115


8123. Probabilistic Representation of Weak Solutions of Partial  
Differential Equations with Polynomial Growth Coefficients
Author(s): Qi Zhang and Huaizhong Zhao

Abstract: In this paper we develop a new weak convergence and compact  
embedding method to study the existence and uniqueness of the  
$L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes  
L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solution of  
backward stochastic differential equations with p-growth coefficients.  
Then we establish the probabilistic representation of the weak  
solution of PDEs with p-growth coefficients via corresponding BSDEs.

http://arxiv.org/abs/0902.1148


8124. On the spread of supercritical random graphs
Author(s): Louigi Addario-Berry and Svante Janson and Colin McDiarmid

Abstract: The spread of a connected graph G was introduced by Alon  
Boppana and Spencer (1998) and measures how tightly connected the  
graph is. It is defined as the maximum over all Lipschitz functions f  
on V(G) of the variance of f(X) when X is uniformly distributed on  
$V(G)$. We investigate the spread of a variety of random graphs, in  
particular the random regular graphs G(n,d), d >= 3, and Erdos-Renyi  
random graphs G_{n,p} in the supercritical range p>1/n. We show that  
if p=c/n with c>1 fixed then with high probability the spread is  
bounded, and prove similar statements for G(n,d), d >= 3. We also  
prove lower bounds on the spread in the barely supercritical case p-1/ 
n = o(1). Finally, we show that for d large the spread of G(n,d)  
becomes arbitrarily close to that of the complete graph K_n.

http://arxiv.org/abs/0902.1156


8125. Asymptotic Expansions for the Sojourn Time Distribution in the  
$M/G/1$-PS Queue
Author(s): Qiang Zhen and Charles Knessl

Abstract: We consider the $M/G/1$ queue with a processor sharing  
server. We study the conditional sojourn time distribution,  
conditioned on the customer's service requirement, as well as the  
unconditional distribution, in various asymptotic limits. These  
include large time and/or large service request, and heavy traffic,  
where the arrival rate is only slightly less than the service rate.  
Our results demonstrate the possible tail behaviors of the  
unconditional distribution, which was previously known in the cases  
$G=M$ and $G=D$ (where it is purely exponential). We assume that the  
service density decays at least exponentially fast. We use various  
methods for the asymptotic expansion of integrals, such as the Laplace  
and saddle point methods.

http://arxiv.org/abs/0902.1199


8126. On Sojourn Times in the $M/M/1$-PS Model, Conditioned on the  
Number of Other Users
Author(s): Qiang Zhen and Charles Knessl

Abstract: We consider the $M/M/1$-PS queue with processor sharing. We  
study the conditional sojourn time distribution of an arriving  
customer, conditioned on the number of other customers present. A new  
formula is obtained for the conditional sojourn time distribution,  
using a discrete Green's function. This is shown to be equivalent to  
some classic results of Pollaczeck and Vaulot from 1946. Then various  
asymptotic limits are studied, including large time and/or large  
number of customers present, and heavy traffic, where the arrival rate  
is only slightly less than the service rate.

http://arxiv.org/abs/0902.1200


8127. Uniform bounds for exponential moments of maximum of Dyck paths
Author(s): O. Khorunzhiy and J.-F. Marckert

Abstract: Let D be a Dyck path chosen uniformly from the set of Dyck  
paths with 2n steps. We prove that the sequence of the exponential  
moments of the maximum of D normalized by the square root of n  
converges in the limit of infinite n, and therefore is bounded  
uniformly in n. This result justifies corresponding assumption used to  
prove certain estimates of high moments of large random matrices.

http://arxiv.org/abs/0902.1229


8128. On Azema-Yor processes, their optimal properties and the  
Bachelier-Drawdown equation
Author(s): Laurent Carraro and Nicole El Karoui and Jan Obloj

Abstract: We study the class of Azema-Yor (AY) processes defined from  
a general semimartingale with a continuous running supremum process.  
We show that they arise as unique strong solutions of the Bachelier  
stochastic differential equation which we prove is equivalent to the  
Drawdown equation. Solutions of the latter have the drawdown property:  
they always stay above a given function of their past supremum. We  
then show that any process which satisfies the drawdown property is in  
fact an AY process. The proofs exploit group structure of the set of  
AY processes, indexed by functions, which we introduce. Further, we  
study in detail AY martingales defined from a non-negative local  
martingale converging to zero at infinity. In particular, we construct  
AY martingales with a given terminal law and this allows us to  
rediscover the AY solution to the Skorokhod embedding problem.  
Finally, we prove new optimal properties of AY martingales relative to  
concave ordering of terminal laws of martingales.

http://arxiv.org/abs/0902.1328


8129. The martingale problem for Markov solutions to the Navier-Stokes  
equations
Author(s): Marco Romito

Abstract: Under suitable assumptions of regularity and non-degeneracy  
on the covariance of the driving additive noise, any Markov solution  
to the stochastic Navier-Stokes equations has an associated generator  
of the diffusion and is the unique solution to the corresponding  
martingale problem. Some elementary examples are discussed to  
interpret these results.

http://arxiv.org/abs/0902.1402


8130. An almost sure energy inequality for Markov solutions to the 3D  
Navier-Stokes equations
Author(s): Marco Romito

Abstract: We prove existence of weak martingale solutions satisfying  
an almost sure version of the energy inequality and which constitute a  
(almost sure) Markov process.

http://arxiv.org/abs/0902.1407


8131. Random Graphons and a Weak Positivstellensatz for Graphs
Author(s): L\'aszl\'o Lov\'asz and Bal\'azs Szegedy

Abstract: In an earlier paper the authors proved that limits of  
convergent graph sequences can be described by various structures,  
including certain 2-variable real functions called graphons, random  
graph models satisfying certain consistency conditions, and  
normalized, multiplicative and reflection positive graph parameters.  
In this paper we show that each of these structures has a related,  
relaxed version, which are also equivalent. Using this, we describe a  
further structure equivalent to graph limits, namely probability  
measures on countable graphs that are ergodic with respect to the  
group of permutations of the nodes. As an application, we prove an  
analogue of the Positivstellensatz for graphs: We show that every  
linear inequality between subgraph densities that holds asymptotically  
for all graphs has a formal proof in the following sense: it can be  
approximated arbitrarily well by another valid inequality that is a  
"sum of squares" in the algebra of partially labeled graphs.

http://arxiv.org/abs/0902.1327


8132. Bilinear and Quadratic Variants on the Littlewood-Offord Problem
Author(s): Kevin P. Costello

Abstract: If f(x_1, x_2, ..., x_n) is a polynomial dependent on a  
large number of independent Bernoulli random variables, what can be  
said about the maximum concentration of f on any single value? For  
linear polynomials, this reduces to one version of the classical  
Littlewood-Offord problem: Given nonzero constants a_1 through a_n,  
what is the maximum number of sums of the form +/- a_1 +/- a_2 +/-...  
+/- a_n which take on any single value? Here we consider the case  
where f is either a bilinear form or a quadratic form. For the  
bilinear case, we show that the only forms having concentration  
significantly larger than n^{-1} are those which are in a certain  
sense very close to being degenerate. For the quadratic case, we show  
that no form having many nonzero coefficients has concentration  
significantly larger than n^{-1/2}. In both cases the results are  
nearly tight.

http://arxiv.org/abs/0902.1538


8133. Homogenization of locally stationary diffusions with possibly  
degenerate diffusion matrix
Author(s): R\'emi Rhodes (CEREMADE)

Abstract: This paper deals with homogenization of second order  
divergence form parabolic operators with locally stationary  
coefficients. Roughly speaking, locally stationary coefficients have  
two evolution scales: both an almost constant microscopic one and a  
smoothly varying macroscopic one. The homogenization procedure aims to  
give a macroscopic approximation that takes into account the  
microscopic heterogeneities. This paper follows "Diffusion in a  
locally stationary random environment" (published in Probability  
Theory and Related Fields) and improves this latter work by  
considering possibly degenerate diffusion matrices. The geometry of  
the homogenized equation shows that the particle is trapped in  
subspace of R^d.

http://arxiv.org/abs/0902.1586


8134. A simple construction of Werner measure from chordal SLE$_{8/3}$
Author(s): Robert O. Bauer

Abstract: We give a direct construction of the conformally invariant  
measure on self-avoiding loops in Riemann surfaces (Werner measure)  
from chordal $\text{SLE}_{8/3}$. We give a new proof of uniqueness of  
the measure and use Schramm's formula to construct a measure on  
boundary bubbles encircling an interior point. After establishing  
covariance properties for this bubble measure, we apply these  
properties to obtain a measure on loops by integrating measures on  
boundary bubbles. We calculate the distribution of the conformal  
radius of boundary bubbles encircling an interior point and deduce  
from it explicit upper and lower bounds for the loop measure.

http://arxiv.org/abs/0902.1626


8135. Gaussian density estimates for solutions to quasi-linear  
stochastic partial differential equations
Author(s): David Nualart and Lluis Quer-Sardanyons

Abstract: In this paper we establish lower and upper Gaussian bounds  
for the solutions to the heat and wave equations driven by an additive  
Gaussian noise, using the techniques of Malliavin calculus and recent  
density estimates obtained by Nourdin and Viens. In particular, we  
deal with the one-dimensional stochastic heat equation in $[0,1]$  
driven by the space-time white noise, and the stochastic heat and wave  
equations in $\mathbb{R}^d$ ($d\geq 1$ and $d\leq 3$, respectively)  
driven by a Gaussian noise which is white in time and has a general  
spatially homogeneous correlation.

http://arxiv.org/abs/0902.1849


8136. The critical Z-invariant Ising model via dimers: locality property
Author(s): C\'edric Boutillier and B\'eatrice de Tili\`ere

Abstract: We study a large class of critical two-dimensional Ising  
models, namely critical Z-invariant Ising models. Fisher [Fis66]  
introduced a correspondence between the Ising model and the dimer  
model on a decorated graph, thus setting dimer techniques as a  
powerful tool for understanding the Ising model. In this paper, we  
give a full description of the dimer model corresponding to the  
critical Z-invariant Ising model, consisting of explicit expressions  
which only depend on the local geometry of the underlying isoradial  
graph. Our main result is an explicit local formula for the inverse  
Kasteleyn matrix, in the spirit of [Ken02], as a contour integral of  
the discrete exponential function of [Mer01a,Ken02] multiplied by a  
local function. Using results of [BdT08] and techniques of  
[dT07b,Ken02], this yields an explicit local formula for a natural  
Gibbs measure, and a local formula for the free energy. As a  
corollary, we recover Baxter's formula for the free energy of the  
critical Z-invariant Ising model [Bax89], and thus a new proof of it.  
The latter is equal, up to a constant, to the logarithm of the  
normalized determinant of the Laplacian obtained in [Ken02].

http://arxiv.org/abs/0902.1882


8137. Randomness on Computable Probability Spaces - A Dynamical Point  
of View
Author(s): Peter Gacs and Mathieu Hoyrup (INRIA Lorraine - LORIA) and  
Cristobal Rojas (CREA)

Abstract: We extend the notion of randomness (in the version  
introduced by Schnorr) to computable Probability Spaces and compare it  
to a dynamical notion of randomness: typicality. Roughly, a point is  
typical for some dynamic, if it follows the statistical behavior of  
the system (Birkhoff's pointwise ergodic theorem). We prove that a  
point is Schnorr random if and only if it is typical for every mixing  
computable dynamics. To prove the result we develop some tools for the  
theory of computable probability spaces (for example, morphisms) that  
are expected to have other applications.

http://arxiv.org/abs/0902.1939


8138. Cover Time and Broadcast Time
Author(s): Robert Els\"asser and Thomas Sauerwald

Abstract: We introduce a new technique for bounding the cover time of  
random walks by relating it to the runtime of randomized broadcast. In  
particular, we strongly confirm for dense graphs the intuition of  
Chandra et al. \cite{CRRST97} that "the cover time of the graph is an  
appropriate metric for the performance of certain kinds of randomized  
broadcast algorithms". In more detail, our results are as follows: For  
any graph $G=(V,E)$ of size $n$ and minimum degree $\delta$, we have $ 
\mathcal{R}(G)= \Oh(\frac{|E|}{\delta} \cdot \log n)$, where $ 
\mathcal{R}(G)$ denotes the quotient of the cover time and broadcast  
time. This bound is tight for binary trees and tight up to logarithmic  
factors for many graphs including hypercubes, expanders and lollipop  
graphs. For any $\delta$-regular (or almost $\delta$-regular) graph $G 
$ it holds that $\mathcal{R}(G) = \Omega(\frac{\delta^2}{n} \cdot  
\frac{1}{\log n})$. Together with our upper bound on $\mathcal{R}(G)$,  
this lower bound strongly confirms the intuition of Chandra et al. for  
graphs with minimum degree $\Theta(n)$, since then the cover time  
equals the broadcast time multiplied by $n$ (neglecting logarithmic  
factors). Conversely, for any $\delta$ we construct almost $\delta$- 
regular graphs that satisfy $\mathcal{R}(G) = \Oh(\max \{\sqrt{n}, 
\delta \} \cdot \log^2 n)$. Since any regular expander satisfies $ 
\mathcal{R}(G) = \Theta(n)$, the strong relationship given above does  
not hold if $\delta$ is polynomially smaller than $n$. Our bounds also  
demonstrate that the relationship between cover time and broadcast  
time is much stronger than the known relationships between any of them  
and the mixing time (or the closely related spectral gap).

http://arxiv.org/abs/0902.1735


8139. Mesoscopic fluctuations of the zeta zeros
Author(s): Paul Bourgade

Abstract: We prove a multidimensional extension of Selberg's central  
limit theorem for $\log\zeta$, in which non-trivial correlations  
appear. In particular, this answers a question by Coram and Diaconis  
about the mesoscopic fluctuations of the zeros of the Riemann zeta  
function. Similar results are given in the context of random matrices  
from the unitary group. This shows the correspondence $n  
\leftrightarrow \log t$ not only between the dimension of the matrix  
and the height on the critical line, but also, in a local scale, for  
small deviations from the critical axis or the unit circle.

http://arxiv.org/abs/0902.1757


8140. On the diameter of the set of satisfying assignments in random  
satisfiable k-CNF formulas
Author(s): Uriel Feige and Abraham D. Flaxman and Dan Vilenchik

Abstract: It is known that random k-CNF formulas have a so-called  
satisfiability threshold at a density (namely, clause-variable ratio)  
of roughly 2^k\ln 2: at densities slightly below this threshold almost  
all k-CNF formulas are satisfiable whereas slightly above this  
threshold almost no k-CNF formula is satisfiable. In the current work  
we consider satisfiable random formulas, and inspect another parameter  
-- the diameter of the solution space (that is the maximal Hamming  
distance between a pair of satisfying assignments). It was previously  
shown that for all densities up to a density slightly below the  
satisfiability threshold the diameter is almost surely at least  
roughly n/2 (and n at much lower densities). At densities very much  
higher than the satisfiability threshold, the diameter is almost  
surely zero (a very dense satisfiable formula is expected to have only  
one satisfying assignment). In this paper we show that for all  
densities above a density that is slightly above the satisfiability  
threshold (more precisely at ratio (1+ \eps)2^k \ln 2, \eps=\eps(k)  
tending to 0 as k grows) the diameter is almost surely O(k2^{-k}n).  
This shows that a relatively small change in the density around the  
satisfiability threshold (a multiplicative (1 + \eps) factor), makes a  
dramatic change in the diameter. This drop in the diameter cannot be  
attributed to the fact that a larger fraction of the formulas is not  
satisfiable (and hence have diameter 0), because the non-satisfiable  
formulas are excluded from consideration by our conditioning that the  
formula is satisfiable.

http://arxiv.org/abs/0902.2012


8141. Batch queues, reversibility and first-passage percolation
Author(s): James B. Martin

Abstract: We consider a model of queues in discrete time, with batch  
services and arrivals. The case where arrival and service batches both  
have Bernoulli distributions corresponds to a discrete-time M/M/1  
queue, and the case where both have geometric distributions has also  
been previously studied. We describe a common extension to a more  
general class where the batches are the product of a Bernoulli and a  
geometric, and use reversibility arguments to prove versions of  
Burke's theorem for these models. Extensions to models with continuous  
time or continuous workload are also described. As an application, we  
show how these results can be combined with methods of Seppalainen and  
O'Connell to provide exact solutions for a new class of first-passage  
percolation problems.

http://arxiv.org/abs/0902.2026


8142. Transportation-information inequalities for Markov processes  
(II) : relations with other functional inequalities
Author(s): Arnaud Guillin and Christian Leonard (CMAP and MODAL'X) and  
Feng-Yu Wang and Liming Wu

Abstract: We continue our investigation on the transportation- 
information inequalities $W_pI$ for a symmetric markov process,  
introduced and studied in \cite{GLWY}. We prove that $W_pI$ implies  
the usual transportation inequalities $W_pH$, then the corresponding  
concentration inequalities for the invariant measure $\mu$. We give  
also a direct proof that the spectral gap in the space of Lipschitz  
functions for a diffusion process implies $W_1I$ (a result due to  
\cite{GLWY}) and a Cheeger type's isoperimetric inequality. Finally we  
exhibit relations between transportation-information inequalities and  
a family of functional inequalities (such as $\Phi$-log Sobolev or $ 
\Phi$-Sobolev).

http://arxiv.org/abs/0902.2101


8143. A Single Server Retrial Queue with Different Types of Server  
Interruptions
Author(s): Tewfik Kernane

Abstract: We consider a single server retrial queue with the server  
subject to interruptions and classical retrial policy for the access  
from the orbit to the server. We analyze the equilibrium distribution  
of the system and obtain the generating functions of the limiting  
distribution.

http://arxiv.org/abs/0902.2110


8144. Burkholder-Davis-Gundy type Inequalities of the It\^o stochastic  
integral with respect to Levy noise on Banach spaces
Author(s): Erika Hausenblas

Abstract: The aim of this note is to give some Burkholder-Davis-Gundy  
type inequalities which are valid for the Ito stochastic integral with  
respect to Banach valued Levy noise.

http://arxiv.org/abs/0902.2114


8145. Stochastic approach for the subordination in Bochner sense
Author(s): Nicolas Bouleau (CERMA)

Abstract: It is possible to construct a double indexed process with  
sample paths a surface of a family of subordinators obtained by  
subordination. We study here a branch of this subordination process.  
This opens martingale methods on symbolic calculus questions.

http://arxiv.org/abs/0902.2133


8146. A new look at the Heston characteristic function
Author(s): Sebastian del Ba\~no Rollin and Albert Ferreiro-Castilla  
and Frederic Utzet

Abstract: A new expression for the characteristic function of log-spot  
in Heston model is presented. This expression more clearly exhibits  
its properties as an analytic characteristic function and allows us to  
compute the exact domain of the moment generating function. This  
result is then applied to the volatility smile at extreme strikes and  
to the control of the moments of spot. We also give a factorization of  
the moment generating function as product of Bessel type factors, and  
an approximating sequence to the law of log-spot is deduced.

http://arxiv.org/abs/0902.2154


8147. Heavy-traffic analysis of the maximum of an asymptotically  
stable random walk
Author(s): Seva Shneer and Vitali Wachtel

Abstract: For families of random walks $\{S_k^{(a)}\}$ with $\mathbf E  
S_k^{(a)} = -ka < 0$ we consider their maxima $M^{(a)} = \sup_{k \ge  
0} S_k^{(a)}$. We investigate the asymptotic behaviour of $M^{(a)}$ as  
$a \to 0$ for asymptotically stable random walks. This problem  
appeared first in the 1960's in the analysis of a single-server queue  
when the traffic load tends to 1 and since then is referred to as the  
heavy-traffic approximation problem. Kingman and Prokhorov suggested  
two different approaches which were later followed by many authors. We  
give two elementary proofs of our main result, using each of these  
approaches. It turns out that the main technical difficulties in both  
proofs are rather similar and may be resolved via a generalisation of  
the Kolmogorov inequality to the case of an infinite variance. Such a  
generalisation is also obtained in this note.

http://arxiv.org/abs/0902.2185


8148. M/M/1 Queueing System with Non-preemptive Priority
Author(s): Zhao Guo-xi and Hu Qi-Zhou

Abstract: The performance of non-preemptive M/M/1 queueing system with  
two priority is analyzed. By using complementary variable method to  
make vector Markov process and analyzing the state-change equations of  
the queueing system, the generating function of two kinds of  
customers'length distribution are derived under non-preemptive  
priority .Through further discussion, the probability of the server  
that it is working or free and average length of two kinds of  
customers are also derived.

http://arxiv.org/abs/0902.2086


8149. Distribution-valued heavy-traffic limits for the $G/GI/\infty$  
queue
Author(s): Rishi Talreja and Josh Reed

Abstract: We study the $G/GI/\infty$ queue from two different  
perspectives in the same heavy-traffic regime. First, we represent the  
dynamics of the system using a measure-valued process that keeps track  
of the age of each customer in the system. Using the continuous- 
mapping approach together with the martingale functional central limit  
theorem, we obtain fluid and diffusion limits for this process in a  
space of distribution-valued processes. Next, we study a measure- 
valued process that keeps track of the residual service time of each  
customer in the system. In this case, using the functional central  
limit theorem and the random time change theorem together with the  
continuous-mapping approach, we again obtain fluid and diffusion  
limits in our space of distribution-valued processes. In both cases,  
we find that our diffusion limits may be characterized as distribution- 
valued Ornstein-Uhlenbeck processes. Further, these diffusion limits  
can be analyzed using standard results from the theory of Markov  
processes.

http://arxiv.org/abs/0902.2236


8150. A note on the Poisson boundary of lamplighter random walks
Author(s): Ecaterina Sava

Abstract: The main goal of this paper is to determine the Poisson  
boundary of lamplighter random walks over a general class of discrete  
groups $\Gamma$ endowed with a rich boundary. The starting point is  
the Strip Criterion of identification of the Poisson boundary for  
random walks on discrete groups due to Kaimanovich. A geometrical  
method for constructing the strip as a subset of the lamplighter group  
starting with a smaller strip in the base group $\Gamma$ is developed.  
Then, this method is applied to several classes of base groups $\Gamma 
$: groups with infinitely many ends, hyperbolic groups in the sense of  
Gromov, and Euclidean lattices. We show that under suitable hypothesis  
the Poisson boundary for a class of random walks on lamplighter groups  
is the space of infinite limit configurations.

http://arxiv.org/abs/0902.2285


8151. Limit theorems for Parrondo's paradox
Author(s): S. N. Ethier and Jiyeon Lee

Abstract: That two losing games can be combined to form a winning game  
is known as Parrondo's paradox. We establish a strong law of large  
numbers and a central limit theorem for the Parrondo player's sequence  
of profits, both in a one-parameter family of profit-dependent games  
and in a two-parameter family of history-dependent games, with the  
potentially winning game being either a random mixture or a nonrandom  
pattern of the two losing games. We derive formulas for the mean and  
variance parameters of the central limit theorem in nearly all such  
scenarios; formulas for the mean permit an analysis of when the  
Parrondo effect is present.

http://arxiv.org/abs/0902.2368


8152. The determinacy of infinite games with eventual perfect monitoring
Author(s): Eran Shmaya

Abstract: An infinite two-player zero-sum game with a Borel winning  
set, in which the opponent's actions are monitored eventually but not  
necessarily immediately after they are played, admits a value. The  
proof relies on a representation of the game as a stochastic game with  
perfect information, in which Nature operates as a delegate for the  
players and performs the randomizations for them.

http://arxiv.org/abs/0902.2254


8153. Some results on random circulant matrices
Author(s): Mark W. Meckes

Abstract: This paper considers random (non-Hermitian) circulant  
matrices, and proves several results analogous to recent theorems on  
non-Hermitian random matrices with independent entries. In particular,  
the limiting spectral distribution of a random circulant matrix is  
shown to be complex normal, and bounds are given for the probability  
that a circulant sign matrix is singular.

http://arxiv.org/abs/0902.2472


8154. Heat kernel analysis on semi-infinite Lie groups
Author(s): Tai Melcher

Abstract: This paper studies Brownian motion and heat kernel measure  
on a class of infinite dimensional Lie groups. We prove a Cameron- 
Martin type quasi-invariance theorem for the heat kernel measure and  
give estimates on the $L^p$ norms of the Radon-Nikodym derivatives. We  
also prove that a logarithmic Sobolev inequality holds in this setting.

http://arxiv.org/abs/0902.2500


8155. Expansions for Gaussian processes and Parseval frames
Author(s): Harald Luschgy and Gilles Pag\`es (PMA)

Abstract: We derive a precise link between series expansions of  
Gaussian random vectors in a Banach space and Parseval frames in their  
reproducing kernel Hilbert space. The results are applied to pathwise  
continuous Gaussian processes and a new optimal expansion for  
fractional Ornstein-Uhlenbeck processes is derived. In the end an  
extension of this result to Gaussian stationary processes with convex  
covariance function is established.

http://arxiv.org/abs/0902.2563


8156. Integral Equations and the First Passage Time of Brownian Motions
Author(s): Sebastian Jaimungal and Alex Krenin and and Angelo Valov

Abstract: The first passage time problem for Brownian motions hitting  
a barrier has been extensively studied in the literature. In  
particular, many incarnations of integral equations which link the  
density of the hitting time to the equation for the barrier itself  
have appeared. Most interestingly, Peskir(2002b) demonstrates that a  
master integral equation can be used to generate a countable number of  
new equations via differentiation or integration by parts. In this  
article, we generalize Peskir's results and provide a more powerful  
unifying framework for generating integral equations through a new  
class of martingales. We obtain a continuum of Volterra type integral  
equations of the first kind and prove uniqueness for a subclass.  
Furthermore, through the integral equations, we demonstrate how  
certain functional transforms of the boundary affect the density  
function. Finally, we demonstrate a fundamental connection between the  
Volterra integral equations and a class of Fredholm integral equations.

http://arxiv.org/abs/0902.2569


8157. The Policy Iteration Algorithm for Average Continuous Control of  
Piecewise Deterministic Markov Processes
Author(s): O.L.V. Costa and F. Dufour

Abstract: The main goal of this paper is to apply the so-called policy  
iteration algorithm (PIA) for the long run average continuous control  
problem of piecewise deterministic Markov processes (PDMP's) taking  
values in a general Borel space and with compact action space  
depending on the state variable. In order to do that we first derive  
some important properties for a pseudo-Poisson equation associated to  
the problem. In the sequence it is shown that the convergence of the  
PIA to a solution satisfying the optimality equation holds under some  
classical hypotheses and that this optimal solution yields to an  
optimal control strategy for the average control problem for the  
continuous-time PDMP in a feedback form.

http://arxiv.org/abs/0902.2673


8158. Li-Yau Type Gradient Estimates and Harnack Inequalities by  
Stochastic Analysis
Author(s): Marc Arnaudon (LMA) and Anton Thalmaier

Abstract: In this paper we use methods from Stochastic Analysis to  
establish Li-Yau type estimates for positive solutions of the heat  
equation. In particular, we want to emphasize that Stochastic Analysis  
provides natural tools to derive local estimates in the sense that the  
gradient bound at given point depends only on universal constants and  
the geometry of the Riemannian manifold locally about this point.

http://arxiv.org/abs/0902.2681


8159. Existence of an Optimal Control for Stochastic Systems with  
Nonlinear Cost Functional
Author(s): Rainer Buckdahn (LM) and Boubakeur Labed and Catherine  
Rainer (LM) and Lazhar Tamer

Abstract: We consider a stochastic control problem which is composed  
of a controlled stochastic differential equation, and whose associated  
cost functional is defined through a controlled backward stochastic  
differential equation. Under appropriate convexity assumptions on the  
coefficients of the forward and the backward equations we prove the  
existence of an optimal control on a suitable reference stochastic  
system. The proof is based on an approximation of the stochastic  
control problem by a sequence of control problems with smooth  
coefficients, admitting an optimal feedback control. The quadruplet  
formed by this optimal feedback control and the associated solution of  
the forward and the backward equations is shown to converge in law, at  
least along a subsequence. The convexity assumptions on the  
coefficients then allow to construct from this limit an admissible  
control process which, on an appropriate reference stochastic system,  
is optimal for our stochastic control problem.

http://arxiv.org/abs/0902.2693


8160. Regularity of the Optimal Stopping Problem for Levy Processes  
with Non-Degenerate Diffusions
Author(s): Erhan Bayraktar and Hao Xing

Abstract: The value function of an optimal stopping problem for a  
process with Levy jumps is known to be a generalized solution of a  
variational inequality. Assuming the diffusion component of the  
process is non-degenerate and a mild assumption on the singularity of  
the Levy measure, this paper shows that the value function is smooth  
in the continuation region for problems with either finite or infinite  
variation jumps. Moreover, the smooth-fit property is shown via the  
global regularity of the value function. This paper confirms the  
intuition that the non-degenerate diffusion component dictates the  
regularity of the value function in the optimal stopping problem for  
jump processes.

http://arxiv.org/abs/0902.2479


8161. A Simulation Approach to Optimal Stopping Under Partial  
Information
Author(s): Mike Ludkovski

Abstract: We study the numerical solution of nonlinear partially  
observed optimal stopping problems. The system state is taken to be a  
multi-dimensional diffusion and drives the drift of the observation  
process, which is another multi-dimensional diffusion with correlated  
noise. Such models where the controller is not fully aware of her  
environment are of interest in applied probability and financial  
mathematics. We propose a new approximate numerical algorithm based on  
the particle filtering and regression Monte Carlo methods. The  
algorithm maintains a continuous state-space and yields an integrated  
approach to the filtering and control sub-problems. Our approach is  
entirely simulation-based and therefore allows for a robust  
implementation with respect to model specification. We carry out the  
error analysis of our scheme and illustrate with several computational  
examples. An extension to discretely observed stochastic volatility  
models is also considered.

http://arxiv.org/abs/0902.2518


8162. A presentation of the category of stochastic matrices
Author(s): Tobias Fritz

Abstract: This note gives generators and relations for the strict  
monoidal category of probabilistic maps on finite cardinals (i.e.,  
stochastic matrices).

http://arxiv.org/abs/0902.2554


8163. Random Walks in the Quarter Plane Absorbed at the Boundary :  
Exact and Asymptotic
Author(s): Kilian Raschel

Abstract: Nearest neighbor random walks in the quarter plane that are  
absorbed when reaching the boundary are studied. The cases of positive  
and zero drift are considered. Absorption probabilities at a given  
time and at a given site are made explicit. The following asymptotics  
for these random walks starting from a given point $(n_0,m_0)$ are  
computed : that of probabilities of being absorbed at a given site $(i, 
0)$ [resp. $(0,j)$] as $i\to \infty$ [resp. $j \to \infty$], that of  
the distribution's tail of absorption time at x-axis [resp. y-axis],  
that of the Green functions at site $(i,j)$ when $i,j\to \infty$ and  
$j/i \to \tan \gamma$ for $\gamma \in [0, \pi/2]$. These results give  
the Martin boundary of the process and in particular the suitable Doob  
$h$-transform in order to condition the process never to reach the  
boundary. They also show that this $h$-transformed process is equal in  
distribution to the limit as $n\to \infty$ of the process conditioned  
by not being absorbed at time $n$. The main tool used here is complex  
analysis.

http://arxiv.org/abs/0902.2785


8164. Continuous Model for Homopolymers
Author(s): M. Cranston and L. Koralov and S. Molchanov and B. Vainberg

Abstract: We consider the model for the distribution of a long  
homopolymer in a potential field. The typical shape of the polymer  
depends on the temperature parameter. We show that at a critical value  
of the temperature the transition occurs from a globular to an  
extended phase. For various values of the temperature, including those  
at or near the critical value, we consider the limiting behavior of  
the polymer when its size tends to infinity.

http://arxiv.org/abs/0902.2830


8165. Fractional multiplicative processes
Author(s): Julien Barral and Benoit Mandelbrot

Abstract: Statistically self-similar measures on $[0,1]$ are limit of  
multiplicative cascades of random weights distributed on the $b$-adic  
subintervals of $[0,1]$. These weights are i.i.d, positive, and of  
expectation $1/b$. We extend these cascades naturally by allowing the  
random weights to take negative values. This yields martingales taking  
values in the space of continuous functions on $[0,1]$. Specifically,  
we consider for each $H\in (0,1)$ the martingale $(B_{n})_{n\geq1}$  
obtained when the weights take the values $-b^{-H}$ and $b^{-H}$, in  
order to get $B_n$ converging almost surely uniformly to a  
statistically self-similar function $B$ whose H\"{o}lder regularity  
and fractal properties are comparable with that of the fractional  
Brownian motion of exponent $H$. This indeed holds when $H\in(1/2,1)$.  
Also the construction introduces a new kind of law, one that it is  
stable under random weighted averaging and satisfies the same  
functional equation as the standard symmetric stable law of index $1/H 
$. When $H\in(0,1/2]$, to the contrary, $B_n$ diverges almost surely.  
However, a natural normalization factor $ a_n$ makes the normalized  
correlated random walk $ B_n / a_n$ converge in law, as $n$ tends to $ 
\infty$, to the restriction to $[0,1]$ of the standard Brownian  
motion. Limit theorems are also associated with the case $H>1/2$.

http://arxiv.org/abs/0902.2902


8166. Random repeated quantum interactions and random invariant states
Author(s): Ion Nechita (ICJ) and Cl\'ement Pellegrini

Abstract: We consider a generalized model of repeated quantum  
interactions, where a system $\mathcal{H}$ is interacting in a random  
way with a sequence of independent quantum systems $\mathcal{K}_n, n  
\geq 1$. Two types of randomness are studied in detail. One is  
provided by considering Haar-distributed unitaries to describe each  
interaction between $\mathcal{H}$ and $\mathcal{K}_n$. The other  
involves random quantum states describing each copy $\mathcal{K}_n$.  
In the limit of a large number of interactions, we present convergence  
results for the asymptotic state of $\mathcal{H}$. This is achieved by  
studying spectral properties of (random) quantum channels which  
guarantee the existence of unique invariant states. Finally this  
allows to introduce a new physically motivated ensemble of random  
density matrices called the \emph{asymptotic induced ensemble}.

http://arxiv.org/abs/0902.2634


8167. Bounds on the Location of the Maximum Stirling Numbers of the  
Second Kind
Author(s): Yaming Yu

Abstract: Let K_n denote the smaller mode of the nth row of Stirling  
numbers of the second kind S(n, k). Using a probablistic argument, it  
is shown that for all n>=2, [exp(w(n))]-2<=K_n<=[exp(w(n))]+1, where  
[x] denotes the integer part of x, and w(n) is Lambert's W-function.

http://arxiv.org/abs/0902.2964


8168. Irreducibility and uniqueness of stationary distribution
Author(s): Ping He and Jiangang Ying

Abstract: In this paper, we shall prove that the irreducibility in the  
sense of fine topology implies the uniqueness of invariant probability  
measures. It is also proven that this irreducibility is strictly  
weaker than the strong Feller property plus irreducibility in the  
sense of original topology, which is the usual uniqueness condition.

http://arxiv.org/abs/0902.3296


8169. Backward SDEs with superquadratic growth
Author(s): Freddy Delbaen (Department of Mathematics) and Ying Hu  
(IRMAR) and Xiaobo Bao (Department of Mathematics)

Abstract: In this paper, we discuss the solvability of backward  
stochastic differential equations (BSDEs) with superquadratic  
generators. We first prove that given a superquadratic generator,  
there exists a bounded terminal value, such that the associated BSDE  
does not admit any bounded solution. On the other hand, we prove that  
if the superquadratic BSDE admits a bounded solution, then there exist  
infinitely many bounded solutions for this BSDE. Finally, we prove the  
existence of a solution for Markovian BSDEs where the terminal value  
is a bounded continuous function of a forward stochastic differential  
equation.

http://arxiv.org/abs/0902.3316


8170. Quenched scaling limits of trap models
Author(s): M. Jara and C. Landim and A. Teixeira

Abstract: Fix a strictly positive measure $W$ on the $d$-dimensional  
torus $\bb T^d$. For an integer $N\ge 1$, denote by $W^N_x$,  
$x=(x_1, ..., x_d)$, $0\le x_i 1$, if $W$ is a finite discrete  
measure, $W=\sum_{i\ge 1} w_i \delta_{x_i}$, we prove that the random  
walk which jumps from $x/N$ uniformly to one of its neighbors at rate $ 
(W^N_x)^{-1}$ has a metastable behavior, as defined in \cite{bl1},  
described by the $K$-process introduced in \cite{fm1}.

http://arxiv.org/abs/0902.3334


8171. Anisotropic Young diagrams and infinite-dimensional diffusion  
processes with the Jack parameter
Author(s): Grigori Olshanski

Abstract: We construct a family of Markov processes with continuous  
sample trajectories on an infinite-dimensional space, the Thoma  
simplex. The family depends on three continuous parameters, one of  
which, the Jack parameter, is similar to the beta parameter in random  
matrix theory. The processes arise in a scaling limit transition from  
certain finite Markov chains, the so called up-down chains on the  
Young graph with the Jack edge multiplicities. Each of the limit  
Markov processes is ergodic and its stationary distribution is a  
symmetrizing measure. The infinitesimal generators of the processes  
are explicitly computed; viewed as selfadjoint operators in the L^2  
spaces over the symmetrizing measures, the generators have purely  
discrete spectrum which is explicitly described. For the special value  
1 of the Jack parameter, the limit Markov processes coincide with  
those of the recent work by Borodin and the author (Prob. Theory Rel.  
Fields 144 (2009), 281--318; arXiv:0810.3751). In the limit as the  
Jack parameter goes to 0, our family of processes degenerates to the  
one-parameter family of diffusions on the Kingman simplex studied long  
ago by Ethier and Kurtz in connection with some models of population  
genetics. The techniques of the paper are essentially algebraic. The  
main computations are performed in the algebra of shifted symmetric  
functions with the Jack parameter and rely on the concept of  
anisotropic Young diagrams due to Kerov.

http://arxiv.org/abs/0902.3395


8172. Effect of Noise on Front Propagation in Reaction-Diffusion  
equations of KPP type
Author(s): Carl Mueller and Leonid Mytnik and Jeremy Quastel

Abstract: We consider reaction-diffusion equations of KPP type in one  
spatial dimension, perturbed by a Fisher-Wright white noise, under the  
assumption of uniqueness in distribution. Examples include the  
randomly perturbed Fisher-KPP equations $ \partial_t u = \partial_x^2  
u + u(1-u) + \epsilon \sqrt{u(1-u)}\dot W, $ and $ \partial_t u =  
\partial_x^2 u + u(1-u) + \epsilon \sqrt{u}\dot W, $ where $\dot W=  
\dot W(t,x)$ is a space-time white noise. We prove the Brunet-Derrida  
conjecture that the speed of traveling fronts is asymptotically $ 2- 
\pi^2 |\log \epsilon^2|^{-2} $ up to a factor of order $ (\log|\log 
\epsilon|)|\log\epsilon|^{-3}$.

http://arxiv.org/abs/0902.3423


8173. Finitely-additive measures on the asymptotic foliations of a  
Markov compactum
Author(s): Alexander I. Bufetov

Abstract: An asymptotic expansion is established for time averages of  
translation flows on flat surfaces. This result, which extends earlier  
work of A.Zorich and G.Forni, yields limit theorems for translation  
flows. The argument, close in spirit to that of G.Forni, uses the  
approximation of ergodic integrals by holonomy-invariant Hoelder  
cocycles on trajectories of the flows. The space of holonomy-invariant  
Hoelder cocycles is finite-dimensional, and is given by an explicit  
construction. First, a symbolic representation for a uniquely ergodic  
translation flow is obtained following S.Ito and A.M. Vershik, and  
then, the space of cocycles is constructed using a family of finitely- 
additive complex-valued holonomy-invariant measures on the asymptotic  
foliations of a Markov compactum.

http://arxiv.org/abs/0902.3303


8174. A (rough) pathwise approach to fully non-linear stochastic  
partial differential equations
Author(s): Michael Caruana and Peter Friz and Harald Oberhauser

Abstract: In a series of papers, starting with [Fully nonlinear  
stochastic partial differential equations. C. R. Acad. Sci. Paris Ser.  
I Math. 326 (1998), no. 9] Lions and Souganidis proposed a (pathwise)  
theory for fully non-linear stochastic partial differential equations.  
We present here a (partial) extension towards certain spatial  
dependence in the noise term. The main novelty is the use of rough  
path theory in this context [Lyons, Terry J.; Differential equations  
driven by rough signals. Rev. Mat. Iberoamericana 14 (1998), no. 2,  
215-310].

http://arxiv.org/abs/0902.3352


8175. Periodic homogenization with an interface: the one-dimensional  
case
Author(s): Martin Hairer and Charles Manson

Abstract: We consider a one-dimensional diffusion process with  
coefficients that are periodic outside of a finite 'interface region'.  
The question investigated in this article is the limiting long time /  
large scale behaviour of such a process under diffusive rescaling. Our  
main result is that it converges weakly to a rescaled version of skew  
Brownian motion, with parameters that can be given explicitly in terms  
of the coefficients of the original diffusion. Our method of proof  
relies on the framework provided by Freidlin and Wentzell for  
diffusion processes on a graph in order to identify the generator of  
the limiting process. The graph in question consists of one vertex  
representing the interface region and two infinite segments  
corresponding to the regions on either side.

http://arxiv.org/abs/0902.3471


8176. Interacting Brownian motions in infinite dimensions with  
logarithmic interaction potentials
Author(s): Hirofumi Osada

Abstract: We investigate the construction of diffusions consisting of  
infinitely numerous Brownian particles moving in $ \Rd $ and  
interacting via logarithmic functions (2D Coulomb potentials). These  
potentials are really strong and long range in nature. The associated  
equilibrium states are no longer Gibbs measures. We present general  
results for the construction of such diffusions and, as applications  
thereof, construct two typical interacting Brownian motions with  
logarithmic interaction potentials, namely the Dyson model in infinite  
dimensions and Ginibre interacting Brownian motions. The former is a  
particle system in $ \R $ while the latter is in $ \R ^2 $. Both  
models are translation and rotation invariant in space, and as such,  
are prototypes of dimensions $ d = 1,2 $, respectively. The  
equilibrium states of the former diffusion model are determinantal  
random point fields with sine kernels. They appear in the  
thermodynamical limits of the spectrum of the ensembles of Gaussian  
random matrices such as GOE, GUE and GSE. The equilibrium states of  
the latter diffusion model are the thermodynamical limits of the  
spectrum of the ensemble of complex non-Hermitian Gaussian random  
matrices known as the Ginibre ensemble.

http://arxiv.org/abs/0902.3561


8177. Large Deviations and Moments for the Euler Characteristic of a  
Random Surface
Author(s): Kevin Fleming and Nicholas Pippenger

Abstract: We study random surfaces constructed by glueing together $N/k 
$ filled $k$-gons along their edges, with all $(N-1)!! = (N-1)(N-3)... 
3\cdot 1$ pairings of the edges being equally likely. (We assume that  
lcm $\{2,k\}$ divides $N$.) The Euler characteristic of the resulting  
surface is related to the number of cycles in a certain random  
permutation of $\{1, ..., N\}$. Gamburd has shown that when 2 lcm $ 
\{2,k\}$ divides $N$, the distribution of this random permutation  
converges to that of the uniform distribution on the alternating group  
$A_N$ in the total-variation distance as $N\to\infty$. We obtain large- 
deviations bounds for the number of cycles that, together with  
Gamburd's result, allow us to derive sharp estimates for the moments  
of the number of cycles. These estimates allow us to confirm certain  
cases of conjectures made by Pippenger and Schleich.

http://arxiv.org/abs/0902.3646


8178. Single-crossover dynamics: finite versus infinite populations
Author(s): Ellen Baake and Inke Herms

Abstract: Populations evolving under the joint influence of  
recombination and resampling (traditionally known as genetic drift)  
are investigated. First, we summarise and adapt a deterministic  
approach, as valid for infinite populations, which assumes continuous  
time and single crossover events. The corresponding nonlinear system  
of differential equations permits a closed solution, both in terms of  
the type frequencies and via linkage disequilibria of all orders. To  
include stochastic effects, we then consider the corresponding finite- 
population model, the Moran model with single crossovers, and examine  
it both analytically and by means of simulations. Particular emphasis  
is on the connection with the deterministic solution. If there is only  
recombination and every pair of recombined offspring replaces their  
pair of parents (i.e., there is no resampling), then the {\em  
expected} type frequencies in the finite population, of arbitrary  
size, equal the type frequencies in the infinite population. If  
resampling is included, the stochastic process converges, in the  
infinite-population limit, to the deterministic dynamics, which turns  
out to be a good approximation already for populations of moderate size.

http://arxiv.org/abs/q-bio/0612024


8179. A better algorithm for random k-SAT
Author(s): Amin Coja-Oghlan

Abstract: Let F be a uniformly distributed random k-SAT formula with n  
variables and m clauses. We present a polynomial time algorithm that  
finds a satisfying assignment of F with high probability for  
constraint densities m/n<(1-eps_k)2^k\ln(k)/k, where eps_k->0.  
Previously no efficient algorithm was known to find solutions with non- 
vanishing probability beyond m/n=1.817.2^k/k [Frieze and Suen, J. of  
Algorithms 1996].

http://arxiv.org/abs/0902.3583


8180. On the re-rooting invariance property of Levy trees
Author(s): Thomas Duquesne and Jean-Francois Le Gall

Abstract: We prove a strong form of the invariance under re-rooting of  
the distribution of the continuous random trees called Levy trees.  
This extends previous results due to several authors.

http://arxiv.org/abs/0902.3735


8181. Thick Points of the Gaussian Free Field
Author(s): Xiaoyu Hu and Jason Miller and Yuval Peres

Abstract: Let $U \subseteq \C$ be a bounded domain with smooth  
boundary and let $F$ be an instance of the continuum Gaussian free  
field on $U$ with respect to the Dirichlet inner product $\int_U  
\nabla f(x) \cdot \nabla g(x) dx$. The set $T(a;U)$ of $a$-thick  
points of $F$ consists of those $z \in U$ such that the average of $F$  
on a disk of radius $r$ centered at $z$ has growth $\sqrt{a/\pi} \log  
\tfrac{1}{r}$ as $r \to 0$. We show that for each $0 \leq a \leq 2$  
the Hausdorff dimension of $T(a;U)$ is almost surely $2-a$ and that  
with probability one $T(a;U)$ is empty when $a > 2$. Furthermore, we  
prove that $T(a;U)$ is invariant under conformal transformations in an  
appropriate sense. The notion of a thick point is connected to the  
Liouville quantum gravity measure with parameter $\gamma$ given  
formally by $\Gamma(dz) = e^{\sqrt{2\pi} \gamma F(z)} dz$ considered  
by Duplantier and Sheffield.

http://arxiv.org/abs/0902.3842


8182. Asymptotic Independence of the Extreme Eigenvalues of GUE
Author(s): Folkmar Bornemann

Abstract: We give a short, operator-theoretic proof of the asymptotic  
independence of the minimal and maximal eigenvalue of the n \times n  
Gaussian Unitary Ensemble in the large matrix limit n \to \infty. This  
is done by representing the joint probability distribution of those  
extreme eigenvalues as the Fredholm determinant of an operator matrix  
that asymptotically becomes diagonal. The method is amenable to  
explicitly establish the leading order term of an asymptotic  
expansion. As a corollary we obtain that the correlation of the  
extreme eigenvalues asymptotically behaves like n^{-2/3}/4\sigma^2,  
where \sigma^2 denotes the variance of the Tracy--Widom distribution.

http://arxiv.org/abs/0902.3870


8183. Equilibrium Fluctuations for the Totally Asymmetric Zero Range  
process
Author(s): Patricia Goncalves

Abstract: We prove a Central Limit Theorem for the empirical measure  
in the one-dimensional Totally Asymmetric Zero-Range Process in the  
hyperbolic scaling $N$, starting from the equilibrium measure $ 
\nu_{\rho}$. We also show that when taking the direction of the  
characteristics, the limit density fluctuation field does not evolve  
in time until $N^{4/3}$, which implies the current across the  
characteristics to vanish in this longer time scale.

http://arxiv.org/abs/0902.3974


8184. Rare event simulation for T-cell activation
Author(s): Florian Lipsmeier and Ellen Baake

Abstract: The problem of \emph{statistical recognition} is considered,  
as it arises in immunobiology, namely, the discrimination of foreign  
antigens against a background of the body's own molecules. The precise  
mechanism of this foreign-self-distinction, though one of the major  
tasks of the immune system, continues to be a fundamental puzzle.  
Recent progress has been made by van den Berg, Rand, and Burroughs  
(2001), who modelled the \emph{probabilistic} nature of the  
interaction between the relevant cell types, namely, T-cells and  
antigen-presenting cells (APCs). Here, the stochasticity is due to the  
random sample of antigens present on the surface of every APC, and to  
the random receptor type that characterises individual T-cells. It has  
been shown previously that this model, though highly idealised, is  
capable of reproducing important aspects of the recognition  
phenomenon, and of explaining them on the basis of stochastic rare  
events. These results were obtained with the help of a refined large  
deviation theorem and were thus asymptotic in nature. Simulations  
have, so far, been restricted to the straightforward simple sampling  
approach, which does not allow for sample sizes large enough to  
address more detailed questions. Building on the available large  
deviation results, we develop an importance sampling technique that  
allows for a convenient exploration of the relevant tail events by  
means of simulation. With its help, we investigate the mechanism of  
statistical recognition in some depth. In particular, we illustrate  
how a foreign antigen can stand out against the self background if it  
is present in sufficiently many copies, although no \emph{a priori}  
difference between self and nonself is built into the model.

http://arxiv.org/abs/0901.2227


8185. Levy flights and Levy -Schroedinger semigroups
Author(s): Piotr Garbaczewski

Abstract: We analyze Levy flights subject to an influence of external  
potentials and/or external conservative forces. Our goal is to clarify  
a discord between two classes of pertinent processes: those driven by  
Langevin equation with Levy noise and those named topological  
processes. Jump intensities of the latter processes are locally  
modified (via multiplicative Gibbs-type factors) by a "potential  
landscape" traveled by the flight and no explicit external forces are  
used to modify (confine) the noise. The discussion is set within the  
general framework of so-called Schrodinger boundary data problem which  
encompasses both Gaussian and non-Gaussian Markov processes.

http://arxiv.org/abs/0902.3536


8186. Space-time covariance functions with compact support
Author(s): Viktor P. Zastavnyi and Emilio Porcu

Abstract: We characterize completely the Gneiting class of space-time  
covariance functions and give more relaxed conditions on the involved  
functions. We then show necessary conditions for the construction of  
compactly supported functions of the Gneiting type. These conditions  
are very general since they do not depend on the Euclidean norm.  
Finally, we discuss a general class of positive definite functions,  
used for multivariate Gaussian random fields. For this class, we show  
necessary criteria for its generator to be compactly supported.

http://arxiv.org/abs/0902.3656


8187. On the Bennett-Hoeffding inequality
Author(s): Iosif Pinelis

Abstract: The well-known Bennett-Hoeffding bound for sums of  
independent random variables is refined, by taking into account  
truncated third moments, and at that also improved by using, instead  
of the class of all increasing exponential functions, the much larger  
class of all generalized moment functions f such that f and f" are  
increasing and convex. It is shown that the resulting bounds have  
certain optimality properties. Comparisons with related known bounds  
are given. The results can be extended in a standard manner to (the  
maximal functions of) (super)martingales.

http://arxiv.org/abs/0902.4058


8188. Positive-part moments via the Fourier-Laplace transform
Author(s): Iosif Pinelis

Abstract: Integral expressions for positive-part moments E X_+^p (p>0)  
of random variables X are presented, in terms of the Fourier-Laplace  
or Fourier transforms of the distribution of X. A necessary and  
sufficient condition for the validity of such an expression is given.  
This study was motivated by extremal problems in probability and  
statistics, where one needs to evaluate such positive-part moments.

http://arxiv.org/abs/0902.4214


8189. On regularity properties of Bessel flow
Author(s): L. Vostrikova

Abstract: We study the differentiability of Bessel flow $\rho : x \to  
\rho ^x_t$, where $(\rho ^x_t)_{t\geq 0}$ is BES $^x(\delta $) process  
of dimension $\delta >1$ starting from $x$. For $\delta \geq 2$ we  
prove the existence of bicontinuous derivatives in P-a.s. sense at $x 
\geq 0$ and we study the asymptotic behaviour of the derivatives at  
$x=0$. For $1< \delta <2$ we prove the existence of a modification of  
Bessel flow having derivatives in probability sense at $x\geq 0$. We  
study the asymptotic behaviour of the derivatives at $t=\tau_0(x)$  
where $\tau_0(x)$ is the first zero of $(\rho ^x_t)_{t\geq 0}$.

http://arxiv.org/abs/0902.4232


8190. Antithetic variates in higher dimensions
Author(s): Sebastian del Ba\~no Rollin and Joan-Andreu L\'azaro-Cam\'i

Abstract: We introduce the concept of multidimensional antithetic as  
the absolute minimum of the covariance function $O(N)\to\mathbb{R}$  
defined by $A\mapsto Cov(f(\xi),f(A\xi))$ where $\xi$ is a standard $N 
$-dimensional normal random variable and $f:\mathbb{R}^{N}\to\mathbb{R} 
$ is an almost everywhere differentiable function. The antithetic  
matrix is designed to optimise the calculation of $E[f(\xi)]$ in a  
Monte Carlo simulation. We present an iterative annealing algorithm  
that dynamically incorporates the estimation of the antithetic matrix  
within the Monte Carlo calculation.

http://arxiv.org/abs/0902.4211


8191. Load optimization in a planar network
Author(s): Charles Bordenave (IMT) and Giovanni Luca Torrisi

Abstract: We analyze the asymptotic properties of an Euclidean  
optimization problem on the plane. Specifically, we consider a network  
with 3 bins and n objects spatially uniformly distributed, each object  
being allocated to a bin at a cost depending on its position. Two  
allocations are considered: the allocation minimizing the bin loads  
and the allocation allocating each object to its less costly bin. We  
analyze the asymptotic properties of these allocations as the number  
of objects grows to infinity. Using the symmetries of the problem, we  
derive a law of large numbers, a central limit theorem and a large  
deviation principle for both loads with explicit expressions. In  
particular, we prove that the two allocations satisfy the same law of  
large numbers, but they do not have the same asymptotic fluctuations  
and rate functions.

http://arxiv.org/abs/0902.4304


8192. Scaling Limit of the Prudent Walk
Author(s): V. Beffara and S. Friedli and Y. Velenik

Abstract: We describe the scaling limit of the nearest neighbour  
prudent walk on the square lattice, which performs steps uniformly in  
directions in which it does not see sites already visited. We show  
that the process eventually settles in one of the quadrants, and  
derive its scaling limit, which can be expressed in terms of a pair of  
independent stable subordinators. We also show that the asymptotic  
speed of the walk is well-defined in the L_1 -norm and equals 3/7.  
This process possesses unusual properties: it is ballistic but does  
not have an asymptotic direction, and several natural observables  
display ageing.

http://arxiv.org/abs/0902.4312


8193. A Note on variational solutions to SPDE perturbed by Gaussian  
noise in a general class
Author(s): Michael R\"ockner and Yi Wang

Abstract: This note deals with existence and uniqueness of  
(variational) solutions to the following type of stochastic partial  
differential equations on a Hilbert space H dX(t) = A(t,X(t))dt +  
B(t,X(t))dW(t) + h(t) dG(t) where A and B are random nonlinear  
operators satisfying monotonicity conditions and G is an infinite  
dimensional Gaussian process adapted to the same filtration as the  
cylindrical Wiener pocess W(t), t >= 0.

http://arxiv.org/abs/0902.4324


8194. General tax structures and the Levy insurance risk model
Author(s): Andreas E.Kyprianou and Xiaowen Zhou

Abstract: In the spirit of previous of Albrecher, Hipp, Renaud and  
Zhou we consider a L\'evy insurance risk model with tax payments of a  
more general structure than in the aforementioned papers that was also  
considered in \cite{ABBR}. In terms of scale functions, we establish  
three fundamental identities of interest which have stimulated a large  
volume of actuarial research in recent years. That is to say, the two  
sided exit problem, the net present value of tax paid until ruin as  
well as a generalized version of the Gerber-Shiu function. The method  
we appeal to differs from former works in that we appeal predominantly  
to excursion theory.

http://arxiv.org/abs/0902.4340


8195. Strong limit theorems for a simple random walk on the 2- 
dimensional comb
Author(s): E. Csaki and M. Csorgo and A. Foldes and P. Revesz

Abstract: We study the path behaviour of a simple random walk on the 2- 
dimensional comb lattice ${\mathbb C}^2$ that is obtained from $ 
{\mathbb Z}^2$ by removing all horizontal edges off the x-axis. In  
particular, we prove a strong approximation result for such a random  
walk which, in turn, enables us to establish strong limit theorems,  
like the joint Strassen type law of the iterated logarithm of its two  
components, as well as their marginal Hirsch type behaviour.

http://arxiv.org/abs/0902.4369


8196. Theory of minimum spanning trees I: Mean-field theory and  
strongly disordered spin-glass model
Author(s): T. S. Jackson and N. Read

Abstract: The minimum spanning tree (MST) is a combinatorial  
optimization problem: given a connected graph with a real weight  
("cost") on each edge, find the spanning tree that minimizes the sum  
of the total cost of the occupied edges. We consider the random MST,  
in which the edge costs are (quenched) independent random variables.  
There is a strongly-disordered spin-glass model due to Newman and  
Stein [Phys. Rev. Lett. 72, 2286 (1994)], which maps precisely onto  
the random MST. We study scaling properties of random MSTs using a  
relation between Kruskal's greedy algorithm for finding the MST, and  
bond percolation. We solve the random MST problem on the Bethe lattice  
(BL) with appropriate wired boundary conditions and calculate the  
fractal dimension D=6 of the connected components. Viewed as a mean- 
field theory, the result implies that on a lattice in Euclidean space  
of dimension d, there are of order W^{d-D} large connected components  
of the random MST inside a window of size W, and that d = d_c = D = 6  
is a critical dimension. This differs from the value 8 suggested by  
Newman and Stein. We also critique the original argument for 8, and  
provide an improved scaling argument that again yields d_c=6. The  
result implies that the strongly-disordered spin-glass model has many  
ground states for d>6, and only of order one below six. The results  
for MSTs also apply on the Poisson-weighted infinite tree, which is a  
mean-field approach to the continuum model of MSTs in Euclidean space,  
and is a limit of the BL. In a companion paper we develop an epsilon=6- 
d expansion for the random MST on critical percolation clusters.

http://arxiv.org/abs/0902.3651


8197. Stationarity, time--reversal and fluctuation theory for a class  
of piecewise deterministic Markov processes
Author(s): Alessandra Faggionato and Davide Gabrielli and Marco  
Ribezzi Crivellari

Abstract: We consider a class of stochastic dynamical systems, called  
piecewise deterministic Markov processes, with states $(x, \s)\in \O 
\times \G$, $\O$ being a region in $\bbR^d$ or the $d$--dimensional  
torus, $\G$ being a finite set. The continuous variable $x$ follows a  
piecewise deterministic dynamics, the discrete variable $\s$ evolves  
by a stochastic jump dynamics and the two resulting evolutions are  
fully--coupled. We study stationarity, reversibility and time-- 
reversal symmetries of the process. Increasing the frequency of the $\s 
$--jumps, we show that the system behaves asymptotically as  
deterministic and we investigate the structure of fluctuations (i.e.  
deviations from the asymptotic behavior), recovering in a non  
Markovian frame results obtained by Bertini et al. \cite{BDGJL1,  
BDGJL2, BDGJL3, BDGJL4}, in the context of Markovian stochastic  
interacting particle systems. Finally, we discuss a Gallavotti--Cohen-- 
type symmetry relation with involution map different from time-- 
reversal. For several examples the above results are recovered by  
explicit computations.

http://arxiv.org/abs/0902.4195


8198. Maximal inequality for high-dimensional cubes: quantitative  
estimates
Author(s): Guillaume Aubrun (ICJ)

Abstract: We present lower estimates for the best constant appearing  
in the weak (1,1) maximal inequality in the space $(\R^n,\|\cdot\| 
_{\infty})$. We show that it grows to infinity faster than $(\log  
n)^{\kappa}$ for any $\kappa <1$. We follow the approach used by J.M.  
Aldaz in a recent paper. The new part of the argument relies on  
Donsker's theorem identifying the Brownian bridge as the limit $(n \to  
\infty)$ of the empirical distribution function associated to  
coordinates of a point randomly chosen in the unit cube $[0,1]^n$.

http://arxiv.org/abs/0902.4305


8199. Connectivity, Percolation, and Information Dissemination in  
Large-Scale Wireless Networks with Dynamic Links
Author(s): Zhenning Kong and Edmund M. Yeh

Abstract: We investigate the problem of disseminating broadcast  
messages in wireless networks with time-varying links from a  
percolation-based perspective. Using a model of wireless networks  
based on random geometric graphs with dynamic on-off links, we show  
that the delay for disseminating broadcast information exhibits two  
behavioral regimes, corresponding to the phase transition of the  
underlying network connectivity. When the dynamic network is in the  
subcritical phase, ignoring propagation delays, the delay scales  
linearly with the Euclidean distance between the sender and the  
receiver. When the dynamic network is in the supercritical phase, the  
delay scales sub-linearly with the distance. Finally, we show that in  
the presence of a non-negligible propagation delay, the delay for  
information dissemination scales linearly with the Euclidean distance  
in both the subcritical and supercritical regimes, with the rates for  
the linear scaling being different in the two regimes.

http://arxiv.org/abs/0902.4449


8200. Strategies of Voting in Stochastic Environment: Egoism and  
Collectivism
Author(s): V.I. Borzenko and Z.M. Lezina and A. K.Loginov and Ya.Yu.  
Tsodikova and and P.Yu. Chebotarev

Abstract: Consideration was given to a model of social dynamics  
controlled by successive collective decisions based on the threshold  
majority procedures. The current system state is characterized by the  
vector of participants' capitals (utilities). At each step, the voters  
can either retain their status quo or accept the proposal which is a  
vector of the algebraic increments in the capitals of the  
participants. In this version of the model, the vector is generated  
stochastically. Comparative utility of two social attitudes--egoism  
and collectivism--was analyzed. It was established that, except for  
some special cases, the collectivists have advantages, which makes  
realizable the following scenario: on the conditions of protecting the  
corporate interests, a group is created which is joined then by the  
egoists attracted by its achievements. At that, group egoism  
approaches altruism. Additionally, one of the considered variants of  
collectivism handicaps manipulation of voting by the organizers.

http://arxiv.org/abs/0902.4460


8201. Asymptotic coupling and a weak form of Harris' theorem with  
applications to stochastic delay equations
Author(s): Martin Hairer and Jonathan C. Mattingly and Michael Scheutzow

Abstract: There are many Markov chains on infinite dimensional spaces  
whose one-step transition kernels are mutually singular when starting  
from different initial conditions. We give results which prove unique  
ergodicity under minimal assumptions on one hand and the existence of  
a spectral gap under conditions reminiscent of Harris' theorem. The  
first uses the existence of couplings which draw the solutions  
together as time goes to infinity. Such "asymptotic couplings" were  
central to recent work on SPDEs on which this work builds. The  
emphasis here is on stochastic differential delay equations.Harris'  
celebrated theorem states that if a Markov chain admits a Lyapunov  
function whose level sets are "small" (in the sense that transition  
probabilities are uniformly bounded from below), then it admits a  
unique invariant measure and transition probabilities converge towards  
it at exponential speed. This convergence takes place in a total  
variation norm, weighted by the Lyapunov function. A second aim of  
this article is to replace the notion of a "small set" by the much  
weaker notion of a "d-small set," which takes the topology of the  
underlying space into account via a distance-like function d. With  
this notion at hand, we prove an analogue to Harris' theorem, where  
the convergence takes place in a Wasserstein-like distance weighted  
again by the Lyapunov function. This abstract result is then applied  
to the framework of stochastic delay equations.

http://arxiv.org/abs/0902.4495


8202. Geometric ergodicity of a bead-spring pair with stochastic  
Stokes forcing
Author(s): Jonathan C. Mattingly and Scott A. McKinley and Natesh S.  
Pillai

Abstract: We consider a simple model for the fluctuating hydrodynamics  
of a flexible polymer in dilute solution, demonstrating geometric  
ergodicity for a pair of particles that interact with each other  
through a nonlinear spring potential while being advected by a  
stochastic Stokes fluid velocity field. This is a generalization of  
previous models which have used linear spring forces as well as white- 
in-time fluid velocity fields. We follow previous work combining  
control theoretic arguments, Lyapunov functions, and hypo-elliptic  
diffusion theory to prove exponential convergence via a Harris chain  
argument. To this, we add the possibility of excluding certain "bad"  
sets in phase space in which the assumptions are violated but from  
which the systems leaves with a controllable probability. This allows  
for the treatment of singular drifts, such as those derived from the  
Lennard-Jones potential, which is an novel feature of this work.

http://arxiv.org/abs/0902.4496


8203. Many-Sources Large Deviations for Max-Weight Scheduling
Author(s): Vijay G. Subramanian and Tara Javidi and Somsak Kittipiyakul

Abstract: In this paper, a many-sources large deviations principle  
(LDP) for the transient workload of a multi-queue single-server system  
is established where the service rates are chosen from a compact,  
convex and coordinate-convex rate region and where the service  
discipline is the max-weight policy. Under the assumption that the  
arrival processes satisfy a many-sources LDP, this is accomplished by  
employing Garcia's extended contraction principle that is applicable  
to quasi-continuous mappings. For the simplex rate-region, an LDP for  
the stationary workload is also established under the additional  
requirements that the scheduling policy be work-conserving and that  
the arrival processes satisfy certain mixing conditions. The LDP  
results can be used to calculate asymptotic buffer overflow  
probabilities accounting for the multiplexing gain, when the arrival  
process is an average of \emph{i.i.d.} processes. The rate function  
for the stationary workload is expressed in term of the rate functions  
of the finite-horizon workloads when the arrival processes have  
\emph{i.i.d.} increments.

http://arxiv.org/abs/0902.4569


8204. The CRT is the scaling limit of unordered binary trees
Author(s): Jean-Fran\c{c}ois Marckert (LaBRI) and Gr\'egory Miermont  
(ENS)

Abstract: We prove that a uniform, rooted unordered binary tree with $n 
$ vertices has the Brownian continuum random tree as its scaling limit  
for the Gromov-Hausdorff topology. The limit is thus, up to a constant  
factor, the same as that of uniform plane trees or labeled trees. Our  
analysis rests on a combinatorial and probabilistic study of  
appropriate trimming procedures of trees.

http://arxiv.org/abs/0902.4570


8205. Criteria for hitting probabilities with applications to systems  
of stochastic wave equations
Author(s): Robert C. Dalang and Marta Sanz-Sol\'e

Abstract: We develop several results on hitting probabilities of  
random fields which highlight the role of the dimension of the  
parameter space. This yields upper and lower bounds in terms of  
Hausdorff measure and Bessel-Riesz capacity, respectively. We apply  
these results to a system of stochastic wave equations in spatial  
dimension $k\ge1$ driven by a $d$-dimensional spatially homogeneous  
additive Gaussian noise that is white in time and colored in space.

http://arxiv.org/abs/0902.4583


8206. Analytical Expression of the Expected Values of Capital at  
Voting in the Stochastic Environment
Author(s): Pavel Chebotarev

Abstract: In the simplest version of the model of group decision  
making in the stochastic environment, the participants are segregated  
into egoists and a group of collectivists. A "proposal of the  
environment" is a stochastically generated vector of algebraic  
increments of capitals. The social dynamics is determined by the  
sequence of proposals accepted by a majority voting (with a threshold)  
of the participants. In this paper, we obtain analytical expressions  
for the expected values of capitals for all the participants,  
including collectivists and egoists. In addition, distinctions between  
some principles of group voting are discussed.

http://arxiv.org/abs/0902.4514


8207. A combinatorial analysis of interacting diffusions
Author(s): Sourav Chatterjee and Soumik Pal

Abstract: We consider a particular class of n-dimensional homogeneous  
diffusions all of which have an identity diffusion matrix and a drift  
function that is piecewise constant and scale invariant. Abstract  
stochastic calculus immediately gives us general results about  
existence and uniqueness in law and invariant probability  
distributions when they exist. These invariant distributions are  
probability measures on the $n$-dimensional space and can be extremely  
resistant to a more detailed understanding. To have a better analysis,  
we construct a polyhedra such that the inward normal at its surface is  
given by the drift function and show that the finer structures of the  
invariant probability measure is intertwined with the geometry of the  
polyhedra. We show that several natural interacting Brownian particle  
models can thus be analyzed by studying the combinatorial fan  
generated by the drift function, particularly when these are  
simplicial. This is the case when the polyhedra is a polytope that is  
invariant under a Coxeter group action, which leads to an explicit  
description of the invariant measures in terms of iid Exponential  
random variables. Another class of examples is furnished by  
interactions indexed by weighted graphs all of which generate  
simplicial polytopes with $n !$ faces. We show that the proportion of  
volume contained in each component simplex corresponds to a  
probability distribution on the group of permutations, some of which  
have surprising connections with the classical urn models.

http://arxiv.org/abs/0902.4762


8208. Relative frequencies in multitype branching processes
Author(s): Andrei Y. Yakovlev and Nikolay M. Yanev

Abstract: This paper considers the relative frequencies of distinct  
types of individuals in multitype branching processes. We prove that  
the frequencies are asymptotically multivariate normal when the  
initial number of ancestors is large and the time of observation is  
fixed. The result is valid for any branching process with a finite  
number of types; the only assumption required is that of independent  
individual evolutions. The problem under consideration is motivated by  
applications in the area of cell biology. Specifically, the reported  
limiting results are of advantage in cell kinetics studies where the  
relative frequencies but not the absolute cell counts are accessible  
to measurement. Relevant statistical applications are discussed in the  
context of asymptotic maximum likelihood inference for multitype  
branching processes.

http://arxiv.org/abs/0902.4773


8209. Degenerate diffusions arising from gene duplication models
Author(s): Rick Durrett and Lea Popovic

Abstract: We consider two processes that have been used to study gene  
duplication, Watterson's [Genetics 105 (1983) 745--766] double  
recessive null model and Lynch and Force's [Genetics 154 (2000)  
459--473] subfunctionalization model. Though the state spaces of these  
diffusions are two and six-dimensional, respectively, we show in each  
case that the diffusion stays close to a curve. Using ideas of  
Katzenberger [Ann. Probab. 19 (1991) 1587--1628] we show that one- 
dimensional projections converge to diffusion processes, and we obtain  
asymptotics for the time to loss of one gene copy. As a corollary we  
find that the probability of subfunctionalization decreases  
exponentially fast as the population size increases. This rigorously  
confirms a result Ward and Durrett [Theor. Pop. Biol. 66 (2004)  
93--100] found by simulation that the likelihood of  
subfunctionalization for gene duplicates decays exponentially fast as  
the population size increases.

http://arxiv.org/abs/0902.4780


8210. Integrated functionals of normal and fractional processes
Author(s): Boris Buchmann and Ngai Hang Chan

Abstract: Consider $Z^f_t(u)=\int_0^{tu}f(N_s) ds$, $t>0$, $u 
\in[0,1]$, where $N=(N_t)_{t\in\mathbb{R}}$ is a normal process and $f 
$ is a measurable real-valued function satisfying $Ef(N_0)^2<\infty$  
and $Ef(N_0)=0$. If the dependence is sufficiently weak Hariz [J.  
Multivariate Anal. 80 (2002) 191--216] showed that $Z_t^f/t^{1/2}$  
converges in distribution to a multiple of standard Brownian motion as  
$t\to\infty$. If the dependence is sufficiently strong, then $Z_t/ 
(EZ_t(1)^2)^{1/2}$ converges in distribution to a higher order Hermite  
process as $t\to\infty$ by a result by Taqqu [Wahrsch. Verw. Gebiete  
50 (1979) 53--83]. When passing from weak to strong dependence, a  
unique situation encompassed by neither results is encountered. In  
this paper, we investigate this situation in detail and show that the  
limiting process is still a Brownian motion, but a nonstandard norming  
is required. We apply our result to some functionals of fractional  
Brownian motion which arise in time series. For all Hurst indices $H 
\in(0,1)$, we give their limiting distributions. In this context, we  
show that the known results are only applicable to $H<3/4$ and  
$H>3/4$, respectively, whereas our result covers $H=3/4$.

http://arxiv.org/abs/0902.4784


8211. A Berry--Esseen theorem for sample quantiles under weak dependence
Author(s): S. N. Lahiri and S. Sun

Abstract: This paper proves a Berry--Esseen theorem for sample  
quantiles of strongly-mixing random variables under a polynomial  
mixing rate. The rate of normal approximation is shown to be  
$O(n^{-1/2})$ as $n\to\infty$, where $n$ denotes the sample size. This  
result is in sharp contrast to the case of the sample mean of strongly- 
mixing random variables where the rate $O(n^{-1/2})$ is not known even  
under an exponential strong mixing rate. The main result of the paper  
has applications in finance and econometrics as financial time series  
data often are heavy-tailed and quantile based methods play an  
important role in various problems in finance, including hedging and  
risk management.

http://arxiv.org/abs/0902.4796


8212. The calculation of expectations for classes of diffusion  
processes by Lie symmetry methods
Author(s): Mark Craddock and Kelly A. Lennox

Abstract: This paper uses Lie symmetry methods to calculate certain  
expectations for a large class of It\^{o} diffusions. We show that if  
the problem has sufficient symmetry, then the problem of computing  
functionals of the form $E_x(e^{-\lambda X_t-\int_0^tg(X_s) ds})$ can  
be reduced to evaluating a single integral of known functions. Given a  
drift $f$ we determine the functions $g$ for which the corresponding  
functional can be calculated by symmetry. Conversely, given $g$, we  
can determine precisely those drifts $f$ for which the transition  
density and the functional may be computed by symmetry. Many examples  
are presented to illustrate the method.

http://arxiv.org/abs/0902.4806




-----------------------------------
Stefano M. Iacus
Department of Economics,
Business and Statistics
University of Milan
Via Conservatorio, 7
I-20123 Milan - Italy
Ph.: +39 02 50321 461
Fax: +39 02 50321 505
http://www.economia.unimi.it/iacus
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From pas at lists.imstat.org  Mon May 11 04:58:16 2009
From: pas at lists.imstat.org (Probability Abstract Service)
Date: Mon, 11 May 2009 11:58:16 +0200
Subject: [PAS] Probability Abstracts 109
Message-ID: <A8B34539-4382-4F74-A799-1D5A392629F9@unimi.it>

Probability Abstracts 109
This document contains abstracts 8213-8462
from Mar-1-2009 to April-30-2009.
They have been mailed on May 11, 2009.


This letter can be also found on line at
http://pas.imstat.org/Letters/letter_109.shtml


-----------------------------------------------


8213. Martin boundary of a killed random walk on a quadrant
Author(s): Irina Ignatiouk-Robert and Christophe Loree

Abstract: A complete representation of the Martin boundary of killed  
random walks on the quadrant NxN is obtained. It is proved that the  
corresponding full Martin compactification of the quadrant NxN is  
homeomorphic to the closure of the set {w =z/(1+|z|): z in NxN}$ in  
R2. The method is based on a ratio limit theorem for local processes  
and large deviation techniques.

http://arxiv.org/abs/0903.0070


8214. Poisson asymptotics for random projections of points on a high- 
dimensional sphere
Author(s): Itai Benjamini and Oded Schramm and and Sasha Sodin

Abstract: Project a collection of points on the high-dimensional  
sphere onto a random direction. If most of the points are sufficiently  
far from one another in an appropriate sense, the projection is  
locally close in distribution to the Poisson point process.

http://arxiv.org/abs/0903.0107


8215. Large dimensional random k circulants
Author(s): Arup Bose and Joydip Mitra and Arnab Sen

Abstract: Circulant matrices with general shift by k places have been  
studied in the literature and formula for their eigenvalues is known.  
We first reestablish this formula and some further properties of these  
eigenvalues in a manner suitable for our use. We then consider random  
k=k(n) circulants A_{k,n} with $n \to \infty$ and whose input sequence  
{a_i} is independent with mean zero and variance one and $\sup_n  
n^{-1}\sum_{i=1}^n E|a_i|^{2+\delta}< \infty$ for some $\delta > 0$.  
Under suitable restrictions on {k(n)},we show that the limiting  
spectral distribution (LSD) of the empirical distribution of suitably  
scaled eigenvalues exists and identify the limits. As examples, (i) if  
k^g = -1+ s n where $g \ge 1 $ fixed and $s=o(n^{1/3})$, then the LSD  
is $U_1(\prod_{i=1}^g E_i)^{1/2g}$ where E_i are i.i.d. Exp(1) and U_1  
is uniformly distributed over the (2g)th roots of unity, independent  
of the {E_i}, and (ii) if k^g = 1+ sn where $g \ge 2$ is fixed and  
$s=o(n^{\frac{g+1}{g-1}})$ or $s=o(n)$ according as $g \ge 2$ is odd  
or even, then the LSD is $U_2(\prod_{i=1}^g E_i)^{1/2g}$ where {E_i}  
are i.i.d. Exp(1) and U_2 is uniformly distributed over the unit  
circle, independent of the {E_i}. We then consider the limit  
distribution of the spectral norm of A_{k,n}. We show that when  
$n=k^2+1\to \infty$, the spectral norm, with appropriate scaling and  
centering, which we provide explicitly, converges to the Gumbel  
distribution.

http://arxiv.org/abs/0903.0128


8216. Conditioning of quadratic harnesses
Author(s): W. Bryc and J. Wesolowski

Abstract: We describe quadratic harnesses that arise through the  
double sided conditioning of an already known quadratic harness and we  
characterize quadratic harnesses that arise by this construction from  
bridges of Levy processes. We also analyze a construction that  
produces quadratic harnesses by "gluing together" two conditionally- 
independent quadratic harnesses and we show that the only q-Meixner  
processes that can be used in this construction are pairs of Poisson  
processes or pairs of negative binomial processes. Our main tool is a  
deterministic time and space transformation of quadratic harnesses.

http://arxiv.org/abs/0903.0150


8217. Reaching the best possible rate of convergence to equilibrium  
for solutions of Kac's equation via central limit theorem
Author(s): Emanuele Dolera and Ester Gabetta and Eugenio Regazzini

Abstract: Let $f(\cdot,t)$ be the probability density function which  
represents the solution of Kac's equation at time $t$, with initial  
data $f_0$, and let $g_{\sigma}$ be the Gaussian density with zero  
mean and variance $\sigma^2$, $\sigma^2$ being the value of the second  
moment of $f_0$. This is the first study which proves that the total  
variation distance between $f(\cdot,t)$ and $g_{\sigma}$ goes to zero,  
as $t\to +\infty$, with an exponential rate equal to -1/4. In the  
present paper, this fact is proved on the sole assumption that $f_0$  
has finite fourth moment and its Fourier transform $\varphi_0$  
satisfies $|\varphi_0(\xi)|=o(|\xi|^{-p})$ as $|\xi|\to+\infty$, for  
some $p>0$. These hypotheses are definitely weaker than those  
considered so far in the state-of-the-art literature, which in any  
case, obtains less precise rates.

http://arxiv.org/abs/0903.0255


8218. Fluid limits for networks with bandwidth sharing and general  
document size distributions
Author(s): H. Christian Gromoll and Ruth J. Williams

Abstract: We consider a stochastic model of Internet congestion  
control, introduced by Massouli\'{e} and Roberts [Telecommunication  
Systems 15 (2000) 185--201], that represents the randomly varying  
number of flows in a network where bandwidth is shared among document  
transfers. In contrast to an earlier work by Kelly and Williams [Ann.  
Appl. Probab. 14 (2004) 1055--1083], the present paper allows  
interarrival times and document sizes to be generally distributed,  
rather than exponentially distributed. Furthermore, we allow a fairly  
general class of bandwidth sharing policies that includes the weighted  
$\alpha$-fair policies of Mo and Walrand [IEEE/ACM Transactions on  
Networking 8 (2000) 556--567], as well as certain other utility based  
scheduling policies. To describe the evolution of the system, measure  
valued processes are used to keep track of the residual document sizes  
of all flows through the network. We propose a fluid model (or formal  
functional law of large numbers approximation) associated with the  
stochastic flow level model. Under mild conditions, we show that the  
appropriately rescaled measure valued processes corresponding to a  
sequence of such models (with fixed network structure) are tight, and  
that any weak limit point of the sequence is almost surely a fluid  
model solution. For the special case of weighted $\alpha$-fair  
policies, we also characterize the invariant states of the fluid model.

http://arxiv.org/abs/0903.0291


8219. Modified discrete random walk with absorption
Author(s): Theo van Uem

Abstract: We obtain expected number of arrivals, probability of  
arrival, absorption probabilities and expected time before absorption  
for a modified discrete random walk on the (sub)set of integers. In a  
[pqrs] random walk the particle can move one step forward or backward,  
stay for a moment in the same state or it can be absorbed immediately  
in the current state. M[pqrs] is a modified version, where  
probabilities on both sides of a multiple function barrier M are of  
different [pqrs] type.

http://arxiv.org/abs/0903.0364


8220. The Generalized Road Coloring Problem and periodic digraphs
Author(s): Greg Budzban and Philip Feinsilver

Abstract: A proof of the Generalized Road Coloring Problem,  
independent of the recent work by Beal and Perrin, is presented, using  
both semigroup methods and Trakhtman's algorithm. Algebraic properties  
of periodic, strongly connected digraphs are studied in the semigroup  
context. An algebraic condition which characterizes periodic, strongly  
connected digraphs is determined in the context of periodic Markov  
chains.

http://arxiv.org/abs/0903.0192


8221. On the equality of the quenched and averaged large deviation  
rate functions for high-dimensional ballistic random walk in a random  
environment
Author(s): Atilla Yilmaz

Abstract: We consider large deviations for nearest-neighbor random  
walk in a uniformly elliptic i.i.d. environment. It is easy to see  
that the quenched and averaged rate functions are not identically  
equal. When the dimension is at least four and Sznitman's transience  
condition (T) is satisfied, we prove that these rate functions are  
finite and equal on a closed set whose interior contains every nonzero  
velocity at which the rate functions vanish.

http://arxiv.org/abs/0903.0410


8222. Motion in a Random Force Field
Author(s): Dmitry Dolgopyat and Leonid Koralov

Abstract: We consider the motion of a particle in a random isotropic  
force field. Assuming that the force field arises from a Poisson field  
in $\mathbb{R}^d$, $d \geq 4$, and the initial velocity of the  
particle is sufficiently large, we describe the asymptotic behavior of  
the particle.

http://arxiv.org/abs/0903.0425


8223. Nonlinear Stochastic Perturbations of Dynamical Systems and  
Quasi-linear Parabolic PDE's with a Small Parameter
Author(s): M. Freidlin and L. Koralov

Abstract: In this paper we describe the asymptotic behavior, in the  
exponential time scale, of solutions to quasi-linear parabolic  
equations with a small parameter at the second order term and the long  
time behavior of corresponding diffusion processes. In particular, we  
discuss the exit problem and metastability for the processes  
corresponding to quasi-linear initial-boundary value problems.

http://arxiv.org/abs/0903.0428


8224. Metastability for Non-Linear Random Perturbations of Dynamical  
Systems
Author(s): M. Freidlin and L. Koralov

Abstract: In this paper we describe the long time behavior of  
solutions to quasi-linear parabolic equations with a small parameter  
at the second order term and the long time behavior of corresponding  
diffusion processes.

http://arxiv.org/abs/0903.0430


8225. Random Perturbations of 2-dimensional Hamiltonian Flows
Author(s): L. Koralov

Abstract: We consider the motion of a particle in a periodic two  
dimensional flow perturbed by small (molecular) diffusion. The flow is  
generated by a divergence free zero mean vector field. The long time  
behavior corresponds to the behavior of the homogenized process - that  
is diffusion process with the constant diffusion matrix (effective  
diffusivity). We obtain the asymptotics of the effective diffusivity  
when the molecular diffusion tends to zero.

http://arxiv.org/abs/0903.0436


8226. Coupled paraxial wave equations in random media in the white- 
noise regime
Author(s): Josselin Garnier and Knut S{\o}lna

Abstract: In this paper the reflection and transmission of waves by a  
three-dimensional random medium are studied in a white-noise and  
paraxial regime. The limit system derives from the acoustic wave  
equations and is described by a coupled system of random Schr 
\"{o}dinger equations driven by a Brownian field whose covariance is  
determined by the two-point statistics of the fluctuations of the  
random medium. For the reflected and transmitted fields the associated  
Wigner distributions and the autocorrelation functions are determined  
by a closed system of transport equations. The Wigner distribution is  
then used to describe the enhanced backscattering phenomenon for the  
reflected field.

http://arxiv.org/abs/0903.0449


8227. Adaptive independent Metropolis--Hastings
Author(s): Lars Holden and Ragnar Hauge and Marit Holden

Abstract: We propose an adaptive independent Metropolis--Hastings  
algorithm with the ability to learn from all previous proposals in the  
chain except the current location. It is an extension of the  
independent Metropolis--Hastings algorithm. Convergence is proved  
provided a strong Doeblin condition is satisfied, which essentially  
requires that all the proposal functions have uniformly heavier tails  
than the stationary distribution. The proof also holds if proposals  
depending on the current state are used intermittently, provided the  
information from these iterations is not used for adaption. The  
algorithm gives samples from the exact distribution within a finite  
number of iterations with probability arbitrarily close to 1. The  
algorithm is particularly useful when a large number of samples from  
the same distribution is necessary, like in Bayesian estimation, and  
in CPU intensive applications like, for example, in inverse problems  
and optimization.

http://arxiv.org/abs/0903.0483


8228. Macroscopic stability for nonfinite range kernels
Author(s): Tom S. Mountford (EPFL) and K. Ravishankar (SUNY) and Ellen  
Saada (LMRS)

Abstract: We extend the strong macroscopic stability introduced in  
Bramson & Mountford (2002) for one-dimensional asymmetric exclusion  
processes with finite range to a large class of one-dimensional  
conservative attractive models (including misanthrope process) for  
which we relax the requirement of finite range kernels. A key  
motivation is extension of constructive hydrodynamics result of  
Bahadoran et al. (2002, 2006, 2008) to nonfinite range kernels.

http://arxiv.org/abs/0903.0498


8229. Crested products of Markov chains
Author(s): Daniele D'Angeli and Alfredo Donno

Abstract: In this work we define two kinds of crested product for  
reversible Markov chains, which naturally appear as a generalization  
of the case of crossed and nested product, as in association schemes  
theory, even if we do a construction that seems to be more general and  
simple. Although the crossed and nested product are inspired by the  
study of Gelfand pairs associated with the direct and the wreath  
product of two groups, the crested products are a more general  
construction, independent from the Gelfand pairs theory, for which a  
complete spectral theory is developed. Moreover, the $k$-step  
transition probability is given. It is remarkable that these Markov  
chains describe some classical models (Ehrenfest diffusion model,  
Bernoulli--Laplace diffusion model, exclusion model) and give some  
generalization of them. As a particular case of nested product, one  
gets the classical Insect Markov chain on the ultrametric space.  
Finally, in the context of the second crested product, we present a  
generalization of this Markov chain to the case of many insects and  
give the corresponding spectral decomposition.

http://arxiv.org/abs/0903.0513


8230. ROC and the bounds on tail probabilities via theorems of Dubins  
and F. Riesz
Author(s): Eric Clarkson and J. L. Denny and Larry Shepp

Abstract: For independent $X$ and $Y$ in the inequality $P(X\leq Y+\mu) 
$, we give sharp lower bounds for unimodal distributions having finite  
variance, and sharp upper bounds assuming symmetric densities bounded  
by a finite constant. The lower bounds depend on a result of Dubins  
about extreme points and the upper bounds depend on a symmetric  
rearrangement theorem of F. Riesz. The inequality was motivated by  
medical imaging: find bounds on the area under the Receiver Operating  
Characteristic curve (ROC).

http://arxiv.org/abs/0903.0518


8231. Random matrices: The distribution of the smallest singular values
Author(s): Terence Tao and Van Vu

Abstract: Let $\a$ be a real-valued random variable of mean zero and  
variance 1. Let $M_n(\a)$ denote the $n \times n$ random matrix whose  
entries are iid copies of $\a$ and $\sigma_n(M_n(\a))$ denote the  
least singular value of $M_n(\a)$. ($\sigma_n(M_n(\a))^2$ is also  
usually interpreted as the least eigenvalue of the Wishart matrix $M_n  
M_n^{\ast}$.) We show that (under a finite moment assumption) the  
probability distribution $n \sigma_n(M_n(\a))^2$ is {\it universal} in  
the sense that it does not depend on the distribution of $\a$. In  
particular, it converges to the same limiting distribution as in the  
special case when $a$ is real gaussian. (The limiting distribution was  
computed explicitly in this case by Edelman.) We also proved a similar  
result for complex-valued random variables of mean zero, with real and  
imaginary parts having variance 1/2 and covariance zero. Similar  
results are also obtained for the joint distribution of the bottom $k$  
singular values of $M_n(\a)$ for any fixed $k$ (or even for $k$  
growing as a small power of $n$) and for rectangular matrices. Our  
approach is motivated by the general idea of ``property testing'' from  
combinatorics and theoretical computer science. This seems to be a new  
approach in the study of spectra of random matrices and combines tools  
from various areas of mathematics.

http://arxiv.org/abs/0903.0614


8232. Coupling, Attractiveness and Hydrodynamics for Conservative  
Particle Systems
Author(s): Thierry Gobron (LPTM) and Ellen Saada (LMRS)

Abstract: Attractiveness is a fundamental tool to study interacting  
particle systems and the basic coupling construction is a usual route  
to prove this property, as for instance in simple exclusion. The  
derived Markovian coupled process $(\xi_t,\zeta_t)_{t\geq 0}$  
satisfies: (A) if $\xi_0\leq\zeta_0$ (coordinate-wise), then for all $t 
\geq 0$, $\xi_t\leq\zeta_t$ a.s. In this paper, we consider  
generalized misanthrope models which are conservative particle systems  
on $\Z^d$ such that, in each transition, $k$ particles may jump from a  
site $x$ to another site $y$, with $k\geq 1$. These models include  
simple exclusion for which $k=1$, but, beyond that value, the basic  
coupling construction is not possible and a more refined one is  
required. We give necessary and sufficient conditions on the rates to  
insure attractiveness; we construct a Markovian coupled process which  
both satisfies (A) and makes discrepancies between its two marginals  
non-increasing. We determine the extremal invariant and translation  
invariant probability measures under general irreducibility  
conditions. We apply our results to examples including a two-species  
asymmetric exclusion process with charge conservation (for which $k\le  
2$) which arises from a Solid-on-Solid interface dynamics, and a stick  
process (for which $k$ is unbounded) in correspondence with a  
generalized discrete Hammersley-Aldous-Diaconis model. We derive the  
hydrodynamic limit of these two one-dimensional models.

http://arxiv.org/abs/0903.0316


8233. The Existence of Pair Potential Corresponding to Specified  
Density and Pair Correlation
Author(s): L. Koralov

Abstract: Given a potential of pair interaction and a value of  
activity, one can consider the Gibbs distribution in a finite domain $ 
\Lambda \subset \mathbb{Z}^d$. It is well known that for small values  
of activity there exist the infinite volume ($\Lambda \to \mathbb{Z}^d 
$) limiting Gibbs distribution and the infinite volume correlation  
functions. In this paper we consider the converse problem - we show  
that given $\rho_1$ and $\rho_2(x)$, where $\rho_1$ is a constant and $ 
\rho_2(x)$ is a function on $\mathbb{Z}^d$, which are sufficiently  
small, there exist a pair potential and a value of activity, for which  
$\rho_1$ is the density and $\rho_2(x)$ is the pair correlation  
function.

http://arxiv.org/abs/0903.0432


8234. An Inverse Problem for Gibbs Fields with Hard Core Potential
Author(s): L. Koralov

Abstract: It is well known that for a regular stable potential of pair  
interaction and a small value of activity one can define the  
corresponding Gibbs field (a measure on the space of configurations of  
points in $\mathbb{R}^d$). In this paper we consider a converse  
problem. Namely, we show that for a sufficiently small constant $ 
\overline{\rho}_1$ and a sufficiently small function $ 
\overline{\rho}_2(x)$, $x \in \mathbb{R}^d$, that is equal to zero in  
a neighborhood of the origin, there exist a hard core pair potential,  
and a value of activity, such that $\overline{\rho}_1$ is the density  
and $\overline{\rho}_2$ is the pair correlation function of the  
corresponding Gibbs field.

http://arxiv.org/abs/0903.0433


8235. Some Diffusion Processes Associated With Two Parameter Poisson- 
Dirichlet Distribution and Dirichlet Process
Author(s): Shui Feng and Wei Sun

Abstract: The two parameter Poisson-Dirichlet distribution $PD(\alpha, 
\theta)$ is the distribution of an infinite dimensional random  
discrete probability. It is a generalization of Kingman's Poisson- 
Dirichlet distribution. The two parameter Dirichlet process $ 
\Pi_{\alpha,\theta,\nu_0}$ is the law of a pure atomic random measure  
with masses following the two parameter Poisson-Dirichlet  
distribution. In this article we focus on the construction and the  
properties of the infinite dimensional symmetric diffusion processes  
with respective symmetric measures $PD(\alpha,\theta)$ and $ 
\Pi_{\alpha,\theta,\nu_0}$. The methods used come from the theory of  
Dirichlet forms.

http://arxiv.org/abs/0903.0623


8236. Products of random matrices: Dimension and growth in norm
Author(s): Vladislav Kargin

Abstract: Suppose that X_1, X_2, ... are independent, identically- 
distributed, rotationally invariant N-by-N matrices. Let P_n be the  
product X_n...X_1. It is known that log|P_n|/n converges to a non- 
random limit. We prove that under certain additional assumptions on  
matrices X_i the speed of convergence to this limit does not decrease  
when the size of matrices, N, grows.

http://arxiv.org/abs/0903.0632


8237. Loss networks
Author(s): Stan Zachary and Ilze Ziedins

Abstract: We review the theory of loss networks, including recent  
results on their dynamical behaviour. We give also some new results.

http://arxiv.org/abs/0903.0640


8238. SPDEs in divergence form with VMO coefficients and filtering  
theory of partially observable diffusion processes with Lipschitz  
coefficients
Author(s): N.V. Krylov

Abstract: We present several results on the smoothness in $L_{p}$  
sense of filtering densities under the Lipschitz continuity assumption  
on the coefficients of a partially observable diffusion processes. We  
obtain them by rewriting in divergence form filtering equation which  
are usually considered in terms of formally adjoint to operators in  
nondivergence form.

http://arxiv.org/abs/0903.0877


8239. Optimal investment with counterparty risk: a default-density  
modeling approach
Author(s): Ying Jiao (PMA) and Huyen Pham (PMA)

Abstract: We consider a financial market with a stock exposed to a  
counterparty risk inducing a drop in the price, and which can still be  
traded after this default time. We use a default-density modeling  
approach, and address in this incomplete market context the expected  
utility maximization from terminal wealth. We show how this problem  
can be suitably decomposed in two optimization problems in complete  
market framework: an after-default utility maximization and a global  
before-default optimization problem involving the former one. These  
two optimization problems are solved explicitly, respectively by  
duality and dynamic programming approaches, and provide a fine  
understanding of the optimal strategy. We give some numerical results  
illustrating the impact of counterparty risk and the loss given  
default on optimal trading strategies, in particular with respect to  
the Merton portfolio selection problem.

http://arxiv.org/abs/0903.0909


8240. Zero bias transformation and asymptotic expansions
Author(s): Ying Jiao (PMA)

Abstract: We apply the zero bias transformation to deduce a recursive  
asymptotic expansion formula for expectation of functions of sum of  
independent random variables in terms of normal expectations and we  
discuss the remainder term estimations.

http://arxiv.org/abs/0903.0910


8241. Convergence, Strong Law of Large Numbers, and Measurement Theory  
in the Language of Fuzzy Variables
Author(s): Adam Bzowski and Michal K. Urbanski

Abstract: In the paper we define the convergence of compact fuzzy sets  
as a convergence of alpha-cuts in the topology of compact subsets of a  
metric space. Furthermore we define typical convergences of fuzzy  
variables and show relations with convergence of their fuzzy  
distributions. In this context we prove a general formulation of the  
Strong Law of Large Numbers for fuzzy sets and fuzzy variables with  
Archimedean t-norms. Next we dispute a structure of fuzzy logics and  
postulate a new definition of necessity measures. Finally, we prove  
fuzzy version of the Glivenko-Cantelli theorem and use it for a  
construction of a complete fuzzy measure theory.

http://arxiv.org/abs/0903.0959


8242. Transformations des lois multivari\'ees avec queues r\'eguli\`eres
Author(s): Youri Davydov and Shuyan Liu

Abstract: Let $X$ be a random vector in $\rd$ with a regularly varying  
tail. We consider two transformations $\|X\|f(\frac{X}{\|X\|})$, $f:  
\sd\to\sd$, and $Xf(\frac{X}{\|X\|})$, $f: \sd\to \mathbb{R}_+$. Some  
sufficient conditions for preserving the property of regularity of the  
tail for this kind of transformations are given.

http://arxiv.org/abs/0903.1005


8243. Strong Convergence on Weakly Logarithmic Combinatorial Assemblies
Author(s): Eugenijus Manstavi\v{c}ius

Abstract: We deal with the random combinatorial structures called  
assemblies. By weakening the logarithmic condition which assures  
regularity of the number of components of a given order, we extend the  
notion of logarithmic assemblies. Using the author's analytic  
approach, we generalize the so-called Fundamental Lemma giving  
independent process approximation in the total variation distance of  
the component structure of an assembly. To evaluate the influence of  
strongly dependent large components, we obtain estimates of the  
appropriate conditional probabilities by unconditioned ones. These  
estimates are applied to examine additive functions defined on such a  
class of structures. Some analogs of Major's and Feller's theorems  
which concern almost sure behavior of sums of independent random  
variables are proved.

http://arxiv.org/abs/0903.1051


8244. A functional approach for random walks in random sceneries
Author(s): Cl\'ement Dombry (LMA) and Nadine Guillotin-Plantard (UCB  
and ICJ)

Abstract: A functional approach for the study of the random walks in  
random sceneries (RWRS) is proposed. Under fairly general assumptions  
on the random walk and on the random scenery, functional limit  
theorems are proved. The method allows to study separately the  
convergence of the walk and of the scenery: on the one hand, a general  
criterion for the convergence of the local time of the walk is  
provided, on the other hand, the convergence of the random measures  
associated with the scenery is studied. This functional approach is  
robust enough to recover many of the known results on RWRS as well as  
new ones, including the case of many walkers evolving in the same  
scenery.

http://arxiv.org/abs/0903.1071


8245. On the Traces of symmetric stable processes on Lipschitz domains
Author(s): Rodrigo Banuelos and Tadeusz Kulczycki and Bartlomiej Siudeja

Abstract: It is shown that the second term in the asymptotic expansion  
as $t\to 0$ of the trace of the semigroup of symmetric stable  
processes (fractional powers of the Laplacian) of order $\alpha$, for  
any $0<\alpha<2$, in Lipschitz domains is given by the surface area of  
the boundary of the domain. This brings the asymptotics for the trace  
of stable processes in domains of Euclidean space on par with those of  
Brownian motion (the Laplacian), as far as boundary smoothness is  
concerned.

http://arxiv.org/abs/0903.1198


8246. Power law Polya's urn and fractional Brownian motion
Author(s): Alan Hammond and Scott Sheffield

Abstract: We introduce a natural family of random walks on the set of  
integers that scale to fractional Brownian motion. The increments X_n  
have the property that given {X_k: k < n}, the conditional law of X_n  
is that of X_{n-k_n}, where k_n is sampled independently from a fixed  
law \mu on the positive integers. When \mu has a roughly power law  
decay (precisely, when it lies in the domain of attraction of an  
\alpha stable subordinator, for 0 < \alpha < 1/2) the walk scales to  
fractional Brownian motion with Hurst parameter \alpha + 1/2. The  
walks are easy to simulate and their increments satisfy an FKG  
inequality. In a sense we describe, they are the natural "fractional"  
analogs of simple random walk on Z.

http://arxiv.org/abs/0903.1284


8247. Stochastic ordering of classical discrete distributions
Author(s): Achim Klenke and Lutz Mattner

Abstract: For several pairs $(P,Q)$ of classical distributions on $ 
\N_0$, we show that their stochastic ordering $P\leq_{st} Q$ can be  
characterized by their extreme tail ordering equivalent to $ P(\{k_ 
\ast \})/Q(\{k_\ast\}) \le 1 \le \lim_{k\to k^\ast} P(\{k\})/Q(\{k\}) 
$, with $k_\ast$ and $k^\ast$ denoting the minimum and the supremum of  
the support of $P+Q$, and with the limit to be read as $P(\{k^\ast\})/ 
Q(\{k^\ast\})$ for $k^\ast$ finite. This includes in particular all  
pairs where $P$ and $Q$ are both binomial ($b_{n_1,p_1} \leq_{st}  
b_{n_2,p_2}$ if and only if $n_1\le n_2$ and $(1-p_1)^{n_1}\ge(1- 
p_2)^{n_2}$, or $p_1=0$), both negative binomial ($b^- 
_{r_1,p_1}\leq_{st} b^-_{r_2,p_2}$ if and only if $p_1\geq p_2$ and  
$p_1^{r_1}\geq p_2^{r_2}$), or both hypergeometric with the same  
sample size parameter. The binomial case is contained in a known  
result about Bernoulli convolutions, the other two cases appear to be  
new. The emphasis of this paper is on providing a variety of different  
methods of proofs: (i) half monotone likelihood ratios, (ii) explicit  
coupling, (iii) Markov chain comparison, (iv) analytic calculation,  
and (v) comparison of Levy measures. We give four proofs in the  
binomial case (methods (i)-(iv)) and three in the negative binomial  
case (methods (i), (iv) and (v)). The statement for hypergeometric  
distributions is proved via method (i).

http://arxiv.org/abs/0903.1361


8248. Positive definite functions and multidimensional versions of  
random variables
Author(s): Alexander Koldobsky

Abstract: We say that a random vector $X=(X_1,...,X_n)$ in $R^n$ is an  
$n$-dimensional version of a random variable $Y$ if for any $a\in R^n$  
the random variables $\sum a_iX_i$ and $\gamma(a) Y$ are identically  
distributed, where $\gamma:R^n\to [0,\infty)$ is called the standard  
of $X.$ An old problem is to characterize those functions $\gamma$  
that can appear as the standard of an $n$-dimensional version. In this  
paper, we prove the conjecture of Lisitsky that every standard must be  
the norm of a space that embeds in $L_0.$ This result is almost  
optimal, as the norm of any finite dimensional subspace of $L_p$ with  
$p\in (0,2]$ is the standard of an $n$-dimensional version ($p$-stable  
random vector) by the classical result of P.L\`evy. An equivalent  
formulation is that if a function of the form $f(\|\cdot\|_K)$ is  
positive definite on $R^n,$ where $K$ is an origin symmetric star body  
in $R^n$ and $f:R\to R$ is an even continuous function, then either  
the space $(R^n,\|\cdot\|_K)$ embeds in $L_0$ or $f$ is a constant  
function. Combined with known facts about embedding in $L_0,$ this  
result leads to several generalizations of the solution of  
Schoenberg's problem on positive definite functions.

http://arxiv.org/abs/0903.1433


8249. Smoothness of scale functions for spectrally negative Levy  
processes
Author(s): Terence Chan and Andreas Kyprianou and Mladen Savov

Abstract: Scale functions play a central role in the fluctuation  
theory of spectrally negative L\'evy processes and often appear in the  
context of martingale relations. These relations are often complicated  
to establish requiring excursion theory in favour of It\^o calculus.  
The reason for the latter is that standard It\^o calculus is only  
applicable to functions with a sufficient degree of smoothness and  
knowledge of the precise degree of smoothness of scale functions is  
seemingly incomplete. The aim of this article is to offer new results  
concerning properties of scale functions in relation to the smoothness  
of the underlying L\'evy measure. We place particular emphasis on  
spectrally negative L\'evy processes with a Gaussian component and  
processes of bounded variation. An additional motivation is the very  
intimate relation of scale functions to renewal functions of  
subordinators. The results obtained for scale functions have direct  
implications offering new results concerning the smoothness of such  
renewal functions for which there seems to be very little existing  
literature on this topic.

http://arxiv.org/abs/0903.1467


8250. Sharp thresholds for the random-cluster and Ising models
Author(s): Benjamin Graham and Geoffrey Grimmett

Abstract: A sharp-threshold theorem is proved for box-crossing  
probabilities on the square lattice. The models in question are the  
random-cluster model near the self-dual point $\psd(q)=\sqrt q/(1+ 
\sqrt q)$, the Ising model with external field, and the coloured  
random-cluster model. The principal technique is an extension of the  
influence theorem for monotonic probability measures applied to  
increasing events with no assumption of symmetry.

http://arxiv.org/abs/0903.1501


8251. Discrete approximation of stable white noise - Application to  
spatial linear filtering
Author(s): Cl\'ement Dombry (LMA)

Abstract: Motivated by the simulation of stable random fields, we  
consider the issue of discrete approximations of independently  
scattered stable noise. Two approaches are proposed: grid  
approximations available when the underlying space is $\bbR^d$ and  
shot noise approximations available on more general spaces. Limit  
theorems stating the convergence of discrete random noises to stable  
white noise are proved. These results are then applied to study moving  
average spatial random fields with heavy-tailed innovations and  
related limit theorems. A second application deals with discrete  
approximation for Brownian L\'evy motion on the sphere or on the  
euclidean space.

http://arxiv.org/abs/0903.1552


8252. Deducing the Density Hales-Jewett Theorem from an infinitary  
removal lemma
Author(s): Tim Austin (UCLA)

Abstract: We offer a new proof of Furstenberg and Katznelson's density  
version of the Hales-Jewett Theorem: For any \delta > 0 there is some  
N_0 \geq 1 such that whenever A \subseteq [k]^N with N \geq N_0 and |A| 
\geq \delta k^N, A contains a combinatorial line: that is, for some I  
\subseteq [N] nonempty and w_0 \in [k]^{[N]\setminus I} we have A  
\supseteq \{w: w|_{[N]\setminus I} = w_0, w|_I = \rm{const.}\}.  
Following Furstenberg and Katznelson, we first show that this result  
is equivalent to a `multiple recurrence' assertion for a class of  
probability measures enjoying a certain kind of stationarity. However,  
we then give a quite different proof of this latter assertion through  
a reduction to an infinitary removal lemma in the spirit of recent  
work of Tao (and also its recent re-interpretation by the author to  
give a proof of the multidimensional Szemeredi Theorem), and resting  
crucially on an observation that arose during ongoing work by a  
collaborative team of authors to give a purely finitary proof of the  
above theorem.

http://arxiv.org/abs/0903.1633


8253. The Central Limit Theorem for uniformly strong mixing measures
Author(s): Nicolai T A Haydn

Abstract: The theorem of Shannon-McMillan-Breiman states that for  
every generating partition on an ergodic system, the exponential decay  
rate of the measure of cylinder sets equals the metric entropy almost  
everywhere (provided the entropy is finite). In this paper we prove  
that the measure of cylinder sets are lognormally distributed for  
strongly mixing systems and infinite partitions and show that the rate  
of convergence is polynomial provided the fourth moment of the  
information function is finite. Also, unlike previous results by  
Ibragimov and others which only apply to finite partitions, here we do  
not require any regularity of the conditional entropy function. We  
also obtain the law of the iterated logarithm and the weak invariance  
principle for the information function.

http://arxiv.org/abs/0903.1325


8254. A Lower Bound on Arbitrary $f$--Divergences in Terms of the  
Total Variation
Author(s): Jochen Br\"ocker

Abstract: An important tool to quantify the likeness of two  
probability measures are f-divergences, which have seen widespread  
application in statistics and information theory. An example is the  
total variation, which plays an exceptional role among the f- 
divergences. It is shown that every f-divergence is bounded from below  
by a monotonous function of the total variation. Under appropriate  
regularity conditions, this function is shown to be monotonous.  
Remark: The proof of the main proposition is relatively easy, whence  
it is highly likely that the result is known. The author would be very  
grateful for any information regarding references or related work.

http://arxiv.org/abs/0903.1765


8255. Definition of evidence fusion rules on the basis of Referee  
Functions
Author(s): Frederic Dambreville (DGA/Cta/DT/Gip)

Abstract: This chapter defines a new concept and framework for  
constructing fusion rules for evidences. This framework is based on a  
referee function, which does a decisional arbitrament conditionally to  
basic decisions provided by the several sources of information. A  
simple sampling method is derived from this framework. The purpose of  
this sampling approach is to avoid the combinatorics which are  
inherent to the definition of fusion rules of evidences. This  
definition of the fusion rule by the means of a sampling process makes  
possible the construction of several rules on the basis of an  
algorithmic implementation of the referee function, instead of a  
mathematical formulation. Incidentally, it is a versatile and  
intuitive way for defining rules. The framework is implemented for  
various well known evidence rules. On the basis of this framework, new  
rules for combining evidences are proposed, which takes into account a  
consensual evaluation of the sources of information.

http://arxiv.org/abs/0903.1451


8256. Laws of Large Numbers for the Occupation Time of an Age- 
Dependent Critical Binary Branching System
Author(s): Jos\'e Alfredo L\'opez-Mimbela and Antonio Murillo Salas

Abstract: The occupation time of an age-dependent branching particle  
system in $\Rd$ is considered, where the initial population is a  
Poisson random field and the particles are subject to symmetric $\alpha 
$-stable migration, critical binary branching and random lifetimes.  
Two regimes of lifetime distributions are considered: lifetimes with  
finite mean and lifetimes belonging to the normal domain of attraction  
of a $\gamma$-stable law, $\gamma\in(0,1)$. It is shown that in  
dimensions $d>\alpha\gamma$ for the heavy-tailed lifetimes case, and  
$d>\alpha$ for finite mean lifetimes, the occupation time proccess  
satisfies a strong law of large numbers.

http://arxiv.org/abs/0903.1871


8257. Invariance principles for linear processes. Application to  
isotonic regression
Author(s): J. Dedecker and F. Merlev\`ede and M. Peligrad

Abstract: In this paper we prove maximal inequalities and study the  
functional central limit theorem for the partial sums of linear  
processes generated by dependent innovations. Due to the general  
weights these processes can exhibit long range dependence and the  
limiting distribution is a fractional Brownian motion. The proofs are  
based on new approximations by a linear process with martingale  
difference innovations. The results are then applied to study an  
estimator of the isotonic regression when the error process is a  
(possibly long range dependent) time series.

http://arxiv.org/abs/0903.1951


8258. $\kappa$-exponential models from the geometrical viewpoint
Author(s): Giovanni Pistone

Abstract: We discuss the use of Kaniadakis' $\kappa$-exponential in  
the construction of a statistical manifold modelled on Lebesgue spaces  
of real random variables. Some algebraic features of the deformed  
exponential models are considered. A chart is defined for each  
strictly positive densities; every other strictly positive density in  
a suitable neighborhood of the reference probability is represented by  
the centered $\Kln$ likelihood

http://arxiv.org/abs/0903.2012


8259. Numerical method for optimal stopping of piecewise deterministic  
Markov processes
Author(s): B. de Saporta and F. Dufour and K. Gonzalez

Abstract: We propose a numerical method to approximate the value  
function for the optimal stopping problem of a piecewise deterministic  
Markov process (PDMP). Our approach is based on quantization of the  
post jump location -- inter-arrival time Markov chain naturally  
embedded in the PDMP, and path-adapted time discretization grids. It  
allows us to derive bounds for the convergence rate of the algorithm  
and to provide a computable epsilon-optimal stopping time. The paper  
is illustrated by a numerical example.

http://arxiv.org/abs/0903.2114


8260. Heat kernel of fractional Laplacian in cones
Author(s): Krzysztof Bogdan and Tomasz Grzywny

Abstract: We give sharp estimates for the transition density of the  
isotropic stable L\'evy process killed when leaving a right circular  
cone.

http://arxiv.org/abs/0903.2269


8261. Quantitative estimates of the convergence of the empirical  
covariance matrix in Log-concave Ensembles
Author(s): Rados{\l}aw Adamczak and Alexander E. Litvak and Alain  
Pajor and Nicole Tomczak-Jaegermann

Abstract: Let $K$ be an isotropic convex body in $\R^n$. Given $ 
\eps>0$, how many independent points $X_i$ uniformly distributed on $K 
$ are needed for the empirical covariance matrix to approximate the  
identity up to $\eps$ with overwhelming probability? Our paper answers  
this question posed by Kannan, Lovasz and Simonovits. More precisely,  
let $X\in\R^n$ be a centered random vector with a log-concave  
distribution and with the identity as covariance matrix. An example of  
such a vector $X$ is a random point in an isotropic convex body. We  
show that for any $\eps>0$, there exists $C(\eps)>0$, such that if $N 
\sim C(\eps) n$ and $(X_i)_{i\le N}$ are i.i.d. copies of $X$, then $  
\Big\|\frac{1}{N}\sum_{i=1}^N X_i\otimes X_i - \Id\Big\| \le \epsilon,  
$ with probability larger than $1-\exp(-c\sqrt n)$.

http://arxiv.org/abs/0903.2323


8262. Large deviations for singular and degenerate diffusion models in  
adaptive evolution
Author(s): Nicolas Champagnat (INRIA Sophia Antipolis / INRIA  
Lorraine / IECN)

Abstract: In the course of Darwinian evolution of a population,  
punctualism is an important phenomenon whereby long periods of genetic  
stasis alternate with short periods of rapid evolutionary change. This  
paper provides a mathematical interpretation of punctualism as a  
sequence of change of basin of attraction for a diffusion model of the  
theory of adaptive dynamics. Such results rely on large deviation  
estimates for the diffusion process. The main difficulty lies in the  
fact that this diffusion process has degenerate and non-Lipschitz  
diffusion part at isolated points of the space and non-continuous  
drift part at the same points. Nevertheless, we are able to prove  
strong existence and the strong Markov property for these diffusions,  
and to give conditions under which pathwise uniqueness holds. Next, we  
prove a large deviation principle involving a rate function which has  
not the standard form of diffusions with small noise, due to the  
specific singularities of the model. Finally, this result is used to  
obtain asymptotic estimates for the time needed to exit an attracting  
domain, and to identify the points where this exit is more likely to  
occur.

http://arxiv.org/abs/0903.2345


8263. A Mean Field Approach for Optimization in Particles Systems and  
Applications
Author(s): Nicolas Gast (INRIA Rh\^one-Alpes / LIG laboratoire  
d'Informatique de Grenoble), Bruno Gaujal (INRIA Rh\^one-Alpes / LIG  
laboratoire d'Informatique de Grenoble)

Abstract: This paper investigates the limit behavior of Markov  
Decision Processes (MDPs) made of independent particles evolving in a  
common environment, when the number of particles goes to infinity. In  
the finite horizon case or with a discounted cost and an infinite  
horizon, we show that when the number of particles becomes large, the  
optimal cost of the system converges almost surely to the optimal cost  
of a discrete deterministic system (the "optimal mean field").  
Convergence also holds for optimal policies. We further provide  
insights on the speed of convergence by proving several central limits  
theorems for the cost and the state of the Markov decision process  
with explicit formulas for the variance of the limit Gaussian laws.  
Then, our framework is applied to a brokering problem in grid  
computing. The optimal policy for the limit deterministic system is  
computed explicitly. Several simulations with growing numbers of  
processors are reported. They compare the performance of the optimal  
policy of the limit system used in the finite case with classical  
policies (such as Join the Shortest Queue) by measuring its asymptotic  
gain as well as the threshold above which it starts outperforming  
classical policies.

http://arxiv.org/abs/0903.2352


8264. Random Marked Sets
Author(s): Felix Ballani and Zakhar Kabluchko and Martin Schlather

Abstract: We introduce a new class of stochastic processes which are  
defined on a random set in R^d. These processes can be seen as a link  
between random fields and marked point processes. Unlike for random  
fields, the mark covariance function need in general not be positive  
definite. This implies that in many situations the use of simple  
geostatistical methods appears to be questionable. Surprisingly, for a  
special class of processes based on Gaussian random fields, we do have  
positive definiteness for the corresponding mark covariance function  
and mark correlation function.

http://arxiv.org/abs/0903.2388


8265. Polynomial bounds in the Ergodic Theorem for positive recurrent  
one-dimensional diffusions and integrability of hitting times
Author(s): Dasha Loukianova and Oleg Loukianov and Eva Loecherbach

Abstract: Let $X$ be a one dimensional positive recurrent diffusion  
with invariant measure $\mu.$ We say that the degree of recurrence of  
$X$ is polynomial of order $p\geq 1$, if for all $x,a$ we have $ 
\E_xT_a^p<\infty$ and $\E_xT_a^{p+1}=\infty$, where $T_a$ is the  
hitting time of $a.$ We give sufficient conditions on the coefficients  
of $X$ in order to have a degree of recurrence at least equal to $p$.  
For such a diffusion, we derive non asymptotic deviation bounds $$ 
\P_{\nu} (|\frac1t\int_0^tf(X_s)ds-\mu(f)|\geq\ge)\leq K(p)\frac1{t^{p/ 
2}}\frac 1{\ge^p}A(f)^p$$ where $\nu$ is an initial distribution, $f$  
bounded or bounded and compactly supported and $A(f)=\|f\|_{\infty}$  
when $f$ is bounded and $A(f)=\mu(|f|)$ when $f$ is bounded and  
compactly supported. We also give a polynomial control of $\E_xT_a^p$  
from above and below.

http://arxiv.org/abs/0903.2405


8266. Moderate deviations for centered additive functionals of  
recurrent Harris processes having general state space
Author(s): Dasha Loukianova and Eva Loecherbach

Abstract: Let $X$ be a Harris recurrent strong Markov process with  
general Polish state space $E,$ having invariant measure $\mu .$ In  
this paper we derive non asymptotic deviation bounds for $$P_{x} (| 
\int_0^tf(X_s)ds|\geq t^{\frac12 + \eta} \ge)$$ in the positive  
recurrent case, for nice functions $f$ with $\mu (f) =0 .$ We  
generalize these bounds to the fully null-recurrent case where we  
obtain an exponential rate of convergence which is expressed in terms  
of the deterministic equivalent of the process. The main ingredient of  
the proof is Nummelin splitting in continuous time which allows to  
introduce regeneration times for the process.

http://arxiv.org/abs/0903.2408


8267. Outliers in INAR(1) models
Author(s): Matyas Barczy and Marton Ispany and Gyula Pap and Manuel  
Scotto and Maria Eduarda Silva

Abstract: In this paper the integer-valued autoregressive model of  
order one, contaminated with additive or innovational outliers is  
studied in some detail, parameter estimation is also addressed. In  
particular, the asymptotic behavior of conditional least squares (CLS)  
estimators is analyzed. We suppose that the time points of the  
outliers are known, but their sizes are unknown. It is proved that the  
CLS estimators of the offspring and innovation means are strongly  
consistent, but the CLS estimators of the sizes of the outliers are  
not strongly consistent; nevertheless, they converge to a random limit  
with probability 1. This random limit depends on the values of the  
process at the outliers' time points and on the values at the  
preceding time points and in case of additive outliers also on the  
values at the following time points. We also prove that the joint CLS  
estimator of the offspring and innovation means is asymptotically  
normal. Conditionally on the above described values of the process,  
the joint CLS estimator of the sizes of the outliers is also  
asymptotically normal.

http://arxiv.org/abs/0903.2421


8268. Non uniqueness of stationary measures for self-stabilizing  
processes
Author(s): Samuel Herrmann Julian Tugaut

Abstract: We investigate the existence of invariant measures for self- 
stabilizing diffusions. These stochastic processes represent roughly  
the behavior of some Brownian particle moving in a double-well  
landscape and attracted by its own law. This specific self-interaction  
leads to nonlinear stochastic differential equations and permits to  
point out singular phenomenons like non uniqueness of associated  
stationary measures. The existence of several invariant measures is  
essentially based on the non convex environment and requires  
generalized Laplace's method approximations.

http://arxiv.org/abs/0903.2460


8269. On the usefulness of persistent excitation in ARX adaptive  
tracking
Author(s): Bernard Bercu and Victor Vazquez

Abstract: The usefulness of persistent excitation is well-known in the  
control community. Thanks to a persistently excited adaptive tracking  
control, we show that it is possible to avoid the strong  
controllability assumption recently proposed in the multidimensional  
ARX framework. We establish the almost sure convergence for both least  
squares and weighted least squares estimators of the unknown  
parameters. A central limit theorem and a law of iterated logarithm  
are also provided. All this asymptotical analysis is related to the  
Schur complement of a suitable limiting matrix.

http://arxiv.org/abs/0903.2572


8270. A Quantitative Arrow Theorem
Author(s): Elchanan Mossel

Abstract: Arrow's Impossibility Theorem states that any constitution  
which satisfies Independence of Irrelevant Alternatives (IIA) and  
Unanimity and is not a Dictator has to be non-transitive. In this  
paper we study quantitative versions of Arrow theorem. Consider $n$  
voters who vote independently at random, each following the uniform  
distribution over the 6 rankings of 3 alternatives. Arrow's theorem  
implies that any constitution which satisfies IIA and Unanimity and is  
not a dictator has a probability of at least $6^{-n}$ for a non- 
transitive outcome. When $n$ is large, $6^{-n}$ is a very small  
probability, and the question arises if for large number of voters it  
is possible to avoid paradoxes with probability close to 1. Here we  
give a negative answer to this question by proving that for every $ 
\eps > 0$, there exists a $\delta = \delta(\eps) > 0$, which depends  
on $\eps$ only, such that for all $n$, and all constitutions on 3  
alternatives, if the constitution satisfies: The IIA condition. For  
every pair of alternatives $a,b$, the probability that the  
constitution ranks $a$ above $b$ is at least $\eps$. For every voter $i 
$, the probability that the social choice function agrees with a  
dictatorship on $i$ at most $1-\eps$. Then the probability of a non- 
transitive outcome is at least $\delta$.

http://arxiv.org/abs/0903.2574


8271. A Polynomial Number of Random Points does not Determine the  
Volume of a Convex Body
Author(s): Ronen Eldan

Abstract: We show that there is no algorithm which, provided a  
polynomial number of random points uniformly distributed over a convex  
body in R^n, can approximate the volume of the body up to a constant  
factor with high probability.

http://arxiv.org/abs/0903.2634


8272. Free point processes and free extreme values
Author(s): G. Ben Arous and V. Kargin

Abstract: We continue here the study of free extreme values begun in  
Ben Arous and Voiculescu (2006). We study the convergence of the free  
point processes associated with free extreme values to a free Poisson  
random measure (Voiculescu (1998), Barndorff-Nielsen and Thorbjornsen  
(2005)). We relate this convergence to the free extremal laws  
introduced in Ben Arous and Voiculescu (2006) and give the limit laws  
for free order statistics.

http://arxiv.org/abs/0903.2672


8273. Random Walks on Dicyclic Group
Author(s): Songzi Du

Abstract: This paper works out the rate of convergence of two  
"natural" random walks on the dicyclic group.

http://arxiv.org/abs/0903.2692


8274. The local time of a random walk on growing hypercubes
Author(s): Pierre Andreoletti (MAPMO)

Abstract: We study a random walk in a random environment (RWRE) on $ 
\Z^d$, $1 \leq d < +\infty$. The main assumptions are that  
conditionned on the environment the random walk is reversible.  
Moreover we construct our environment in such a way that the walk  
can't be trapped on a single point like in some particular RWRE but in  
some specific d-1 surfaces. These surfaces are basic surfaces with  
deterministic geometry. We prove that the local time in the  
neighborhood of these surfaces is driven by a function of the (random)  
reversible measure. As an application we get the limit law of the  
local time as a process on these surfaces.

http://arxiv.org/abs/0903.2696


8275. An explicit rough path construction for continuous paths with  
arbitrary H\"older exponent
Author(s): J. Unterberger

Abstract: We construct in this article an explicit geometric rough  
path over arbitrary $d$-dimensional paths with finite $1/\alpha$- 
variation for any $\alpha\in(0,1)$. The method is a rather  
straightforward extension of that used in a previous article  
\cite{Unt09} for multi-dimensional fractional Brownian motion. It may  
be coined as 'Fourier normal ordering' since it consists in a  
regularization obtained after permuting the order of integration in  
iterated integrals so that innermost integrals have highest Fourier  
frequencies. In doing so, there appear non-trivial tree combinatorics,  
which are best understood by using the Hopf algebra structure of  
decorated rooted trees. The algorithm of regularization follows very  
closely the BPHZ algorithm for the renormalization of Feynmann  
diagrams in quantum field theory. The new feature here (compared to  
\cite{Unt09}) is the use of Besov norms to prove H\"older continuity.

http://arxiv.org/abs/0903.2716


8276. Stationary systems of Gaussian processes
Author(s): Zakhar Kabluchko

Abstract: We describe all countable particle systems on $\mathbb R$  
which have the following three properties: independence, Gaussianity,  
and stationarity. More precisely, we consider particles on the real  
line starting at the points of a Poisson point process with intensity  
measure $m$ and moving independently of each other according to the  
law of some Gaussian process $\xi$. We describe all pairs $(m,\xi)$  
generating a stationary particle system, obtaining three families of  
examples. One of these families appeared in connection with extremes  
of independent Gaussian processes in [Z. Kabluchko, M. Schlather, L.  
de Haan, Stationary max-stable fields associated to negative definite  
functions, Ann. Probab. (2009), in press].

http://arxiv.org/abs/0903.2738


8277. Sharp thresholds for constraint satisfaction problems and  
homomorphisms
Author(s): Hamed Hatami and Michael Molloy

Abstract: We determine under which conditions certain natural models  
of random constraint satisfaction problems have sharp thresholds of  
satisfiability. These models include graph and hypergraph  
homomorphism, the $(d,k,t)$-model, and binary constraint satisfaction  
problems with domain size three.

http://arxiv.org/abs/0903.2579


8278. Exact Thresholds for Ising-Gibbs Samplers on General Graphs
Author(s): Elchanan Mossel and Allan Sly

Abstract: We establish tight results for rapid mixing of Gibbs  
Samplers for the Ferromagnetic Ising model on general graphs. We show  
that if $(d-1) \tanh \beta < 1$, then there exists a constant $C$ such  
that the discrete time mixing time of Gibbs Samplers for the  
Ferromagnetic Ising model on {\em any} graph of $n$ vertices and  
maximal degree $d$, where all interactions are bounded by $\beta$, and  
arbitrary external fields is bounded by $C n \log n$. We further show  
the when $d \tanh \beta < 1$, with high probability over the Erd\H{o}s- 
R\'enyi random graph on $n$ vertices with average degree $d$, it holds  
that the mixing time of Gibbs Samplers is $n^{1+\Theta(\frac{1}{\log  
\log n})}$. Both result are tight as it is known that the mixing time  
for random regular and Erd\H{o}s-R\'enyi random graphs is, with high  
probability, exponential in $n$ when if $(d-1) \tanh \beta > 1$ and $d  
\tanh \beta > 1$ respectively.

http://arxiv.org/abs/0903.2906


8279. A symmetric entropy bound on the non-reconstruction regime of  
Markov chains on Galton-Watson trees
Author(s): M. Formentin and C. Kuelske

Abstract: We give a criterion of the form Q(d)c(M)<1 for the non- 
reconstructability of tree-indexed q-state Markov chains obtained by  
broadcasting a signal from the root with a given transition matrix M.  
Here c(M) is an explicit constant defined in terms of a (q-1)- 
dimensional variational problem over symmetric entropies, and Q(d) is  
the expected number of offspring on the Galton-Watson tree. This  
result is equivalent to proving the extremality of the free boundary  
condition-Gibbs measure within the corresponding Gibbs-simplex. Our  
theorem holds for possibly non-reversible M and its proof is based on  
a general 'Magic Recursion Formula' for expectations of a symmetrized  
relative entropy function, which invites their use as a Lyapunov  
function. In the case of the Potts model, the present theorem  
reproduces earlier results of the authors, with a simplified proof. In  
the case of the Ising model (where the method produces the correct  
reconstruction threshold) the argument becomes similar to the approach  
of Pemantle and Peres.

http://arxiv.org/abs/0903.2962


8280. Knights, spies, games and ballot sequences
Author(s): Mark Wildon

Abstract: This paper presents a solution to the Knights and Spies  
Problem: In a room there are n people, each labelled with a unique  
number between 1 and n. A person may either be a knight or a spy.  
Knights always tell the truth, while spies may either lie or tell the  
truth, as they see fit. Each person in the room knows the identity of  
everyone else. Apart from this, all that is known is that strictly  
more knights than spies are present. Asking only questions of the  
form: `Person i, what is the identity of person j?', what is the least  
number of questions that will guarantee to find the true identities of  
all n people? The analysis of a related two-player game is critical to  
the proof. Some probabilistic aspects are also explored. The paper  
ends by presenting three open questions concerned with generalisations  
of the problem.

http://arxiv.org/abs/0903.2869


8281. Metastability in the generalized Hopfield model with finitely  
many patterns
Author(s): Mykhaylo Shkolnikov

Abstract: This paper continues the study of metastable behaviour in  
disordered mean field models initiated in [2], [3]. We consider the  
generalized Hopfield model with finitely many independent patterns $ 
\xi_1,...,\xi_P$ where the patterns have i.i.d. components and the  
components of patterns $\xi_1,...\xi_p$ have absolutely continuous  
distributions on $[-1,1]$ whereas the components of patterns $\xi_{p 
+1},...,\xi_P$ have discrete distributions on $[-1,1]$ with no atom at  
0. We show that metastable behaviour occurs if there is at least one  
pattern of each type and $2p+7

http://arxiv.org/abs/0903.3050


8282. On normal approximations to $U$-statistics
Author(s): V. Bentkus and B.-Y. Jing and W. Zhou

Abstract: Let ${X_1,...,X_n}$ be i.i.d. random observations. Let $ 
{\Sta =\Lr+\T}$ be a $U$-statistic of order $k \ge 2$, where $\Lr$ is  
a linear statistic having asymptotic normal distribution, and $\T$ is  
a stochastically smaller statistic. We show that the rate of  
convergence to normality for $\Sta$ can be simply expressed as the  
rate of convergence to normality for the linear part $\Lr$ plus a  
correction term, $(\var \T) \ln^2 (\var \T)$, under the condition ${\E  
\T^2 < \infty}$. An optimal bound without this $\log$ factor is  
obtained under a lower moment assumption ${\E |\T |^\alpha < \infty}$  
for ${\alpha<2}$. Some other related results are also obtained in the  
paper. Our results extend, refine and yield a number of related known  
results in the literature.

http://arxiv.org/abs/0903.3081


8283. Amenability of horocyclic products of percolation trees
Author(s): Florian Sobieczky

Abstract: For horocyclic products of percolation subtrees of regular  
trees, we show almost sure amenability. Under a symmetry condition  
concerning the growth of the two percolation trees, we show the  
existence of an increasing Foelner sequence (which we call strong  
amenability).

http://arxiv.org/abs/0903.3140


8284. Note on the Heat-Kernel Decay for Random Walk among Random  
Conductances with Heavy Tail
Author(s): Omar Boukhadra

Abstract: We study models of discrete-time, symmetric, $\Z^{d}$-valued  
random walks in random environments, driven by a field of i.i.d.  
random nearest-neighbor conductances $\omega_{xy}\in[0,1]$, with  
polynomial tail near 0 with exponent $\gamma>0$. We study the decay of  
the $2n$-step return probability $P_\omega^{2n}(0,0)$. For all $d 
\geq4$, we prove that the decay of $P^{2n}_\omega(0,0)$ is as close as  
we want to the standard decay $n^{-d/2}$ for large values of the  
parameter $\gamma$.

http://arxiv.org/abs/0903.3157


8285. Entropy of Random Walk Range
Author(s): Itai Benjamini and Gady Kozma and Ariel Yadin and Amir  
Yehudayoff

Abstract: We study the entropy of the set traced by an $n$-step random  
walk on $\Z^d$. We show that for $d \geq 3$, the entropy is of order $n 
$. For $d = 2$, the entropy is of order $n/\log^2 n$. These values are  
essentially governed by the size of the boundary of the trace.

http://arxiv.org/abs/0903.3179


8286. Time Allocation of a Set of Radars in a Multitarget Environment
Author(s): Emmanuel Duflos (INRIA Futurs) and Marie De Vilmorin  
(LGI2A) and Philippe Vanheeghe (INRIA Futurs)

Abstract: The question tackled here is the time allocation of radars  
in a multitarget environment. At a given time radars can only observe  
a limited part of the space; it is therefore necessary to move their  
axis with respect to time, in order to be able to explore the overall  
space facing them. Such sensors are used to detect, to locate and to  
identify targets which are in their surrounding aerial space. In this  
paper we focus on the detection schema when several targets need to be  
detected by a set of delocalized radars. This work is based on the  
modelling of the radar detection performances in terms of probability  
of detection and on the optimization of a criterion based on detection  
probabilities. This optimization leads to the derivation of allocation  
strategies and is made for several contexts and several hypotheses  
about the targets locations.

http://arxiv.org/abs/0903.3100


8287. Continuity of large closed queueing networks with bottlenecks
Author(s): Vyacheslav M. Abramov

Abstract: This paper studies a closed queueing network containing a  
hub (a state dependent queueing system with service depending on the  
number of units residing here) and $k$ satellite stations, which are  
$GI/M/1$ queueing systems. The number of units in the system, $N$, is  
assumed to be a large number. After service completion in the hub, a  
unit visit the satellite station $j$ with probability $p_j$, and after  
the service completion returns to the hub. The parameters of service  
times in the satellite stations and in the hub are proportional to $ 
\frac{1}{N}$. One of the satellite stations is assumed to be a  
bottleneck station, while others are non-bottleneck. The paper  
establishes the continuity of the queue-length processes in non- 
bottleneck satellite stations of the network when the service times in  
the hub are close in certain sense (exactly defined in the paper) to  
the exponential distribution.

http://arxiv.org/abs/0903.3259


8288. Well-posedness and ergodicity for stochastic reaction-diffusion  
equations with multiplicative Poisson noise
Author(s): Carlo Marinelli and Michael R\"ockner

Abstract: We establish well-posedness in the mild sense for a class of  
stochastic semilinear evolution equations with a polynomially growing  
quasi-monotone nonlinearity and multiplicative Poisson noise. We also  
study existence and uniqueness of invariant measures for the  
associated semigroup in the Markovian case. A key role is played by a  
new maximal inequality for stochastic convolutions in $L_p$ spaces.

http://arxiv.org/abs/0903.3299


8289. New Maximally Stable Gaussian Partitions with Discrete  
Applications
Author(s): Marcus Isaksson and Elchanan Mossel

Abstract: Gaussian noise stability results have recently played an  
important role in proving fundamental results in hardness of  
approximation in computer science and in the study of voting schemes  
in social choice. We propose two Gaussian noise stability conjectures  
and derive consequences of the conjectures in hardness of  
approximation and social choice. Both conjectures generalize  
isoperimetric results by Borell on the heat kernel. One of the  
conjectures may be also be viewed as a generalization of the "Double  
Bubble" theorem. The applications of the conjectures include an  
optimality result for majority in the context of Condorcet voting and  
a proof that the Frieze-Jerrum SDP for MAX-q-CUT achieves the optimal  
approximation factor assuming the Unique Games Conjecture. We finally  
derive a short proof of the first conjecture based on the extended  
Riesz inequality.

http://arxiv.org/abs/0903.3362


8290. Constrained Backward SDEs with Jumps: Application to Optimal  
Switching
Author(s): Romuald Elie (CREST and Ceremade) and Idris Kharroubi  
(CREST and Pma)

Abstract: In this paper, we introduce a new class of BSDE generalizing  
and offering a unifying framework to represent the constrained ones  
presented in [16] or [12] as well as the oblique reflected ones  
studied by [11] and [9]. Via a penalization procedure, we provide an  
existence and uniqueness result for this new class of so-called  
constrained BSDEs with jumps. Remarkably, these BSDEs appear to be  
very convenient to represent the solution to eventually non-Markovian  
switching problems. As a by-product, we enlarge the class of obliquely  
reflected BSDE's, allowing to represent switching problems with  
controlled underlined diffusion.

http://arxiv.org/abs/0903.3372


8291. Off-Critical SLE(2) and SLE(4): a Field Theory Approach
Author(s): Michel Bauer and Denis Bernard and Luigi Cantini

Abstract: Using their relationship with the free boson and the free  
symplectic fermion, we study the off-critical perturbation of SLE(4)  
and SLE(2) obtained by adding a mass term to the action. We compute  
the off-critical statistics of the source in the Loewner equation  
describing the two dimensional interfaces. In these two cases we show  
that ratios of massive by massless partition functions, expressible as  
ratios of regularised determinants of massive and massless Laplacians,  
are (local) martingales for the massless interfaces. The off-critical  
drifts in the stochastic source of the Loewner equation are  
proportional to the logarithmic derivative of these ratios. We also  
show that massive correlation functions are (local) martingales for  
the massive interfaces. In the case of massive SLE(4), we use this  
property to prove a factorisation of the free boson measure.

http://arxiv.org/abs/0903.1023


8292. Exactly Solvable Birth and Death Processes
Author(s): Ryu Sasaki

Abstract: Many examples of exactly solvable birth and death processes,  
a typical stationary Markov chain, are presented together with the  
explicit expressions of the transition probabilities. They are derived  
by similarity transforming exactly solvable `matrix' quantum  
mechanics, which is recently proposed by Odake and the author. The ($q 
$-)Askey-scheme of hypergeometric orthogonal polynomials of a discrete  
variable and their dual polynomials play a central role. The most  
generic solvable birth/death rates are rational functions of $q^x$ ($x 
$ being the population) corresponding to the $q$-Racah polynomial.

http://arxiv.org/abs/0903.3097


8293. The heat semigroup and Brownian motion on strip complexes
Author(s): Alexander Bendikov and Laurent Saloff-Coste and Maura  
Salvatori and and Wolfgang Woess

Abstract: We introduce the notion of strip complex. A strip complex is  
a special type of complex obtained by gluing "strips" along their  
natural boundaries according to a given graph structure. The most  
familiar example is the one dimensional complex classically associated  
with a graph, in which case the strips are simply copies of the unit  
interval (our setup actually allows for variable edge length). A  
leading key example is treebolic space, a geometric object studied in  
a number of recent articles, which arises as a horocyclic product of a  
metric tree with the hyperbolic plane. In this case, the graph is a  
regular tree, the strips are the closed unit interval times the real  
line, and each strip is equipped with the hyperbolic geometry of a  
specific strip in upper half plane. We consider natural families of  
Dirichlet forms on a general strip complex and show that the  
associated heat kernels and harmonic functions have very strong  
smoothness properties. We study questions such as essential  
selfadjointness of the underlying differential operator acting on a  
suitable space of smooth functions satisfying a Kirchoff type  
condition at points where the strip complex bifurcates. Compatibility  
with projections that arise from proper group actions is also  
considered.

http://arxiv.org/abs/0903.3518


8294. Spectrum of large random reversible Markov chains - heavy-tailed  
weights on the complete graph
Author(s): Charles Bordenave (IMT) and Pietro Caputo and Djalil Chafai  
(IMT and UPTE)

Abstract: We consider the random reversible Markov kernel K on the  
complete graph with n vertices obtained by putting i.i.d. positive  
weights of law L on the n(n+1)/2 edges of the graph and normalizing  
each weight by the corresponding row sum. We have already shown in a  
previous work that if L has finite second moment then, as n goes to  
infinity, the limiting spectral distribution of n^{1/2} K is Wigner's  
semi-circle law. In the present work, we consider the case where L  
belongs to the domain of attraction of a stable law of index a. When  
1< a <2, we show that for a suitable regularly varying sequence k_n of  
index 1 - 1/a, the limiting spectral distribution of k_n K coincides  
with the one of the random symmetric matrix of the un-normalized  
weights (i.i.d. entries). In contrast, when 0< a <1, we show that the  
empirical spectral distribution of K converges, without any rescaling,  
to a non-trivial law supported on [-1,1], whose moments are the return  
probabilities of the random walk on a suitable Poisson weighted  
infinite tree of Aldous. The limiting operator is naturally linked  
with the Poisson-Dirichlet distribution PD(a,0). The "critical" cases  
a=1 and a=2 are not solved here.

http://arxiv.org/abs/0903.3528


8295. Spectra of large random trees
Author(s): Shankar Bhamidi and Steven N. Evans and Arnab Sen

Abstract: We analyze the eigenvalues of the adjacency matrices of a  
wide variety of random trees. Using general, broadly applicable  
arguments based on the interlacing inequalities for the eigenvalues of  
a principal submatrix of a Hermitian matrix and a suitable notion of  
local weak convergence for an ensemble of random trees, we show that  
the empirical spectral distributions for each of a number of random  
tree models converge to a deterministic (model dependent) limit as the  
number of vertices goes to infinity. We conclude for ensembles such as  
the linear preferential attachment models, random recursive trees, and  
the uniform random trees that the limiting spectral distribution has a  
set of atoms that is dense in the real line. We obtain precise  
asymptotics on the mass assigned to zero by the empirical spectral  
measures via the connection with the cardinality of a maximal  
matching. Moreover, we show that the total weight of a weighted  
matching is asymptotically equivalent to a constant multiple of the  
number of vertices when the edge weights are independent, identically  
distributed, non-negative random variables with finite expected value.  
We greatly extend a celebrated result obtained by Schwenk for the  
uniform random trees by showing that, under mild conditions, with  
probability converging to one, the spectrum of a realization is shared  
by at least one other tree. For the the linear preferential attachment  
model with parameter $a > -1$, we show that the suitably rescaled $k$  
largest eigenvalues converge jointly.

http://arxiv.org/abs/0903.3589


8296. On the Structure and Representations of Max--Stable Processes
Author(s): Yizao Wang and Stilian A. Stoev

Abstract: We develop classification results for max--stable processes,  
based on their spectral representations. The structure of max--linear  
isometries and minimal spectral representations play important roles.  
We propose a general classification strategy for measurable max-- 
stable processes based on the notion of co--spectral functions. In  
particular, we discuss the spectrally continuous--discrete, the  
conservative--dissipative, and positive--null decompositions. For  
stationary max--stable processes, the latter two decompositions arise  
from connections to non--singular flows and are closely related to the  
classification of stationary sum--stable processes. The interplay  
between the introduced decompositions of max--stable processes is  
further explored. As an example, the Brown--Resnick stationary  
processes, driven by fractional Brownian motions, are shown to be  
dissipative. A result on general Gaussian processes with stationary  
increments and continuous paths is obtained.

http://arxiv.org/abs/0903.3594


8297. Adversarial Smoothed Analysis
Author(s): Felipe Cucker and Raphael Hauser and Martin Lotz

Abstract: The purpose of this note is to extend the results on uniform  
smoothed analysis of condition numbers from \cite{BuCuLo:07} to the  
case where the perturbation follows a radially symmetric probability  
distribution. In particular, we will show that the bounds derived in  
\cite{BuCuLo:07} still hold in the case of distributions whose density  
has a singularity at the center of the perturbation, which we call  
{\em adversarial}.

http://arxiv.org/abs/0903.3499


8298. Some annealed bounds for renewal pinning polymer models with  
weakly dependent disorder
Author(s): Julien Poisat (ICJ)

Abstract: The aim of this paper is to provide some estimates on the  
critical curve of a renewal pinning polymer model in the general case  
of ergodic disorder. More precisely, annealed bounds are given when  
the disorder sequence is no longer i.i.d but has still some nice  
mixing properties.

http://arxiv.org/abs/0903.3704


8299. Invariance principles for local times at the supremum of random  
walks and L\'evy processes
Author(s): Lo\"ic Chaumont (LAREMA) and Ron Arthur Doney

Abstract: We prove that when a sequence of L\'evy processes $X^{(n)}$  
or a normed sequence of random walks $S^{(n)}$ converges a.s. on the  
Skorokhod space toward a L\'evy process $X$, the sequence $L^{(n)}$ of  
local times at the supremum of $X^{(n)}$ converges uniformly on  
compact sets in probability toward the local time at the supremum of $X 
$. A consequence of this result is that the sequence of  
(quadrivariate) ladder processes (both ascending and descending)  
converges jointly in law towards the ladder processes of $X$. As an  
application, we show that in general, the sequence $S^{(n)}$  
conditioned to stay positive converges weakly, jointly with its local  
time at the future minimum, towards the corresponding functional for  
the limiting process $X$. From this we deduce an invariance principle  
for the meander which extends known results for the case of attraction  
to a stable law.

http://arxiv.org/abs/0903.3705


8300. Num\'eraire-invariant preferences in financial modeling
Author(s): Constantinos Kardaras

Abstract: We provide an axiomatic foundation for the representation of  
numeraire-invariant preferences of agents acting in a financial  
market. In a static environment, the simple axioms turn out to be  
equivalent to the following choice rule: the agent prefers one outcome  
over another if and only if the expected (under the agent's subjective  
probability) relative rate of return of the latter outcome with  
respect to the former is nonpositive. With the addition of a  
transitivity requirement, this last preference relation is extended to  
expected logarithmic utility maximization. We also discuss the  
previous in a dynamic environment, where consumption streams are the  
objects of choice. There, a novel result concerning a canonical  
representation of optional measures with unit mass enables one to  
explicitly solve the investment-consumption problem by completely  
separating the two aspects of investment and consumption. Finally, we  
give an application to the problem of optimal numeraire investment  
with a random-time horizon.

http://arxiv.org/abs/0903.3736


8301. Two-parameter stochastic calculus and Malliavin's integration-by- 
parts formula on Wiener space
Author(s): J. R. Norris

Abstract: The integration-by-parts formula discovered by Malliavin for  
the Ito map on Wiener space is proved using the two-parameter  
stochastic calculus. It is also shown that the solution of a one- 
parameter stochastic differential equation driven by a two-parameter  
semimartingale is itself a two-parameter semimartingale.

http://arxiv.org/abs/0903.3855


8302. Entropy, Invertibility and Variational Calculus of the Adapted  
Shifts on Wiener space
Author(s): Ali S\"uleyman \"Ust\"unel

Abstract: In this work we study the necessary and sufficient  
conditions for a positive random variable whose expectation under the  
Wiener measure is one, to be represented as the Radon-Nikodym  
derivative of the image of the Wiener measure under an adapted  
perturbation of identity with the help of the associated innovation  
process. We prove that the innovation conjecture holds if and only if  
the original process is almost surely invertible. We also give  
variational characterizations of the invertibility of the  
perturbations of identity and the representability of a positive  
random variable whose total mass is equal to unity. We prove in  
particular that an adapted perturbation of identity $U=I_W+u$  
satisfying the Girsanov theorem, is invertible if and only if the  
kinetic energy of $u$ is equal to the entropy of the measure induced  
with the action of $U$ on the Wiener measure $\mu$, in other words $U$  
is invertible iff $$ \half \int_W|u|_H^2d\mu=\int_W \frac{dU\mu}{d\mu} 
\log\frac{dU\mu}{d\mu}d\mu >. $$ otherwise the l.h.s. is always  
strictly greater than the r.h.s. The relations with the Monge- 
Kantorovitch measure transportation are also studied. An application  
of these results to a variational problem related to large deviations  
is also given.

http://arxiv.org/abs/0903.3891


8303. Exit time for anchored expansion
Author(s): T. Delmotte and C. Rau

Abstract: Let $(X_n)_{n\geq 0}$ be a reversible random walk on a graph  
$G$ satisfying an anchored isoperimetric inequality. We give upper  
bounds for exit time (and occupation time in transient case) by X of  
any set which contains the root. As an application, we consider random  
environments of $\Z^d$.

http://arxiv.org/abs/0903.3892


8304. Exponential rate of L_p-convergence of intrinsic martingales in  
supercritical branching random walks
Author(s): Gerold Alsmeyer and Alex Iksanov and Sergej Polotsky and  
Uwe Roesler

Abstract: Let $W_n, n\in\mn_{0}$ be an intrinsic martingale with  
almost sure limit $W$ in a supercritical branching random walk. We  
provide criteria for the $L_p$-convergence of the series $\sum_{n\ge  
0} e^{an}(W-W_n)$ for $p>1$ and $a>0$. The result may be viewed as a  
statement about the exponential rate of convergence of $\me |W-W_n|^p$  
to zero.

http://arxiv.org/abs/0903.3935


8305. Fixed point theorems on partial randomness
Author(s): Kohtaro Tadaki

Abstract: In our former work [K. Tadaki, Local Proceedings of CiE  
2008, pp.425-434, 2008], we developed a statistical mechanical  
interpretation of algorithmic information theory by introducing the  
notion of thermodynamic quantities at temperature T, such as free  
energy F(T), energy E(T), and statistical mechanical entropy S(T),  
into the theory. These quantities are real functions of real argument  
T>0. We then discovered that, in the interpretation, the temperature T  
equals to the partial randomness of the values of all these  
thermodynamic quantities, where the notion of partial randomness is a  
stronger representation of the compression rate by program-size  
complexity. Furthermore, we showed that this situation holds for the  
temperature itself as a thermodynamic quantity. Namely, the  
computability of the value of partition function Z(T) gives a  
sufficient condition for T in (0,1) to be a fixed point on partial  
randomness. In this paper, we show that the computability of each of  
all the thermodynamic quantities above gives the sufficient condition  
also. Moreover, we show that the computability of F(T) gives  
completely different fixed points from the computability of Z(T).

http://arxiv.org/abs/0903.3433


8306. Statistical RIP and Semi-Circle Distribution of Incoherent  
Dictionaries
Author(s): Shamgar Gurevich (Berkeley) and Ronny Hadani (Chicago)

Abstract: In this paper we formulate and prove a statistical version  
of the Candes-Tao restricted isometry property (SRIP for short) which  
holds in general for any incoherent dictionary which is a disjoint  
union of orthonormal bases. In addition, we prove that, under  
appropriate normalization, the eigenvalues of the associated Gram  
matrix fluctuate around 1 according to the Wigner semicircle  
distribution. The result is then applied to various dictionaries that  
arise naturally in the setting of finite harmonic analysis, giving, in  
particular, a better understanding on a remark of Applebaum-Howard- 
Searle-Calderbank concerning RIP for the Heisenberg dictionary of  
chirp like functions.

http://arxiv.org/abs/0903.3627


8307. Mass Transportation Proofs of Free Functional Inequalities, and  
Free Poincare Inequalities
Author(s): Michel Ledoux and Ionel Popescu

Abstract: This work is devoted to direct mass transportation proofs of  
families of functional inequalities in the context of one-dimensional  
free probability, avoiding random matrix approximation. The  
inequalities include the free form of the transportation, Log-Sobolev,  
HWI interpolation and Brunn-Minkowski inequalities for strictly convex  
potentials. Sharp constants and some extended versions are put  
forward. The paper also addresses two versions of free Poincar\'e  
inequalities and their interpretation in terms of spectral properties  
of Jacobi operators. The last part establishes the corresponding  
inequalities for measures on $\R_{+}$ with the reference example of  
the Marcenko-Pastur distribution.

http://arxiv.org/abs/0903.3761


8308. Dislocation measure of the fragmentation of a general L\'evy tree
Author(s): Guillaume Voisin (MAPMO)

Abstract: Given a general critical or sub-critical branching mechanism  
and its associated L\'evy continuum random tree, we consider a pruning  
procedure on this tree using a Poisson snake. It defines a  
fragmentation process on the tree. We compute the family of  
dislocation measures associated with this fragmentation. This work  
generalizes the work made for a Brownian tree [Abraham, Serlet] and  
for a tree without Brownian part [Abraham, Delmas].

http://arxiv.org/abs/0903.4024


8309. On the Stability and Ergodicity of an Adaptive Scaling  
Metropolis Algorithm
Author(s): Matti Vihola

Abstract: This paper considers the stability and ergodicity of an  
adaptive random walk Metropolis algorithm. The algorithm adjusts the  
scale of the symmetric proposal distribution continuously, based on  
the observed acceptance probability. A strong law of large numbers is  
shown to hold for functionals bounded on compact sets and growing at  
most exponentially as $\|x\|\to\infty$, assuming that the target  
density is smooth enough and has either compact support or super- 
exponentially decaying tails.

http://arxiv.org/abs/0903.4061


8310. Asymptotic exponential bounds for MLE deviation under minimal  
conditions via classical and generic chaining methods
Author(s): E. Ostrovsky and E. Rogover

Abstract: In this paper non-asymptotic exact exponential estimates are  
derived (under minimal conditions) for the tail of deviation of the  
MLE distribution in the so-called natural terms: natural function,  
natural distance, metric entropy, Banach spaces of random variables,  
contrast function, majorizing measures or, equally, generic chaining.

http://arxiv.org/abs/0903.4062


8311. Random graph asymptotics on high-dimensional tori. II. Volume,  
diameter and mixing time
Author(s): Markus Heydenreich and Remco van der Hofstad

Abstract: For critical (bond-) percolation on general high-dimensional  
torus, this paper answers the following questions: What is the  
diameter of the largest cluster? What is the mixing time of simple  
random walk on the largest cluster? The answer is the same as for  
critical Erdos-Renyi random graphs, and extends an earlier result by  
Nachmias and Peres (2008). We further improve our bound on the size of  
the largest cluster in Heydenreich and van der Hofstad (2007), and  
extend the results on the largest clusters in Borgs, Chayes, van der  
Hofstad, Slade and Spencer (2005a,b) to any finite number of the  
largest clusters. Finally, we show that any weak limit of the largest  
connected component is non-degenerate, which can be viewed as a  
significant sign of critical behavior. This result further justifies  
that the critical value defined in Borgs et al. is appropriate in our  
rather general setting of random subgraphs of high-dimensional tori.

http://arxiv.org/abs/0903.4279


8312. A note on the distribution of the maximum of a set of Poisson  
random variables
Author(s): K. M. Briggs and L. Song and T. Prellberg

Abstract: Given a set of independent Poisson random variables with  
common mean, we study the distribution of their maximum and obtain an  
accurate asymptotic formula to locate the most probable value of the  
maximum. We verify our analytic results with very precise numerical  
computations.

http://arxiv.org/abs/0903.4373


8313. Ballisticity conditions for random walk in random environment
Author(s): Alexander Drewitz and Alejandro F. Ram\'irez

Abstract: Consider a random walk in a uniformly elliptic i.i.d. random  
environment in dimensions $d\ge 2$. In 2002, Sznitman introduced for  
each $\gamma\in (0,1)$ the ballisticity conditions $(T)_\gamma$ and $ 
(T'),$ the latter being defined as the fulfilment of $(T)_\gamma$ for  
all $\gamma\in (0,1).$ He proved that $(T')$ implies ballisticity and  
that for each $\gamma\in (0.5,1),$ $(T)_\gamma$ is equivalent to $(T') 
$. It is conjectured that this equivalence holds for all $\gamma\in  
(0,1).$ Here we prove that for $\gamma\in (\gamma_d,1),$ where $ 
\gamma_d$ is a dimension dependent constant taking values in the  
interval $(0.366,0.388),$ $(T)_\gamma$ is equivalent to $(T').$ This  
is achieved by a detour along the effective criterion, the fulfilment  
of which we establish by a combination of techniques developed by  
Sznitman giving a control on the occurrence of atypical quenched exit  
distributions through boxes.

http://arxiv.org/abs/0903.4465


8314. Outlets of 2D invasion percolation and multiple-armed incipient  
infinite clusters
Author(s): Michael Damron and Artem Sapozhnikov

Abstract: We study invasion percolation in two dimensions, focusing on  
properties of the outlets of the invasion and their relation to  
critical percolation and to incipient infinite clusters (IICs). First  
we compute the exact decay rate of the distribution of both the weight  
of the kth outlet and the volume of the kth pond. Next we prove bounds  
for all moments of the distribution of the number of outlets in an  
annulus. This result leads to almost sure bounds for the number of  
outlets in a box B(2^n) and for the decay rate of the weight of the  
kth outlet to p_c. We then prove existence of multiple-armed IIC  
measures for any number of arms and for any color sequence. We use  
these measures to study the invaded region near outlets and near edges  
in the invasion backbone far from the origin.

http://arxiv.org/abs/0903.4496


8315. Existence, uniqueness and convergence of a particle  
approximation for the Adaptive Biasing Force process
Author(s): Benjamin Jourdain (CERMICS) and Tony Leli\`evre (CERMICS)  
and Rapha\"el Roux (CERMICS)

Abstract: We prove existence and uniqueness for some non linear  
stochastic differential equation used in molecular dynamics, whose non  
linearity comes from a conditional expectation term. We also introduce  
an interacting particle system in order to approximate this  
conditional expectation, providing a discretization scheme for this  
equation.

http://arxiv.org/abs/0903.4518


8316. Phase Transitions in Gravitational Allocation
Author(s): Sourav Chatterjee and Ron Peled and Yuval Peres and Dan Romik

Abstract: Given a Poisson point process of unit masses (``stars'') in  
dimension d>=3, Newtonian gravity partitions space into domains of  
attraction (cells) of equal volume. In earlier work, we showed the  
diameters of these cells have exponential tails. Here we analyze the  
quantitative geometry of the cells and show that their large  
deviations occur at the stretched-exponential scale. More precisely,  
the probability that mass exp(-R^gamma) in a cell travels distance R  
decays like exp(-R^f_d(gamma)) where we identify the functions f_d  
exactly. These functions are piecewise smooth and the discontinuities  
of f_d' represent phase transitions. In dimension d=3, the large  
deviation is due to a ``distant attracting galaxy'' but a phase  
transition occurs when f_3(gamma)=1 (at that point, the fluctuations  
due to individual stars dominate). When d>=5, the large deviation is  
due to a thin tube (a ``wormhole'') along which the star density  
increases monotonically, until the point f_d(gamma)=1 (where again  
fluctuations due to individual stars dominate). In dimension 4 we find  
a double phase transition, where the transition between low- 
dimensional behavior (attracting galaxy) and high-dimensional behavior  
(wormhole) occurs at gamma=4/3. As consequences, we determine the tail  
behavior of the distance from a star to a uniform point in its cell,  
and prove a sharp lower bound for the tail probability of the cell's  
diameter, matching our earlier upper bound.

http://arxiv.org/abs/0903.4647


8317. On the Goodness-of-Fit Testing for Ergodic Diffusion Processes
Author(s): Yury A. Kutoyants

Abstract: We consider the goodness of fit testing problem for ergodic  
diffusion processes. The basic hypothesis is supposed to be simple.  
The diffusion coefficient is known and the alternatives are described  
by the different trend coefficients. We study the asymptotic  
distribution of the Cramer-von Mises type tests based on the empirical  
distribution function and local time estimator of the invariant  
density. At particularly, we propose a transformation which makes  
these tests asymptotically distribution free. We discuss the  
modifications of this test in the case of composite basic hypothesis.

http://arxiv.org/abs/0903.4550


8318. Goodness-of-Fit Tests for Perturbed Dynamical Systems
Author(s): Yury A. Kutoyants

Abstract: We consider the goodness of fit testing problem for  
stochastic differential equation with small diffiusion coefficient.  
The basic hypothesis is always simple and it is described by the known  
trend coefficient. We propose several tests of the type of Cramer-von  
Mises, Kolmogorov-Smirnov and Chi-Square. The power functions of these  
tests we study for a special classes of close alternatives. We discuss  
the construction of the goodness of fit test based on the local time  
and the possibility of the construction of asymptotically distribution  
free tests in the case of composite basic hypothesis.

http://arxiv.org/abs/0903.4612


8319. Correlations, Scale Invariance, and the Riemann Hypothesis
Author(s): B. Holdom

Abstract: Negative correlations in the distribution of prime numbers  
are found to display a scale invariance. There are similarities and  
differences when compared to the scale invariant correlations of  
fractional Brownian motion. We conjecture that a violation of the  
Riemann hypothesis is equivalent to a breakdown of the scale invariance.

http://arxiv.org/abs/0903.2592


8320. Simple Universal Bounds for Chebyshev-Type Quadratures
Author(s): Ron Peled

Abstract: A Chebyshev-type quadrature for a probability measure sigma  
is a distribution which is uniform on n points and has the same first  
k moments as sigma. We give bounds for the smallest possible n  
required to achieve a certain degree k. In contrast to previous  
results of this type, our bounds use only simple properties of sigma  
and are thus applicable in wide generality. In particular, it is shown  
that whenever sigma has bounded density on a finite interval, n may  
increase at most exponentially with k. Examples are given illustrating  
the tightness of our bounds, and applications are given to special  
local constructions on the sphere and cylinder and to an apparently  
new result on Gaussian quadrature. We also introduce the concept of  
random Chebyshev-type quadratures, the case in which nodes are chosen  
by independent random samples from sigma. The concept is discussed and  
some preliminary results are proven. These results were recently  
applied to understand how well can a Poisson process approximate  
certain continuous distributions. We conclude with a list of open  
questions.

http://arxiv.org/abs/0903.4625


8321. On the moments of the meeting time of independent random walks  
in random environment
Author(s): Christophe Gallesco

Abstract: We consider, in the continuous time version, $\gamma$  
independent random walks on $\mathbb{Z_+}$ in random environment in  
the Sinai's regime. Let $T_\gam$ be the first meeting time of one pair  
of the $\gamma$ random walks starting at different positions. We first  
show that the tail of the quenched distribution of $T_\gamma$, after a  
suitable rescaling, converges in probability, to some functional of  
the Brownian motion. Then we compute the law of this functional.  
Eventually, we obtain results about the moments of this meeting time.  
Being $\Eo$ the quenched expectation, we show that, for almost all  
environments $\omega$, $\Eo[T_\gamma^{c}]$ is finite for $c< 
\gamma(\gamma-1)/2$ and infinite for $c>\gamma(\gamma-1)/2$.

http://arxiv.org/abs/0903.4697


8322. The continuum limit of critical random graphs
Author(s): Louigi Addario-Berry and Nicolas Broutin and Christina  
Goldschmidt

Abstract: We consider the Erdos--Renyi random graph G(n,p) inside the  
critical window, that is when p=1/n+\lambda n^{-4/3}, for some fixed  
\lambda in R. Then, as a metric space with the graph distance rescaled  
by n^{1/3}, the sequence of connected components G(n,p) converges  
towards a sequence of continuous compact metric spaces. The result  
relies on a bijection between graphs and certain marked random walks,  
and the theory of continuum random trees. Our result gives access to  
the answers to a great many questions about distances in critical  
random graphs. In particular, we deduce that the diameter of G(n,p)  
rescaled by n^{1/3} converges in distribution to an absolutely  
continuous random variable with finite mean.

http://arxiv.org/abs/0903.4730


8323. Central limit theorems for eigenvalues of deformations of Wigner  
matrices
Author(s): Mireille Capitaine and Catherine Donati-Martin (PMA) and  
Delphine F\'eral (IMB)

Abstract: In this paper, we explain the dependance of the fluctuations  
of the largest eigenvalues of a Deformed Wigner model with respect to  
the eigenvectors of the perturbation matrix. We exhibit quite general  
situations that will give rise to universality or non universality of  
the fluctuations.

http://arxiv.org/abs/0903.4740


8324. Three problems for the clairvoyant demon
Author(s): Geoffrey Grimmett

Abstract: A number of tricky problems in probability are discussed,  
having in common one or more infinite sequences of coin tosses, and a  
representation as a problem in dependent percolation. Three of these  
problems are of `Winkler' type, that is, they ask about what can be  
achieved by a clairvoyant demon.

http://arxiv.org/abs/0903.4749


8325. The Arcsine law as the limit of the internal DLA cluster  
generated by Sinai's walk
Author(s): N. Enriquez and C. Lucas and F. Simenhaus

Abstract: We identify the limit of the internal DLA cluster generated  
by Sinai's walk as the law of a functional of a Brownian motion which  
turns out to be a new interpretation of the Arcsine law.

http://arxiv.org/abs/0903.4831


8326. Recovering a time-homogeneous stock price process from perpetual  
option prices
Author(s): Erik Ekstrom and David Hobson

Abstract: It is well-known how to determine the price of perpetual  
American options if the underlying stock price is a time-homogeneous  
diffusion. In the present paper we consider the inverse problem, i.e.  
given prices of perpetual American options for different strikes we  
show how to construct a time-homogeneous model for the stock price  
which reproduces the given option prices.

http://arxiv.org/abs/0903.4833


8327. Lyapunov exponents of Green's functions for random potentials  
tending to zero
Author(s): Elena Kosygina and Thomas S. Mountford and Martin P. W.  
Zerner

Abstract: We consider quenched and annealed Lyapunov exponents for the  
Green's function of $-\Delta+\gamma V$, where the potentials $V(x), x 
\in\Z^d$, are i.i.d. nonnegative random variables and $\gamma>0$ is a  
scalar. We present a probabilistic proof that both Lyapunov exponents  
scale like $c\sqrt{\gamma}$ as $\gamma$ tends to 0. Here the constant  
$c$ is the same for the quenched as for the annealed exponent and is  
computed explicitly. This improves results obtained previously by Wei- 
Min Wang. We also consider other ways to send the potential to zero  
than multiplying it by a small number.

http://arxiv.org/abs/0903.4928


8328. Hydrodynamic limit of gradient exclusion processes with  
conductances on $\bb Z^d$
Author(s): Fabio J. Valentim

Abstract: Fix a smooth function $\Phi : [l,r] \to \bb R$, defined on  
some interval $[l,r]$ of $\bb R$, such that $0

http://arxiv.org/abs/0903.4993


8329. A Probabilistic Characterization of Random Proximity Catch  
Digraphs and the Associated Tools
Author(s): Elvan Ceyhan

Abstract: Proximity catch digraphs (PCDs) are based on proximity maps  
which yield proximity regions and are special types of proximity  
graphs. PCDs are based on the relative allocation of points from two  
or more classes in a region of interest and have applications in  
various fields. In this article, we provide auxiliary tools for and  
various characterizations of PCDs based on their probabilistic  
behavior. We consider the cases in which the vertices of the PCDs come  
from uniform and non-uniform distributions in the region of interest.  
We also provide some of the newly defined proximity maps as  
illustrative examples.

http://arxiv.org/abs/0903.5005


8330. Convergence to equilibrium of biased plane partitions
Author(s): Pietro Caputo and Fabio Martinelli and Fabio Lucio Toninelli

Abstract: We study a single-flip dynamics for the monotone surface in  
(2+1) dimensions obtained from a boxed plane partition. The surface is  
analyzed as a system of non-intersecting simple paths. When the flips  
have a non-zero bias we prove that there is a positive spectral gap  
uniformly in the boundary conditions and in the size of the system.  
Under the same assumptions, for a system of size $M$, the mixing time  
is shown to be of order $M$ up to logarithmic corrections.

http://arxiv.org/abs/0903.5079


8331. Regularity Properties for a System of Interacting Bessel Processes
Author(s): Sebastian Andres and Max-K. von Renesse

Abstract: We study the regularity of a diffusion on a simplex with  
singular drift and reflecting boundary condition which describes a  
finite system of particles on an interval with Coulomb interaction and  
reflection between nearest neighbors. As our main result we establish  
the Feller property for the process in both cases of repulsion and  
attraction. In particular the system can be started from any initial  
state, including multiple point configurations. Moreover we show that  
the process is a Euclidean semi-martingale if and only if the  
interaction is repulsive. Hence, contrary to classical results about  
reflecting Brownian motion in smooth domains, in the attractive regime  
a construction via a system of Skorokhod SDEs is impossible. Finally,  
we establish exponential heat kernel gradient estimates in the  
repulsive regime. The main proof for the attractive case is based on  
potential theory in Sobolev spaceswith Muckenhoupt weights.

http://arxiv.org/abs/0903.5085


8332. First passage percolation on random graphs with finite mean  
degrees
Author(s): S. Bhamidi and R. van der Hofstad and G. Hooghiemstra

Abstract: We study first passage percolation on the configuration  
model. Assuming that each edge has an independent exponentially  
distributed edge weight, we derive explicit distributional asymptotics  
for the minimum weight between two randomly chosen connected vertices  
in the network, as well as for the number of edges on the least weight  
path, the so-called hopcount. We analyze the configuration model with  
degree power-law exponent \tau >2, in which the degrees are assumed to  
be i.i.d. with a tail distribution which is either of power-law form  
with exponent \tau-1>1, or has even thinner tails (\tau=\infty). In  
this model, the degrees have a finite first moment, while the variance  
is finite for \tau>3, but infinite for \tau\in (2,3). We prove a  
central limit theorem for the hopcount, with asymptotically equal  
means and variances equal to \alpha\log{n}, where \alpha\in (0,1) for  
\tau\in (2,3), while \alpha>1 for \tau>3. Here n denotes the size of  
the graph. For \tau\in (2,3), it is known that the graph distance  
between two randomly chosen connected vertices is proportional to \log 
\log{n} (van der Hofstad, Hooghiemtra and Znamenski (2007), i.e.,  
distances are ultra small. Thus, the addition of edge weights causes a  
marked change in the geometry of the network. We further study the  
weight of the least weight path, and prove convergence in distribution  
of an appropriately centered version.

http://arxiv.org/abs/0903.5136


8333. Approximation of Stable-dominated Semigroups
Author(s): Pawe/l Sztonyk

Abstract: We consider Feller semigroups of operators determinated by  
systems of jumps dominated by the rotation invariant stable L\'evy  
measure. Using an approximation schema we prove the existence and  
obtain estimates of corresponding heat kernels.

http://arxiv.org/abs/0903.5294


8334. Asymptotics of The Hole Probability for Zeros of Random Entire  
Functions
Author(s): Alon Nishry

Abstract: We study the hole probability of Gaussian random entire  
functions. More specifically, we work with the flat model (the zero  
set of this function has a distribution which is invariant with  
respect to the plane isometries). A hole is the event where the  
function has no zeros in a disc of radius r. We show that the  
logarithm of the probability of the hole event decays asymptotically  
like -3/4 * e^2 * r^4 + o(r^4). We also study the behavior of the hole  
probability with other types of random coefficients.

http://arxiv.org/abs/0903.4970


8335. Maximum entropy Gaussian approximation for the number of integer  
points and volumes of polytopes
Author(s): Alexander Barvinok and John Hartigan

Abstract: We describe a maximum entropy approach for computing volumes  
and counting integer points in polyhedra. To estimate the number of  
points from a particular set X from R^n in a polyhedron P in R^n we  
construct a probability distribution on the set X by solving a certain  
entropy maximization problem such that a) the probability mass  
function is constant on the intersection of P and X and b) the  
expectation of the distribution lies in P. This allows us to apply  
Central Limit Theorem type arguments to deduce computationally  
efficient approximations for the number of integer points, volumes,  
and the number of 0-1 vectors in the polytope in a number of cases.  
Examples include polytopes of doubly stochastic matrices and  
polystochastic tensors, polytopes defined by totally unimodular  
matrices of constraints, and polytopes associated to some covering  
problems.

http://arxiv.org/abs/0903.5223


8336. Unspecified distribution in single disorder problem
Author(s): Wojciech Sarnowski and Krzysztof Szajowski

Abstract: We register a stochastic sequence affected by one disorder.  
Monitoring of the sequence is made in the circumstances when not full  
information about distributions before and after the change is  
available. The initial problem of disorder detection is transformed to  
optimal stopping of observed sequence. Formula for optimal decision  
functions is derived.

http://arxiv.org/abs/0903.5341


8337. Exact Non-Parametric Bayesian Inference on Infinite Trees
Author(s): Marcus Hutter

Abstract: Given i.i.d. data from an unknown distribution, we consider  
the problem of predicting future items. An adaptive way to estimate  
the probability density is to recursively subdivide the domain to an  
appropriate data-dependent granularity. A Bayesian would assign a data- 
independent prior probability to "subdivide", which leads to a prior  
over infinite(ly many) trees. We derive an exact, fast, and simple  
inference algorithm for such a prior, for the data evidence, the  
predictive distribution, the effective model dimension, moments, and  
other quantities. We prove asymptotic convergence and consistency  
results, and illustrate the behavior of our model on some prototypical  
functions.

http://arxiv.org/abs/0903.5342


8338. Analytic and asymptotic properties of multivariate generalized  
Linnik's probability densities
Author(s): S.C. Lim and L.P. Teo

Abstract: This paper studies the properties of the probability density  
function $p_{\alpha,\nu, n}(\mathbf{x})$ of the $n$-variate  
generalized Linnik distribution whose characteristic function $ 
\varphi_{\alpha,\nu,n}(\boldsymbol{t})$ is given by \varphi_{\alpha, 
\nu,n}(\boldsymbol{t})=\frac{1} {(1+\Vert\boldsymbol{t} 
\Vert^{\alpha})^{\nu}}, \alpha\in (0,2], \nu>0, \boldsymbol{t}\in  
\mathbb{R}^n, where $\Vert\boldsymbol{t}\Vert$ is the Euclidean norm  
of $\boldsymbol{t}\in\mathbb{R}^n$. Integral representations of  
$p_{\alpha,\nu, n}(\mathbf{x})$ are obtained and used to derive the  
asymptotic expansions of $p_{\alpha,\nu, n}(\mathbf{x})$ when $\Vert 
\mathbf{x}\Vert\to 0$ and $\Vert\mathbf{x}\Vert\to \infty$  
respectively. It is shown that under certain conditions which are  
arithmetic in nature, $p_{\alpha,\nu, n}(\mathbf{x})$ can be  
represented in terms of entire functions.

http://arxiv.org/abs/0903.5344


8339. Dutch Books and Combinatorial Games
Author(s): Peter Harremoes

Abstract: The theory of combinatorial game (like board games) and the  
theory of social games (where one looks for Nash equilibria) are  
normally considered as two separate theories. Here we shall see what  
comes out of combining the ideas. The central idea is Conway's  
observation that real numbers can be interpreted as special types of  
combinatorial games. Therefore the payoff function of a social game is  
a combinatorial game. Probability theory should be considered as a  
safety net that prevents inconsistent decisions via the Dutch Book  
Argument. This result can be extended to situations where the payoff  
function is a more general game than a real number. The main  
difference between number valued payoff and game valued payoff is that  
a probability distribution that gives non-negative mean payoff does  
not ensure that the game will be lost due to the existence of  
infinitisimal games. Also the Ramsay/de Finetti theorem on exchangable  
sequences is discussed.

http://arxiv.org/abs/0903.5429


8340. Random walks in $(\mathbb{Z}_+)^2$ with non-zero drift absorbed  
at the axes
Author(s): Irina Kurkova and Kilian Raschel

Abstract: Spatially homogeneous random walks in $(\mathbb{Z}_{+})^{2}$  
with non-zero jump probabilities at distance at most 1, with non-zero  
drift in the interior of the quadrant and absorbed when reaching the  
axes are studied. Absorption probabilities generating functions are  
obtained and the asymptotic of absorption probabilities along the axes  
is made explicit. The asymptotic of the Green functions is computed  
along all different infinite paths of states, in particular along  
those approaching the axes.

http://arxiv.org/abs/0903.5486


8341. Convergence of delay differential equations driven by fractional  
Brownian motion
Author(s): Marco Ferrante Carles Rovira

Abstract: In this note we prove an existence and uniqueness result of  
solution for stochastic differential delay equations with hereditary  
drift driven by a fractional Brownian motion with Hurst parameter $H >  
1/2$. Then, we show that, when the delay goes to zero, the solutions  
to these equations converge, almost surely and in $L^p$, to the  
solution for the equation without delay. The stochastic integral with  
respect to the fractional Brownian motion is a pathwise Riemann- 
Stieltjes integral.

http://arxiv.org/abs/0903.5498


8342. Hydrostatics and dynamical large deviations of boundary driven  
gradient symmetric exclusion
Author(s): Jonathan Farfan and Claudio Landim and Mustapha Mourragui

Abstract: We prove hydrostatics of boundary driven gradient exclusion  
processes, Fick's law and we present a simple proof of the dynamical  
large deviations principle which holds in any dimension

http://arxiv.org/abs/0903.5526


8343. Exact Tail Asymptotics of Dirichlet Distributions
Author(s): Enkelejd Hashorva

Abstract: Let X be a generalised symmetrised Dirichlet random vector  
in R^k, and let t_n be thresholds such that P{X> t_n} tends to 0 as n  
goes infinity. In this paper we derive an exact asymptotic expansion  
of P{X> t_n} assuming that the associated random radius of X has  
distribution function in the Gumbel max-domain of attraction

http://arxiv.org/abs/0904.0144


8344. Noise Correlation Bounds for Uniform Low Degree Functions
Author(s): Per Austrin and Elchanan Mossel

Abstract: We study correlation bounds under pairwise independent  
distributions for functions with no large Fourier coefficients.  
Functions in which all Fourier coefficients are bounded by $\delta$  
are called $\delta$-{\em uniform}. The search for such bounds is  
motivated by their potential applicability to hardness of  
approximation, derandomization, and additive combinatorics. In our  
main result we show that $\E[f_1(X_1^1,...,X_1^n) ...  
f_k(X_k^1,...,X_k^n)]$ is close to 0 under the following assumptions:  
1. The vectors $\{(X_1^j,...,X_k^j) : 1 \leq j \leq n\}$ are i.i.d,  
and for each $j$ the vector $(X_1^j,...,X_k^j)$ has a pairwise  
independent distribution. 2. The functions $f_i$ are uniform. 3. The  
functions $f_i$ are of low degree. We compare our result with recent  
results by the second author for low influence functions and to recent  
results in additive combinatorics using the Gowers norm. Our proofs  
extend some techniques from the theory of hypercontractivity to a  
multilinear setup.

http://arxiv.org/abs/0904.0157


8345. Pointwise ergodic theorems with rate and application to limit  
theorems for stationary processes
Author(s): Christophe Cuny

Abstract: We obtain pointwise ergodic theorems with rate under  
conditions expressed in terms of the convergence of series involving $ 
\|\sum_{k=1} ^nf\circ \theta^k\|_2$, improving previous results. Then,  
using known results on martingale approximation, we obtain some LIL  
for stationary ergodic processes and quenched central limit theorems  
for functional of Markov chains. The proofs are based on the use of  
the spectral theorem and, on a recent work of Zhao-Woodroofe extending  
a method of Derriennic-Lin.

http://arxiv.org/abs/0904.0185


8346. A new model for evolution in a spatial continuum
Author(s): N.H. Barton and A.M. Etheridge and A. Veber

Abstract: We introduce a new model for populations evolving in a  
spatial continuum. This model can be thought of as a spatial version  
of the Lambda-Fleming-Viot process. It explicitly incorporates both  
small scale reproduction events and large scale extinction- 
recolonisation events. The lineages ancestral to a sample from a  
population evolving according to this model can be described in terms  
of a spatial version of the Lambda-coalescent. Using a technique of  
Evans(1997), we prove existence and uniqueness in law for the model.  
We then investigate the asymptotic behaviour of the genealogy of a  
finite number of individuals sampled uniformly at random (or more  
generally `far enough apart') from a two-dimensional torus of side L  
as L tends to infinity. Under appropriate conditions (and on a  
suitable timescale), we can obtain as limiting genealogical processes  
a Kingman coalescent, a more general Lambda-coalescent or a system of  
coalescing Brownian motions (with a non-local coalescence mechanism).

http://arxiv.org/abs/0904.0210


8347. Percolation and Connectivity in AB Random Geometric Graphs
Author(s): Srikanth K. Iyer (INRIA Rocquencourt) and D. Yogeshwaran  
(INRIA Rocquencourt)

Abstract: We study a generalization to the continuum of the $AB$  
percolation model on discrete lattices. Let $\Pl,\Pm$ be independent  
Poisson point processes in $\mR^d$, $d \geq 2,$ of intensities $ 
\lambda, \mu$ respectively. The $AB$ random geometric graph $G(\lam,  
\mu, r)$ is a graph whose vertex set is $\Pl$ with edges between any  
two points $X_i, X_j \in \Pl$ provided there exists a $Y \in \Pm$ such  
that $|X_k - Y| \leq r$, $k=i, j$. We investigate percolation and  
connectivity in $AB$ random geometric graphs.

http://arxiv.org/abs/0904.0223


8348. Self-similarity and random walks
Author(s): Vadim A. Kaimanovich

Abstract: This is an introductory level survey of some topics from a  
new branch of fractal analysis -- the theory of self-similar groups.  
We discuss recent works on random walks on self-similar groups and  
their applications to the problem of amenability for these groups.

http://arxiv.org/abs/0904.0047


8349. Step Size in Stein's Method of Exchangeable Pairs
Author(s): Nathan Ross

Abstract: Stein's method of exchangeable pairs is examined through  
five examples in relation to Poisson and normal distribution  
approximation. In particular, in the case where the exchangeable pair  
is constructed from a reversible Markov chain, we analyze how  
modifying the step size of the chain in a natural way affects the  
error term in the approximation acquired through Stein's method. It  
has been noted for the normal approximation that smaller step sizes  
may yield better bounds, and we obtain the first rigorous results that  
verify this intuition. For the examples associated to the normal  
distribution, the bound on the error is expressed in terms of the  
spectrum of the underlying chain, a characteristic of the chain  
related to convergence rates. The Poisson approximation using  
exchangeable pairs is less studied than the normal, but in the  
examples presented here the same principles hold.

http://arxiv.org/abs/0904.0284


8350. Backward stochastic dynamics on a filtered probability space
Author(s): G. Liang and T. Lyons and Z. Qian (Mathematical Institute  
and University of Oxford) (Oxford-Man Institute, University of Oxford)

Abstract: We consider the following backward stochastic dynamics based  
on a general filtered probability space (\Omega, F, {F_t}_{t\geq  
0},P): dY_t=-f_0(t,Y_t,L(M)_t)dt-\sum_{i=1}^{N}f_i(t,Y_t)dB_t^i+dM_t,  
Y_T=\xi \in F_T where B is an N-dimensional Brownian motion as given,  
and M, a correction term, is a square-integrable martingale to be  
determined. Under adapteness constraints on Y, we prove that the  
equation admits a solution pair (Y,M) which is unique in the sense of  
strict solutions to be introduced in the main text. The martingale  
representation is not required, and in order to prove the existence  
and uniqueness, we establish the existence and uniqueness of a  
functional differential equation, in a form V=\mathbb{L}(V), where  
\mathbb{L} is a non-linear functional. Finally we indicate a  
connection between the backward stochastic equations discussed here  
and a class of non-linear PDE, namely semi-linear parabolic PDE with  
non-local integral term.

http://arxiv.org/abs/0904.0377


8351. Hamilton cycles in 3-out
Author(s): Tom Bohman and Alan Frieze

Abstract: Let G_{\rm 3-out} denote the random graph on vertex set [n]  
in which each vertex chooses 3 neighbors uniformly at random. Note  
that G_{\rm 3-out} has minimum degree 3 and average degree 6. We prove  
that the probability that G_{\rm 3-out} is Hamiltonian goes to 1 as n  
tends to infinity.

http://arxiv.org/abs/0904.0431


8352. Ces\'aro summation for random fields
Author(s): Allan Gut (Uppsala University) and Ulrich Stadtmueller (Ulm  
University)

Abstract: Various methods of summation for divergent series of real  
numbers have been generalized to analogous results for sums of iid  
random variables. The natural extension of results corresponding to Ces 
\`aro summation amounts to proving almost sure convergence of the Ces 
\`aro means. In the present paper we extend such results as well as  
weak laws and results on complete convergence to random fields, more  
specifically to random variables indexed by $\mathbb{Z}_+^2$, the  
positive two-dimensional integer lattice points.

http://arxiv.org/abs/0904.0538


8353. A Large Deviation Principle for Martingales over Brownian  
Filtration
Author(s): Z. Qian and C. Xu (Mathematical Institute and University of  
Oxford)

Abstract: In this article we establish a large deviation principle for  
the family {\nu_{\epsilon}:\epsilon \in (0,1)} of distributions of the  
scaled stochastic processes {P_{-\log\sqrt{\epsilon}}Z_t}_{t\leq 1},  
where (Z_t)_{t\in \lbrack 0,1]} is a square-integrable martingale over  
Brownian filtration and (P_t)_{t\geq 0} is the Ornstein-Uhlenbeck  
semigroup. The rate function is identified as well in terms of the  
Wiener-It\^{o} chaos decomposition of the terminal value Z_{1}. The  
result is established by developing a continuity theorem for large  
deviations, together with two essential tools, the hypercontractivity  
of the Ornstein-Uhlenbeck semigroup and Lyons' continuity theorem for  
solutions of Stratonovich type stochastic differential equations.

http://arxiv.org/abs/0904.0547


8354. A new approach to LIBOR modeling
Author(s): Martin Keller-Ressel and Antonis Papapantoleon and Josef  
Teichmann

Abstract: We provide a general and flexible approach to LIBOR modeling  
based on the class of affine factor processes. Our approach respects  
the basic economic requirement that LIBOR rates are non-negative, and  
the basic requirement from mathematical finance that LIBOR rates are  
analytically tractable martingales with respect to their own forward  
measure. Additionally, and most importantly, our approach also leads  
to analytically tractable expressions of multi-LIBOR payoffs. This  
approach unifies therefore the advantages of well-known forward price  
models with those of classical LIBOR rate models. Several examples are  
added and prototypical volatility smiles are shown. We believe that  
the CIR-process based LIBOR model might be of particular interest for  
applications, since closed form valuation formulas for caps and  
swaptions are derived.

http://arxiv.org/abs/0904.0555


8355. Limit conditional distributions for bivariate vectors with polar  
representation
Author(s): Anne-Laure Foug\`eres (ICJ) and Philippe Soulier (MODAL'X)

Abstract: We investigate conditions for the existence of the limiting  
conditional distribution of a bivariate random vector when one  
component becomes large. We revisit the existing literature on the  
topic, and present some new sufficient conditions. We concentrate on  
the case where the conditioning variable belongs to the maximum domain  
of attraction of the Gumbel law, and we study geometric conditions on  
the joint distribution of the vector. We show that these conditions  
are of a local nature and imply asymptotic independence when both  
variables belong to the domain of attraction of an extreme value  
distribution. The new model we introduce can also be useful for  
simulations.

http://arxiv.org/abs/0904.0580


8356. A limit theorem for trees of alleles in branching processes with  
rare neutral mutations
Author(s): Jean Bertoin (DMA and Pma)

Abstract: We are interested in the genealogical structure of alleles  
for a Bienaym\'e-Galton-Watson branching process with neutral  
mutations (infinite alleles model), in the situation where the initial  
population is large and the mutation rate small. We shall establish  
that for an appropriate regime, the process of the sizes of the  
allelic sub-families converges in distribution to a certain continuous  
state branching process (i.e. a Jirina process) in discrete time. It 
\^o's excursion theory and the L\'eevy-It\^o decomposition of  
subordinators provide fundamental insights for the results.

http://arxiv.org/abs/0904.0581


8357. Prime chains and Pratt trees
Author(s): Kevin Ford and Sergei V. Konyagin and Florian Luca

Abstract: We study the distribution of prime chains, which are  
sequences p_1,...,p_k of primes for which p_{j+1}\equiv 1\pmod{p_j}  
for each j. We first give conditional upper bounds on the length of  
Cunningham chains, chains with p_{j+1}=2p_j+1 for each j. We give  
estimates for the number of chains with p_k\le x (k variable), and the  
number of chains with p_1=p and p_k \le px. The majority of the paper  
concerns the distribution of H(p), the length of the longest chain  
with p_k=p, which is also the height of the Pratt tree for p. We show  
H(p)\ge c\log\log p and H(p)\le (\log p)^{1-c'} for almost all p, with  
c,c' explicit positive constants. We can take, for any \epsilon>0, c=e- 
\epsilon assuming the Elliott-Halberstam conjecture. A stochastic  
model of the Pratt tree, based on a branching random walk, is  
introduced and analyzed. The model suggests that for most p, H(p)  
stays very close to e \log\log p.

http://arxiv.org/abs/0904.0473


8358. Analysis of the market weights under the Volatility-Stabilized  
Market models
Author(s): Soumik Pal

Abstract: We derive the joint density of market weights, at fixed  
times and suitable stopping times, of the Volatility-stabilized market  
models introduced by Fernholz & Karatzas in 2005. The key argument  
involves computing the exit density of a collection of independent  
Bessel-square processes of different dimensions from the unit simplex  
in n-dimension. As a side result, we furnish a novel proof of the  
transition density function of the multi-allele Wright-Fisher model  
which was originally derived by Griffiths by orthogonal series  
expansion.

http://arxiv.org/abs/0904.0656


8359. Optimal Multi-Modes Switching Problem in Infinite Horizon
Author(s): Brahim El Asri

Abstract: This paper studies the problem of the deterministic version  
of the Verification Theorem for the optimal m-states switching in  
infinite horizon under Markovian framework with arbitrary switching  
cost functions. The problem is formulated as an extended impulse  
control problem and solved by means of probabilistic tools such as the  
Snell envelop of processes and reflected backward stochastic  
differential equations. A viscosity solutions approach is employed to  
carry out a finne analysis on the associated system of m variational  
inequalities with inter-connected obstacles. We show that the vector  
of value functions of the optimal problem is the unique viscosity  
solution to the system. This problem is in relation with the valuation  
of firms in a financial market.

http://arxiv.org/abs/0904.0707


8360. On the reversal of radial SLE, I: Commutation Relations in Annuli
Author(s): Dapeng Zhan

Abstract: We aim at finding the reversal of radial SLE and proving the  
reversibility of whole-plane SLE. For this purpose, we define annulus  
SLE$(\kappa,\Lambda)$ processes in doubly connected domains with one  
marked boundary point. We derive some partial differential equation  
for $\Lambda$, which is sufficient for the annulus SLE$(\kappa,\Lambda) 
$ process to satisfy commutation relation. If $\Lambda$ satisfies this  
PDE, then using a coupling technique, we are able to construct a  
global commutation coupling of two annulus SLE$(\kappa,\Lambda)$  
processes. If more conditions are satisfied, the coupling exists in  
the degenerate case, which becomes a coupling of two whole-plane SLE$_ 
\kappa$ processes. The reversibility of whole-plane SLE$_\kappa$  
follows from this coupling together with the assumption that such  
annulus SLE$(\kappa,\Lambda)$ trace ends at the marked point. We then  
conclude that the limit of such annulus SLE$(\kappa,\Lambda)$ trace is  
the reversal of radial SLE$_\kappa$ trace. In the end, we derive some  
particular solutions to the PDE for $\Lambda$.

http://arxiv.org/abs/0904.0808


8361. A Central Limit Theorem and its Applications to Multicolor  
Randomly Reinforced Urns
Author(s): Patrizia Berti and Irene Crimaldi and Luca Pratelli and  
Pietro Rigo

Abstract: We give a central limit theorem, which has applications to  
Bayesian statistics and urn problems. The latter are investigated, by  
paying special attention to multicolor randomly reinforced generalized  
Polya urns.

http://arxiv.org/abs/0904.0932


8362. Regularity of Intersection Local Times of Fractional Brownian  
Motions
Author(s): Dongsheng Wu (University of Alabama in Huntsville) and  
Yimin Xiao (Michigan State University)

Abstract: Let $B^{\alpha_i}$ be an $(N_i,d)$-fractional Brownian  
motion with Hurst index ${\alpha_i}$ ($i=1,2$), and let $B^{\alpha_1}$  
and $B^{\alpha_2}$ be independent. We prove that, if $\frac{N_1} 
{\alpha_1}+\frac{N_2}{\alpha_2}>d$, then the intersection local times  
of $B^{\alpha_1}$ and $B^{\alpha_2}$ exist, and have a continuous  
version. We also establish H\"{o}lder conditions for the intersection  
local times and determine the Hausdorff and packing dimensions of the  
sets of intersection times and intersection points. One of the main  
motivations of this paper is from the results of Nualart and Ortiz- 
Latorre ({\it J. Theor. Probab.} {\bf 20} (2007)), where the existence  
of the intersection local times of two independent $(1,d)$-fractional  
Brownian motions with the same Hurst index was studied by using a  
different method. Our results show that anisotropy brings subtle  
differences into the analytic properties of the intersection local  
times as well as rich geometric structures into the sets of  
intersection times and intersection points.

http://arxiv.org/abs/0904.0949


8363. Exact Asymptotics of Bivariate Scale Mixture Distributions
Author(s): Enkelejd Hashorva

Abstract: Let (RU_1, R U_2) be a given bivariate scale mixture random  
vector, with R>0 being independent of the bivariate random vector  
(U_1,U_2). In this paper we derive exact asymptotic expansions of the  
tail probability P{RU_1> x, RU_2> ax}, a \in (0,1] as x tends infintiy  
assuming that R has distribution function in the Gumbel max-domain of  
attraction and (U_1,U_2) has a specific tail behaviour around some  
absorbing point. As a special case of our results we retrieve the  
exact asymptotic behaviour of bivariate polar random vectors. We apply  
our results to investigate the asymptotic independence and the  
asymptotic behaviour of conditional excess for bivariate scale mixture  
distributions.

http://arxiv.org/abs/0904.0966


8364. On the stability of call/put option's prices in incomplete  
models under statistical estimations
Author(s): L. Vostrikova

Abstract: In exponential semi-martingale setting for risky asset we  
estimate the difference of prices of options when initial physical  
measure $P$ and corresponding martingale measure $Q$ change to $ 
\tilde{P}$ and $\tilde{Q}$ respectively. Then, we estimate $L_1$- 
distance of option's prices for corresponding parametric models with  
known and estimated parameters. The results are applied to exponential  
Levy models with special choice of martingale measure as Esscher  
measure, minimal entropy measure and $f^q$-minimal martingale measure.  
We illustrate our results by considering GMY and CGMY models.

http://arxiv.org/abs/0904.0984


8365. Breaking through the Thresholds: an Analysis for Iterative  
Reweighted $\ell_1$ Minimization via the Grassmann Angle Framework
Author(s): Weiyu Xu and M. Amin Khajehnejad and Salman Avestimehr and  
Babak Hassibi

Abstract: It is now well understood that $\ell_1$ minimization  
algorithm is able to recover sparse signals from incomplete  
measurements [2], [1], [3] and sharp recoverable sparsity thresholds  
have also been obtained for the $\ell_1$ minimization algorithm.  
However, even though iterative reweighted $\ell_1$ minimization  
algorithms or related algorithms have been empirically observed to  
boost the recoverable sparsity thresholds for certain types of  
signals, no rigorous theoretical results have been established to  
prove this fact. In this paper, we try to provide a theoretical  
foundation for analyzing the iterative reweighted $\ell_1$ algorithms.  
In particular, we show that for a nontrivial class of signals, the  
iterative reweighted $\ell_1$ minimization can indeed deliver  
recoverable sparsity thresholds larger than that given in [1], [3].  
Our results are based on a high-dimensional geometrical analysis  
(Grassmann angle analysis) of the null-space characterization for $ 
\ell_1$ minimization and weighted $\ell_1$ minimization algorithms.

http://arxiv.org/abs/0904.0994


8366. Thin Partitions: Isoperimetric Inequalities and Sampling  
Algorithms for some Nonconvex Families
Author(s): Karthekeyan Chandrasekaran and Daniel Dadush and Santosh  
Vempala

Abstract: Star-shaped bodies are an important nonconvex generalization  
of convex bodies (e.g., linear programming with violations). Here we  
present an efficient algorithm for sampling a given star-shaped body.  
The complexity of the algorithm grows polynomially in the dimension  
and inverse polynomially in the fraction of the volume taken up by the  
kernel of the star-shaped body. The analysis is based on a new  
isoperimetric inequality. Our main technical contribution is a tool  
for proving such inequalities when the domain is not convex. As a  
consequence, we obtain a polynomial algorithm for computing the volume  
of such a set as well. In contrast, linear optimization over star- 
shaped sets is NP-hard.

http://arxiv.org/abs/0904.0583


8367. Comportement asymptotique des polyn\^omes orthogonaux associ\'es  
\`a un poids ayant un z\'ero d'ordre fractionnaire sur le cercle.  
Applications aux valeurs propres d'une classe de matrices al\'eatoires  
unitaires
Author(s): Philippe Rambour (LM-Orsay) and Abdellatif Seghier (LM-Orsay)

Abstract: Asymptotic behavior of orthogonal polynomials on the circle,  
with respect to a weight having a fractional zero on the torus.  
Applications to the eigenvalues of certain unitary random matrices.  
This paper is devoted to the orthogonal polynomial on the circle, with  
respect to a weight of type $ f=(1-\cos \theta )^\alpha c$ where $c$  
is a sufficiently smooth function and $\alpha \in ]-{1/2}, {1/2}[$. We  
obtain an asymptotic expansion of the coefficients of this polynomial  
and of $\Phi^{(p)}_{N}(1)$ for all integer $p$. These results allow us  
to obtain an asymptotic expansion of the associated Christofel-Darboux  
kernel, and to compute the distribution of the eigenvalues of a family  
of random unitary matrices. The proof of the resuts related with the  
orthogonal polynomials are essentialy based on the inversion of  
Toeplitz matice associated to the symbol $f$.

http://arxiv.org/abs/0904.0777


8368. Strong law of large numbers on graphs and groups with  
applications -- I
Author(s): Natalia Mosina and Alexander Ushakov

Abstract: We introduce the notion of the mean-set (expectation) of a  
graph-(group-)valued random element $\xi$ and prove a generalization  
of the strong law of large numbers on graphs and groups. Furthermore,  
we prove an analogue of the classical Chebyshev's inequality for $\xi 
$. We show that our generalized law of large numbers, as a new  
theoretical tool, provides a framework for practical applications;  
namely, it has implications for cryptanalysis of group-based  
authentication protocols. In addition, we prove several results about  
configurations of mean-sets in graphs and their applications. In  
particular, we discuss computational problems and methods of computing  
of mean-sets in practice and propose an algorithm for such computation.

http://arxiv.org/abs/0904.1005


8369. Invariance Principles for Homogeneous Sums: Universality of  
Gaussian Wiener Chaos
Author(s): Ivan Nourdin (PMA) and Giovanni Peccati (MODAL'X) and  
Gesine Reinert

Abstract: We compute explicit bounds in the normal and chi-square  
approximations of multilinear homogenous sums (of arbitrary order) of  
general centered independent random variables with unit variance. Our  
techniques combine an invariance principle by Mossel, O'Donnell and  
Oleszkiewicz with a refinement of some recent results by Nourdin and  
Peccati, about the approximation of laws of random variables belonging  
to a fixed (Gaussian) Wiener chaos. In particular, we show that  
chaotic random variables enjoy the following form of  
\textsl{universality}: (a) the normal and chi-square approximations of  
any homogenous sum can be completely characterized and assessed by  
first switching to its Wiener chaos counterpart, and (b) the simple  
upper bounds and convergence criteria available on the Wiener chaos  
extend almost verbatim to the class of homogeneous sums. These results  
partially rely on the notion of "low influences" for functions defined  
on product spaces, and provide a generalization of central and non- 
central limit theorems proved by Nourdin, Nualart and Peccati. They  
also imply a further drastic simplification of the method of moments  
and cumulants -- as applied to the proof of probabilistic limit  
theorems -- and yield substantial generalizations, new proofs and new  
insights into some classic findings by de Jong and Rotar'. Our tools  
involve the use of Malliavin calculus, and of both the Stein's method  
and the Lindeberg invariance principle for probabilistic approximations.

http://arxiv.org/abs/0904.1153


8370. Martingales and Rates of Presence in Homogeneous Fragmentations
Author(s): Nathalie Krell (MAP5) and Alain Rouault (LM-Versailles)

Abstract: In mass-conservative homogeneous fragmentations, sizes of  
the fragments decrease at {\bf asymptotic} exponential rates. Like in  
branching processes, two situations occur: either the number of such  
fragments is exponentially growing - the rate is effective -, or the  
probability of presence of such fragments is exponentially decreasing.  
In a recent paper, N. Krell considers fragments whose sizes decrease  
at {\bf exact} exponential rates. In this new setting, she  
characterizes the effective rates and studies Hausdorff dimension. The  
present paper carries out a detailed analysis of this model and focus  
on presence probabilities, using the spine method and a suitable  
martingale. For the sake of completeness, we compare our results with  
results and methods of the classical model.

http://arxiv.org/abs/0904.1167


8371. Space-time duality for fractional diffusion
Author(s): Boris Baeumer and Mark M. Meerschaert and Erkan Nane

Abstract: Zolotarev proved a duality result that relates stable  
densities with different indices. In this paper, we show how Zolotarev  
duality leads to some interesting results on fractional diffusion.  
Fractional diffusion equations employ fractional derivatives in place  
of the usual integer order derivatives. They govern scaling limits of  
random walk models, with power law jumps leading to fractional  
derivatives in space, and power law waiting times between the jumps  
leading to fractional derivatives in time. The limit process is a  
stable L\'evy motion that models the jumps, subordinated to an inverse  
stable process that models the waiting times. Using duality, we relate  
the density of a spectrally negative stable process with index $1< 
\alpha<2$ to the density of the hitting time of a stable subordinator  
with index $1/\alpha$, and thereby unify some recent results in the  
literature. These results also provide a concrete interpretation of  
Zolotarev duality in terms of the fractional diffusion model.

http://arxiv.org/abs/0904.1176


8372. Optimal Holder exponent for the SLE path
Author(s): Fredrik Johansson and Gregory F. Lawler

Abstract: We prove an upper bound on the optimal H\"older exponent for  
the chordal SLE path parameterized by capacity and thereby establish  
the optimal exponent as conjectured by J. Lind. We also give a new  
proof of the lower bound. Our proofs are based on the sharp estimates  
of moments of the derivative of the inverse map. In particular, we  
improve an estimate of the second author.

http://arxiv.org/abs/0904.1180


8373. Curvature, concentration, and error estimates for Markov chain  
Monte Carlo
Author(s): Ald\'eric Joulin and Yann Ollivier

Abstract: Under a "positive curvature" assumption expressing a kind of  
metric ergodicity, we provide explicit non-asymptotic estimates for  
the rate of convergence of empirical means of Markov chains, together  
with a Gaussian or exponential control on the deviations of empirical  
means.

http://arxiv.org/abs/0904.1312


8374. Limit theorems for nonlinear functionals of Volterra processes  
via white noise analysis
Author(s): S\'ebastien Darses and Ivan Nourdin (PMA) and David Nualart

Abstract: By means of white noise analysis, we prove some limit  
theorems for nonlinear functionals of a given Volterra process. In  
particular, our results apply to fractional Brownian motion (fBm), and  
should be compared with the classical convergence results of the  
eighties by Breuer, Dobrushin, Giraitis, Major, Surgailis and Taqqu,  
as well as the recent advances concerning the construction of a L\'evy  
area for fBm by Coutin, Qian and Unterberger

http://arxiv.org/abs/0904.1401


8375. The Engel algorithm for absorbing Markov chains
Author(s): J. Laurie Snell

Abstract: In this module, suitable for use in an introductory  
probability course, we present Engel's chip-moving algorithm for  
finding the basic descriptive quantities for an absorbing Markov  
chain, and prove that it works. The tricky part of the proof involves  
showing that the initial distribution of chips recurs. At the time of  
writing (circa 1979) no published proof of this was available, though  
Engel had stated that such a proof had been found by L. Scheller.

http://arxiv.org/abs/0904.1413


8376. Asymptotic Normality of Statistics on Permutation Tableaux
Author(s): Pawel Hitczenko and Svante Janson

Abstract: In this paper we use a probabilistic approach to derive the  
expressions for the characteristic functions of basic statistics  
defined on permutation tableaux. Since our expressions are exact, we  
can identify the distributions of basic statistics (like the number of  
unrestricted rows, the number of rows, and the number of 1s in the  
first row) exactly. In all three cases the distributions are known to  
be asymptotically normal after a suitable normalization. We also  
establish the asymptotic normality of the number of superfluous 1s.  
The latter result relies on a bijection between permutation tableaux  
and permutations and on a rather general sufficient condition for the  
central limit theorem for the sums of random variables in terms of  
dependency graph of the summands.

http://arxiv.org/abs/0904.1222


8377. Risk-averse asymptotics for reservation prices
Author(s): Laurence Carassus (PMA) and Miklos Rasonyi (MTA-SZTAKI)

Abstract: An investor's risk aversion is assumed to tend to infinity.  
In a fairly general setting, we present conditions ensuring that the  
respective utility indifference prices of a given contingent claim  
converge to its super replication price.

http://arxiv.org/abs/0904.1480


8378. Interacting Poisson processes and applications to neuronal  
modeling
Author(s): Stefano Cardanobile and Stefan Rotter

Abstract: A family of interacting Poisson processes is introduced.  
Events from a process are assumed to act multiplicatively on the rate  
of the processes to which they are connected. The family can be seen  
as a multivariate Cox process with both excitatory and inhibitory  
connections. The expected intensities of the process are approximated  
by a differential system of first-order and the stability of the  
solutions of this equation is studied. We discuss the applications in  
the neuroscience and the relations to the generalised linear model  
used for the analysis of spike trains.

http://arxiv.org/abs/0904.1505


8379. Kingman's coalescent and Brownian motion
Author(s): J. Berestycki and N. Berestycki

Abstract: We describe a simple construction of Kingman's coalescent in  
terms of a Brownian excursion. This construction is closely related  
to, and sheds some new light on, earlier work by Aldous and Warren.  
Our approach also yields some new results: for instance, we obtain the  
full multifractal spectrum of Kingman's coalescent. This complements  
earlier work on Beta-coalescents by the authors and Schweinsberg.  
Surprisingly, the thick part of the spectrum is not obtained by taking  
the limit as $\alpha \to 2$ in the result for Beta-coalescents  
mentioned above. Other analogies and differences between the case of  
Beta-coalescents and Kingman's coalescent are discussed.

http://arxiv.org/abs/0904.1526


8380. On the Numerical Evaluation of Distributions in Random Matrix  
Theory: A Review with an Invitation to Experimental Mathematics
Author(s): Folkmar Bornemann

Abstract: In this paper we review and compare the numerical evaluation  
of those probability distributions in random matrix theory that are  
analytically represented in terms of Painleve transcendents or  
Fredholm determinants. Concrete examples for the Gaussian and Laguerre  
(Wishart) beta-ensembles and their various scaling limits are  
discussed. We argue that the numerical approximation of Fredholm  
determinants is the conceptually more simple and efficient of the two  
approaches, easily generalized to the computation of joint  
probabilities and correlations. Having the means for extensive  
numerical explorations at hand, we discovered new and surprising  
determinantal formulae for the k-th largest level in the edge scaling  
limit of the Gaussian Orthogonal and Symplectic Ensembles; formulae  
that in turn led to improved numerical evaluations. The paper comes  
with a toolbox of Matlab functions that facilitates further  
mathematical experiments by the reader.

http://arxiv.org/abs/0904.1581


8381. A statistical mechanical interpretation of algorithmic  
information theory
Author(s): Kohtaro Tadaki

Abstract: We develop a statistical mechanical interpretation of  
algorithmic information theory by introducing the notion of  
thermodynamic quantities, such as free energy, energy, statistical  
mechanical entropy, and specific heat, into algorithmic information  
theory. We investigate the properties of these quantities by means of  
program-size complexity from the point of view of algorithmic  
randomness. It is then discovered that, in the interpretation, the  
temperature plays a role as the compression rate of the values of all  
these thermodynamic quantities, which include the temperature itself.  
Reflecting this self-referential nature of the compression rate of the  
temperature, we obtain fixed point theorems on compression rate.

http://arxiv.org/abs/0801.4194


8382. A statistical mechanical interpretation of algorithmic  
information theory III: Composite systems and fixed points
Author(s): Kohtaro Tadaki

Abstract: The statistical mechanical interpretation of algorithmic  
information theory (AIT, for short) was introduced and developed by  
our former works [K. Tadaki, Local Proceedings of CiE 2008, pp. 
425-434, 2008] and [K. Tadaki, Proceedings of LFCS'09, Springer's  
LNCS, vol.5407, pp.422-440, 2009], where we introduced the notion of  
thermodynamic quantities, such as partition function Z(T), free energy  
F(T), energy E(T), and statistical mechanical entropy S(T), into AIT.  
We then discovered that, in the interpretation, the temperature T  
equals to the partial randomness of the values of all these  
thermodynamic quantities, where the notion of partial randomness is a  
stronger representation of the compression rate by means of program- 
size complexity. Furthermore, we showed that this situation holds for  
the temperature itself as a thermodynamic quantity, namely, for each  
of all the thermodynamic quantities above, the computability of its  
value at temperature T gives a sufficient condition for T in (0,1) to  
be a fixed point on partial randomness. In this paper, we develop the  
statistical mechanical interpretation of AIT further and pursue its  
formal correspondence to normal statistical mechanics. The  
thermodynamic quantities in AIT are defined based on the halting set  
of an optimal computer, which is a universal decoding algorithm used  
to define the notion of program-size complexity. We show that there  
are infinitely many optimal computers which give completely different  
sufficient conditions in each of the thermodynamic quantities in AIT.  
We do this by introducing the notion of composition of computers to  
AIT, which corresponds to the notion of composition of systems in  
normal statistical mechanics.

http://arxiv.org/abs/0904.0973


8383. Spatial and Temporal Correlation of the Interference in ALOHA Ad  
Hoc Networks
Author(s): Radha Krishna Ganti and Martin Haenggi

Abstract: Interference is a main limiting factor of the performance of  
a wireless ad hoc network. The temporal and the spatial correlation of  
the interference makes the outages correlated temporally (important  
for retransmissions) and spatially correlated (important for routing).  
In this letter we quantify the temporal and spatial correlation of the  
interference in a wireless ad hoc network whose nodes are distributed  
as a Poisson point process on the plane when ALOHA is used as the  
multiple-access scheme.

http://arxiv.org/abs/0904.1444


8384. Average and deviation for slow-fast stochastic partial  
differential equations
Author(s): W.Wang and A.J. Roberts

Abstract: Averaging is an important method to extract effective  
macroscopic dynamics from complex systems with slow modes and fast  
modes. This article derives an averaged equation for a class of  
stochastic partial differential equations without any Lipschitz  
assumption on the slow modes. The rate of convergence in probability  
is obtained as a byproduct. Importantly, the deviation between the  
original equation and the averaged equation is also studied. A  
martingale approach proves that the deviation is described by a  
Gaussian process. This gives an approximation to errors of $\mathcal{O} 
(\e)$ instead of $\mathcal{O}(\sqrt{\e})$ attained in previous  
averaging.

http://arxiv.org/abs/0904.1462


8385. First hitting time law for some jump-diffusion processes :  
existence of a density
Author(s): Laure Coutin (MAP5) and Diana Dorobantu (SAF - EA2429)

Abstract: Let (Xt, t >= 0) be a diffusion process with jumps, sum of a  
Brownian motion with drift and a compound Poisson process. We consider  
T_x the first hitting time of a fixed level x > 0 by (Xt, t >= 0). We  
prove that the law of T_x has a density (defective when E(X1) < 0)  
with respect to the Lebesgue measure.

http://arxiv.org/abs/0904.1669


8386. Central Limit Theorems for the Brownian motion on large unitary  
groups
Author(s): Florent Benaych-Georges (CMAP and PMA)

Abstract: In this paper, we are concerned with the large N limit of  
linear combinations of entries of Brownian motions on the group of N  
by N unitary matrices. We prove that the process of such a linear  
combination converges to a Gaussian one. Various scales of time are  
concerned, giving rise to various limit processes, in relation to the  
geometric construction of the unitary Brownian motion. As an  
application, we recover certain results about linear combinations of  
the entries of Haar distributed random unitary matrices.

http://arxiv.org/abs/0904.1681


8387. The threshold function for vanishing of the top homology group  
of random $d$-complexes
Author(s): Dmitry N. Kozlov

Abstract: For positive integers $n$ and $d$, and the probability  
function $0\leq p(n)\leq 1$, we let $Y_{n,p,d}$ denote the probability  
space of all at most $d$-dimensional simplicial complexes on $n$  
vertices, which contain the full $(d-1)$-dimensional skeleton, and  
whose $d$-simplices appear with probability $p(n)$. In this paper we  
determine the threshold function for vanishing of the top homology  
group in $Y_{n,p,d}$, for all $d\geq 1$.

http://arxiv.org/abs/0904.1652


8388. Market viability via absence of arbitrages of the first kind
Author(s): Constantinos Kardaras

Abstract: The absence of arbitrages of the first kind, a weakening of  
the "No Free Lunch with Vanishing Risk" condition, is analyzed in a  
general semimartingale financial market model. In the spirit of the  
Fundamental Theorem of Asset Pricing (FTAP), it is shown that there is  
absence of arbitrages of the first kind in the market if and only if  
an equivalent local martingale deflator (ELMD) exists. An ELMD is a  
strictly positive process that, when deflated by it, discounted  
nonnegative wealth processes become local martingales. In terms of  
measures, absence of arbitrages of the first kind is shown to be  
equivalent to the existence of a finitely additive probability, weakly  
equivalent to the original and locally countably additive, under which  
the discounted asset-price process is a "local martingale". Finally,  
the aforementioned results are used to obtain an independent proof of  
the FTAP.

http://arxiv.org/abs/0904.1798


8389. Hitting half-spaces by Bessel-Brownian diffusions
Author(s): T. Byczkowski and J. Malecki and M. Ryznar

Abstract: The purpose of the paper is to find explicit formulas  
describing the joint distributions of the first hitting time and place  
for half-spaces of codimension one for a diffusion in $\R^{n+1}$,  
composed of one-dimensional Bessel process and independent n- 
dimensional Brownian motion. The most important argument is carried  
out for the two-dimensional situation. We show that this amounts to  
computation of distributions of various integral functionals with  
respect to a two-dimensional process with independent Bessel  
components. As a result, we provide a formula for the Poisson kernel  
of a half-space or of a strip for the operator $(I-\Delta)^{\alpha/ 
2}$, $0<\alpha<2$. In the case of a half-space, this result was  
recently found, by different methods, in [6]. As an application of our  
method we also compute various formulas for first hitting places for  
the isotropic stable L\'evy process.

http://arxiv.org/abs/0904.1803


8390. Random Walks on Strict Partitions
Author(s): Leonid Petrov

Abstract: We consider a certain sequence of random walks. The state  
space of the n-th random walk is the set of all strict partitions of n  
(that is, partitions without equal parts). We prove that, as n goes to  
infinity, these random walks converge to a continuous-time Markov  
process. The state space of this process is the infinite-dimensional  
simplex consisting of all nonincreasing infinite sequences of  
nonnegative numbers with sum less than or equal to one. The main  
result about the limit process is the expression of its the pre- 
generator as a formal second order differential operator in a  
polynomial algebra. Of separate interest is the generalization of  
Kerov interlacing coordinates to the case of shifted Young diagrams.

http://arxiv.org/abs/0904.1823


8391. Coupled perfect simulation of infinite range Gibbs measures and  
their finite range approximations
Author(s): Antonio Galves and Eva Loecherbach and Enza Orlandi

Abstract: Consider a Gibbs measure with a pairwise infinite range  
potential and its finite range approximation obtained by truncating  
the pairwise interaction at a certain range. If we make a local  
inspection of a perfect sampling of the finite range approximation,  
how often does it coincide with a sample from the original infinite  
range measure? We address this question by introducing a new coupled  
perfect simulation algorithm for these measures.

http://arxiv.org/abs/0904.1845


8392. On the distribution of the integral of the exponential Brownian  
motion
Author(s): Leonid Tolmatz

Abstract: The density distribution function of the integral of the  
exponential Brownian motion is determined explicitly in the form of a  
rapidly convergent series.

http://arxiv.org/abs/0904.1870


8393. Uniform bounds for norms of sums of independent random functions
Author(s): A. Goldenshluger and O.Lepski

Abstract: In this paper we study a collection of random processes $\ 
{\psi_w, w\in \cW\}$ determined by a sequence of independent random  
elements and parameterized by a set of weight functions $w\in \cW$. We  
develop uniform concentration--type inequalities for a norm $\|\psi_w\| 
$, i.e., we present an explicit upper bound $U_\psi(w)$ on $\|\psi_w\| 
$ and study behavior of \[ \sup_{w\in \cW} \{\|\psi_w\|-U_\psi(w)\}.  
\] Several probability and moment inequalities for this random  
variable are derived and used in order to get some asymptotic results.  
We also consider applications of obtained bounds to many important  
problems arising in modern nonparametric statistics including  
bandwidth selection in multivariate density and regression estimation.

http://arxiv.org/abs/0904.1950


8394. A multiple stochastic integral criterion for almost sure limit  
theorems
Author(s): Bernard Bercu and Ivan Nourdin and Murad S. Taqqu

Abstract: In this paper, we study almost sure central limit theorems  
for multiple stochastic integrals and provide a criterion based on the  
kernel of these multiple integrals. We apply our result to normalized  
partial sums of Hermite polynomials of increments of fractional  
Brownian motion. We obtain almost sure central limit theorems for  
these normalized sums when they converge in law to a normal  
distribution.

http://arxiv.org/abs/0904.2094


8395. Fonctions de Mittag-Leffler et processus de L\'evy stables sans  
saut n\'egatif
Author(s): Thomas Simon

Abstract: It is noticed that a certain transform of the Mittag-Leffler  
function Ea is completely monotone for a in [1,2]. Using the explicit  
expressions of its Bernstein density, an identity in law between  
suprema of completely asymmetric Levy a-stable processes. In the  
spectrally positive case, we retrieve the exact expression of a  
unilateral small deviation constant which had been previously obtained  
by a different method by Bernyk, Dalang and Peskir.

http://arxiv.org/abs/0904.2191


8396. Asymptotics of a Brownian ratchet for Protein Translocation
Author(s): Andrej Depperschmidt and Peter Pfaffelhuber

Abstract: Protein translocation in cells has been modelled by  
\emph{Brownian ratchets}. In such models, the protein diffuses through  
the nanopore by thermal fluctuations. On one side of the pore  
ratcheting molecules bind to the protein and hinder it to diffuse out  
of the pore. We study a simple Brownian ratchet by means of a  
reflected Brownian motion $(X_t)_{t\geq 0}$ with a changing reflection  
point $(R_t)_{t\geq 0}$. The rate of change of $R_t$ is $\gamma(X_t- 
R_t)$ and is distributed uniformly on $[R_t;X_t]$. We show that the  
asymptotic speed of the ratchet scales with $\gamma^{1/3}$ and the  
asymptotic variance is independent of $\gamma$.

http://arxiv.org/abs/0904.2276


8397. Correction to: Branching-coalescing particle systems
Author(s): Siva R. Athreya and Jan M. Swart

Abstract: In the article titled "Branching-Coalescing Particle  
Systems" published in Probability Theory and Related Fields 131(3),  
pages 376-414, (2005), Theorem 7 as stated there is incorrect. Indeed,  
we show by counterexample that the equality that we claimed there to  
hold for all time, in general holds only for almost every time with  
respect to Lebesgue measure. We prove a weaker version of the theorem  
that is still sufficient for our applications in the mentioned paper.

http://arxiv.org/abs/0904.2288


8398. Supremum of Random Dirichlet Polynomials with Sub-multiplicative  
Coefficients
Author(s): Michel Weber

Abstract: We study the supremum of random Dirichlet polynomials  
$D_N(t)=\sum_{n=1}^N\varepsilon_n d(n) n^{- s}$, where $(\varepsilon_n) 
$ is a sequence of independent Rademacher random variables, and $ d $  
is a sub-multiplicative function. The approach is gaussian and  
entirely based on comparison properties of Gaussian processes, with no  
use of the metric entropy method.

http://arxiv.org/abs/0904.2316


8399. Tridiagonal realization of the anti-symmetric Gaussian $\beta$- 
ensemble
Author(s): Ioana Dumitriu and Peter J. Forrester

Abstract: The Householder reduction of a member of the anti-symmetric  
Gaussian unitary ensemble gives an anti-symmetric tridiagonal matrix  
with all independent elements. The random variables permit the  
introduction of a positive parameter $\beta$, and the eigenvalue  
probability density function of the corresponding random matrices can  
be computed explicitly, as can the distribution of $\{q_i\}$, the  
first components of the eigenvectors. Three proofs are given. One  
involves an inductive construction based on bordering of a family of  
random matrices which are shown to have the same distributions as the  
anti-symmetric tridiagonal matrices. This proof uses the Dixon- 
Anderson integral from Selberg integral theory. A second proof  
involves the explicit computation of the Jacobian for the change of  
variables between real anti-symmetric tridiagonal matrices, its  
eigenvalues and $\{q_i\}$. The third proof, which is restricted to $n$  
even, maps matrices from the anti-symmetric Gaussian $\beta$-ensemble  
to those realizing particular examples of the Laguerre $\beta$- 
ensemble. In addition to these proofs, we note some simple properties  
of the shooting eigenvector and associated Pr\"ufer phases of the  
random matrices.

http://arxiv.org/abs/0904.2216


8400. Taylor expansions of solutions of stochastic partial  
differential equations
Author(s): Arnulf Jentzen

Abstract: The solutions of parabolic and hyperbolic stochastic partial  
differential equations (SPDEs) driven by an infinite dimensional  
Brownian motion, which is a martingale, are in general not semi- 
martingales any more and therefore do not satisfy an It\^o formula  
like the solutions of finite dimensional stochastic differential  
equations (SODEs). In particular, it is not possible to derive  
stochastic Taylor expansions as for the solutions of SODEs using an  
iterated application of the It\^o formula. However, in this article we  
introduce Taylor expansions of solutions of SPDEs via an alternative  
approach, which avoids the need of an It\^o formula. The main idea  
behind these Taylor expansions is to use first classical Taylor  
expansions for the nonlinear coefficients of the SPDE and then to  
insert recursively the mild presentation of the solution of the SPDE.  
The iteration of this idea allows us to derive stochastic Taylor  
expansions of arbitrarily high order. Combinatorial concepts of trees  
and woods provide a compact formulation of the Taylor expansions.

http://arxiv.org/abs/0904.2232


8401. Random walk versus random line
Author(s): Joel De Coninck and Francois Dunlop and Thierry Huillet

Abstract: We consider random walks X_n in Z+, obeying a detailed  
balance condition, with a weak drift towards the origin when X_n tends  
to infinity. We reconsider the equivalence in law between a random  
walk bridge and a 1+1 dimensional Solid-On-Solid bridge with a  
corresponding Hamiltonian. Phase diagrams are discussed in terms of  
recurrence versus wetting. A drift -delta/X_n of the random walk  
yields a Solid-On-Solid potential with an attractive well at the  
origin and a repulsive tail delta(delta+2)/(8X_n^2) at infinity,  
showing complete wetting for delta<=1 and critical partial wetting for  
delta>1.

http://arxiv.org/abs/0904.2440


8402. Statistical analysis of single-server loss queueing systems
Author(s): Vyacheslav M. Abramov

Abstract: In this article statistical bounds for certain output  
characteristics of the $M/GI/1/n$ and $GI/M/1/n$ loss queueing systems  
are derived on the basis of large samples of an input characteristic  
of these systems.

http://arxiv.org/abs/0904.2426


8403. Joint Range of R\'enyi Entropies
Author(s): Peter Harremo\"es

Abstract: The exact range of the joined values of several R\'{e}nyi  
entropies is determined. The method is based on topology with special  
emphasis on the orientation of the objects studied. Like in the case  
when only two orders of R\'{e}nyi entropies are studied one can  
parametrize upper and lower bounds but an explicit formula for a tight  
upper or lower bound cannot be given.

http://arxiv.org/abs/0904.2477


8404. A Class of degenerate Stochastic differential equations with non- 
Lipschitz coefficients
Author(s): K. Suresh Kumar

Abstract: We obtain sufficient condition for SDEs to evolve in the  
positive orthant. We use comparison theorem arguments to achieve this.  
As a result we prove the existence of a unique strong solution for a  
class of multidimensional degenerate SDEs with non-Lipschitz diffusion  
coefficients.

http://arxiv.org/abs/0904.2629


8405. Boundary crossing identities for diffusions having the time  
inversion property
Author(s): Larbi Alili and Pierre Patie

Abstract: We review and study a one-parameter family of functional  
transformations, denoted by $(S^{(\beta)})_{\beta\in \R}$, which, in  
the case $\beta<0$, provides a path realization of bridges associated  
to the family of diffusion processes enjoying the time inversion  
property. This family includes the Brownian motion, Bessel processes  
with a positive dimension and their conservative $h$-transforms. By  
means of these transformations, we derive an explicit and simple  
expression which relates the law of the boundary crossing times for  
these diffusions over a given function $f$ to those over the image of  
$f$ by the mapping $S^{(\beta)}$, for some fixed $\beta\in \mathbb{R} 
$. We give some new examples of boundary crossing problems for the  
Brownian motion and the family of Bessel processes. We also provide,  
in the Brownian case, an interpretation of the results obtained by the  
standard method of images and establish connections between the exact  
asymptotics for large time of the densities corresponding to various  
curves of each family.

http://arxiv.org/abs/0904.2680


8406. On the It\^o-Wentzell formula for distribution-valued processes  
and related topics
Author(s): N.V. Krylov

Abstract: We prove the It\^o-Wentzell formula for processes with  
values in the space of generalized functions by using the stochastic  
Fubini theorem and the It\^o-Wentzell formula for real-valued  
processes, appropriate versions of which are also proved.

http://arxiv.org/abs/0904.2752


8407. Horizontal diffusion in $C^1$ path space
Author(s): Marc Arnaudon (LMA) and Abdoulaye Kol\'eh\`e Coulibaly- 
Pasquier (LMA) and Anton Thalmaier

Abstract: We define horizontal diffusion in $C^1$ path space over a  
Riemannian manifold and prove its existence. If the metric on the  
manifold is developing under the forward Ricci flow, horizontal  
diffusion along Brownian motion turns out to be length preserving. As  
application, we prove contraction properties in the Monge-Kantorovich  
minimization problem for probability measures evolving along the heat  
flow. For constant rank diffusions, differentiating a family of  
coupled diffusions gives a derivative process with a covariant  
derivative of finite variation. This construction provides an  
alternative method to filtering out redundant noise.

http://arxiv.org/abs/0904.2762


8408. Random surface growth with a wall and Plancherel measures for  
O(infinity)
Author(s): Alexei Borodin and Jeffrey Kuan

Abstract: We consider a Markov evolution of lozenge tilings of a  
quarter-plane and study its asymptotics at large times. One of the  
boundary rays serves as a reflecting wall. We observe frozen and  
liquid regions, prove convergence of the local correlations to  
translation-invariant Gibbs measures in the liquid region, and obtain  
new discrete Jacobi and symmetric Pearcey determinantal point  
processes near the wall. The model can be viewed as the one-parameter  
family of Plancherel measures for the infinite-dimensional orthogonal  
group, and we use this interpretation to derive the determinantal  
formula for the correlation functions at any finite time moment.

http://arxiv.org/abs/0904.2607


8409. Asymptotic properties of resolvents of large dilute Wigner  
matrices
Author(s): S. Ayadi and O. Khorunzhiy

Abstract: We study the spectral properties of the dilute Wigner random  
real symmetric n-dimensional matrices H such that the entries H(i,j)  
take zero value with probability 1-p/n. We prove that under rather  
general conditions on the probability distribution of H(i,j) the  
semicircle law is valid for the dilute Wigner ensemble in the limit of  
infinite n and p. In the second part of the paper we study the leading  
term of the correlation function of the resolvent G(z) of H with large  
enough Im z in the limit of infinite n and p such that 3/5 log n < log  
n. We show that this leading term, when considered in the local  
spectral scale, converges to the same limit as that of the resolvent  
correlation function of the Wigner ensemble of random matrices. This  
shows that the moderate dilution of the Wigner ensemble does not alter  
its universality class.

http://arxiv.org/abs/0904.2689


8410. Symmetric Jump Processes and their Heat Kernel Estimates
Author(s): Zhen-Qing Chen

Abstract: We survey the recent development of the DeGiorgi-Nash-Moser- 
Aronson type theory for a class of symmetric jump processes(or  
equivalently, a class of symmetric integro-differential operators). We  
focus on the sharp two-sided estimates for the transition density  
functions (or heat kernels) of the processes, a priori Holder estimate  
and parabolic Harnack inequalities for their parabolic functions. In  
contrast to the second order elliptic differential operator case, the  
methods to establish these properties for symmetric integro- 
differential operators are mainly probabilistic.

http://arxiv.org/abs/0904.2796


8411. The Optimal Filtering of Markov Jump Processes in Additive White  
Noise
Author(s): M. Zakai

Abstract: This note is based on Wonham \cite{Wonham}. The differences  
between this note and [Wonham] are discussed in Section VIII.

http://arxiv.org/abs/0904.2888


8412. Viability of infinite-asset financial models where constrained  
agents with limited information act
Author(s): Constantinos Kardaras

Abstract: A study of the boundedness in probability of the set of  
possible wealth outcomes of an economic agent facing constraints, and  
with limited access to information, is undertaken. The wealth-process  
set is abstractly structured with reasonable economic properties,  
instead of the usual practice of taking it to consist of stochastic  
integrals against a semimartingale integrator. We obtain the  
equivalence of (a) the boundedness in probability of wealth outcomes  
with (b) the existence of at least one deflator that make the deflated  
wealth processes have a generalized supermartingale property.  
Specializing in the case of full information, we obtain as a  
consequence that in a viable market all wealth processes have versions  
that are semimartingales.

http://arxiv.org/abs/0904.2913


8413. Limit Distributions for Random Hankel, Toeplitz Matrices and  
Independent Products
Author(s): Dang-Zheng Liu and Zheng-Dong Wang

Abstract: For random selfadjoint (real symmetric, complex Hermitian,  
or quaternion self-dual) Toeplitz matrices and real symmetric Hankel  
matrices, the existence of universal limit distributions for  
eigenvalues and products of several independent matrices is proved.  
The joint moments are the integral sums related to certain pair  
partitions. Our method can apply to random Hankel and Toeplitz band  
matrices, and the similar results are given. In particular, when the  
band width grows slowly as the dimension $N\ra \iy$, the exact limit  
distribution functions are given (N(0,1) for Toeplitz band matrices)  
and some asymptotic commutativity is observed.

http://arxiv.org/abs/0904.2958


8414. Law of the exponential functional of one-sided L\'evy processes  
and Asian options
Author(s): Pierre Patie

Abstract: The purpose of this note is to describe, in terms of a power  
series, the distribution function of the exponential functional, taken  
at some independent exponential time, of a spectrally negative L\'evy  
process \xi with unbounded variation. We also derive a Geman-Yor type  
formula for Asian options prices in a financial market driven by e^\xi.

http://arxiv.org/abs/0904.3000


8415. Quasi-stationary distributions and Fleming-Viot processes for  
finite state Markov processes
Author(s): Amine Asselah and Pablo A. Ferrari and Pablo Groisman

Abstract: Consider a continuous time Markov chain with rates $Q$ in  
the state space $\Lambda\cup\{0\}$ with 0 as an absorbing state. In  
the associated Fleming-Viot process $N$ particles evolve independently  
in $\Lambda$ with rates $Q$ until one of them attempts to jump to the  
absorbing state 0. At this moment the particle comes back to $\Lambda$  
instantaneously, by jumping to one of the positions of the other  
particles, chosen uniformly at random. When $\Lambda$ is finite, we  
show that the empirical distribution of the particles at a fixed time  
converges as $N\to\infty$ to the distribution of a single particle at  
the same time conditioned on non absorption. Furthermore, the  
empirical profile of the unique invariant measure for the Fleming-Viot  
process with $N$ particles converges as $N\to\infty$ to the unique  
quasi-stationary distribution of the one-particle motion. A key  
element of the approach is to show that the two-particle correlations  
is of order $1/N$.

http://arxiv.org/abs/0904.3039


8416. Hoeffding spaces and Specht modules
Author(s): Giovanni Peccati (LSTA and MODAL'X) and Jean-Renaud Pycke  
(DP)

Abstract: It is proved that each Hoeffding space associated with a  
random permutation (or, equivalently, with extractions without  
replacement from a finite population) carries an irreducible  
representation of the symmetric group, equivalent to a two-block  
Specht module.

http://arxiv.org/abs/0904.3086


8417. L1-Penalized Quantile Regression in High-Dimensional Sparse Models
Author(s): Alexandre Belloni and Victor Chernozhukov

Abstract: We consider median regression and, more generally, quantile  
regression in high-dimensional sparse models. In these models the  
overall number of regressors $p$ is very large, possibly larger than  
the sample size $n$, but only $s$ of these regressors have non-zero  
impact on the conditional quantile of the response variable, where $s$  
grows slower than $n$. Since in this case the ordinary quantile  
regression is not consistent, we consider quantile regression  
penalized by the $\ell_1$-norm of coefficients ($\ell_1$-QR). First,  
we show that $\ell_1$-QR is consistent at the rate $\sqrt{s/n}  
\sqrt{\log p}$, which is close to the oracle rate $\sqrt{s/n}$,  
achievable when the minimal true model is known. The overall number of  
regressors $p$ affects the rate only through the $\log p$ factor, thus  
allowing nearly exponential growth in the number of zero-impact  
regressors. The rate result holds under relatively weak conditions,  
requiring that $s/n$ converges to zero at a super-logarithmic speed  
and that regularization parameter satisfies certain theoretical  
constraints. Second, we propose a pivotal, data-driven choice of the  
regularization parameter and show that it satisfies these theoretical  
constraints. Third, we show that $\ell_1$-QR correctly selects the  
true minimal model as a valid submodel, when the non-zero coefficients  
of the true model are well separated from zero. We also show that the  
number of non-zero coefficients in $\ell_1$-QR is of same stochastic  
order as $s$, the number of non-zero coefficients in the minimal true  
model. Fourth, we analyze the rate of convergence of a two-step  
estimator that applies ordinary quantile regression to the selected  
model. Fifth, we evaluate the performance of $\ell_1$-QR in a Monte- 
Carlo experiment, and illustrate its use on an international economic  
growth application.

http://arxiv.org/abs/0904.2931


8418. Asymptotic Properties of Random Matrices of Long-Range  
Percolation Model
Author(s): Slim Ayadi

Abstract: We study the spectral properties of matrices of long-range  
percolation model. These are N\times N random real symmetric matrices  
H=\{H(i,j)\}_{i,j} whose elements are independent random variables  
taking zero value with probability 1-\psi((i-j)/b), b\in  
\mathbb{R}^{+}, where $\psi$ is an even positive function with \psi(t) 
\le{1} and vanishing at infinity. We study the resolvent G(z)=(H- 
z)^{-1}, Imz\neq{0} in the limit N,b\to\infty, b=O(N^{\alpha}), 1/3< 
\alpha<1 and obtain the explicit expression T(z_{1},z_{2}) for the  
leading term of the correlation function of the normalized trace of  
resolvent g_{N,b}(z)=N^{-1}Tr G(z). We show that in the scaling limit  
of local correlations, this term leads to the expression  
(Nb)^{-1}T(\lambda+r_{1}/N+i0,\lambda+r_{2}/N-i0)= b^{-1}\sqrt{N}| 
r_{1}-r_{2}|^{-3/2}(1+o(1)) found earlier by other authors for band  
random matrix ensembles. This shows that the ratio $b^{2}/N$ is the  
correct scale for the eigenvalue density correlation function and that  
the ensemble we study and that of band random matrices belong to the  
same class of spectral universality.

http://arxiv.org/abs/0904.2837


8419. Gibbs random fields with unbounded spins on unbounded degree  
graphs
Author(s): Yuri Kondratiev and Yuri Kozitsky and Tanja Pasurek

Abstract: Gibbs random fields corresponding to systems of real-valued  
spins (e.g. systems of interacting anharmonic oscillators) indexed by  
the vertices of unbounded degree graphs with a certain summability  
property are constructed. It is proven that the set of tempered Gibbs  
random fields is non-void and weakly compact, and that they obey  
uniform exponential integrability estimates. In the second part of the  
paper, a class of graphs is described in which the mentioned  
summability is obtained as a consequence of a property, by virtue of  
which vertices of large degree are located at large distances from  
each other. The latter is a stronger version of a metric property,  
introduced in [Bassalygo, L. A. and Dobrushin, R. L. (1986).  
\textrm{Uniqueness of a Gibbs field with a random potential--an  
elementary approach.}\textit{Theory Probab. Appl.} {\bf 31} 572--589].

http://arxiv.org/abs/0904.3207


8420. Weak Solutions of stochastic recursions: an explicit construction
Author(s): Pascal Moyal

Abstract: We propose an explicit construction of the solution of a  
stationary stochastic recursion of the form $X\circ\theta=\phi(X)$ on  
a semi-ordered Polish space, when the monotonicity of $\phi$ is not  
assumed. This solution exists on an enriched probability space (it is  
said \emph{weak}), provided the recursion is lattice-valued, and  
dominated by a proper monotonic stochastic recursion.

http://arxiv.org/abs/0904.3240


8421. Computations of Greeks in stochastic volatility models via the  
Malliavin calculus
Author(s): Youssef El-Khatib

Abstract: We compute Greeks for stochastic volatility models driven by  
Brownian informations. We use the Malliavin method introduced for  
deterministic volatility models.

http://arxiv.org/abs/0904.3247


8422. On continuity properties for option prices in exponential L\'evy  
models
Author(s): S. Cawston and L. Vostrikova

Abstract: For a converging sequence of exponential L\'evy models, we  
give conditions under which the associated sequence of option prices  
converges. We also study the behaviour of the prices when no such  
convergence holds. We then consider two special cases, first when the  
martingale measure is chosen by minimisation of entropy and then when  
it minimises Hellinger integrals.

http://arxiv.org/abs/0904.3274


8423. Large Deviation Principle for Semilinear Stochastic Evolution  
Equations with Monotone Nonlinearity and Multiplicative Noise
Author(s): Hassan Dadashi-Arani and Bijan Z. Zangeneh

Abstract: Using a recently developed method, weak convergence method,  
in dealing with the large deviation principle, we demonstrate the  
large deviation principle property for mild solutions of stochastic  
evolution equations with monotone nonlinearity and multiplicative  
noise. An It^o-type inequality is a main tool in the proofs. We also  
give two examples to illustrate the applications of the theorems.

http://arxiv.org/abs/0904.3305


8424. Posterior Inference in Curved Exponential Families under  
Increasing Dimensions
Author(s): Alexandre Belloni and Victor Chernozhukov

Abstract: The goal of this work is to study the large sample  
properties of the posterior-based inference in the curved exponential  
family under increasing dimension. The curved structure arises from  
the imposition of various restrictions, such as moment restrictions,  
on the model, and plays a fundamental role in various branches of data  
analysis. We establish conditions under which the posterior  
distribution is approximately normal, which in turn implies various  
good properties of estimation and inference procedures based on the  
posterior. In the process we revisit and improve upon previous results  
for the exponential family under increasing dimension by making use of  
concentration of measure. We also discuss a variety of applications  
including the multinomial model with moment restrictions, seemingly  
unrelated regression equations, and single structural equation models.  
In our analysis, both the parameter dimension and the number of  
moments are increasing with the sample size.

http://arxiv.org/abs/0904.3132


8425. Quasi-stationary distributions for structured birth and death  
processes with mutations
Author(s): Pierre Collet (CPHT) and Servet Martinez and Sylvie M\'el 
\'eard (CMAP) and Jaime San Martin

Abstract: We study the probabilistic evolution of a birth and death  
continuous time measure-valued process with mutations and ecological  
interactions. The individuals are characterized by (phenotypic) traits  
that take values in a compact metric space. Each individual can die or  
generate a new individual. The birth and death rates may depend on the  
environment through the action of the whole population. The offspring  
can have the same trait or can mutate to a randomly distributed trait.  
We assume that the population will be extinct almost surely. Our goal  
is the study, in this infinite dimensional framework, of quasi- 
stationary distributions when the process is conditioned on non- 
extinction. We firstly show in this general setting, the existence of  
quasi-stationary distributions. This result is based on an abstract  
theorem proving the existence of finite eigenmeasures for some  
positive operators. We then consider a population with constant birth  
and death rates per individual and prove that there exists a unique  
quasi-stationary distribution with maximal exponential decay rate. The  
proof of uniqueness is based on an absolute continuity property with  
respect to a reference measure.

http://arxiv.org/abs/0904.3468


8426. An excursion approach to maxima of the Brownian Bridge
Author(s): Mihael Perman (Institute for Mathematics and Physics and  
Mechanics and Ljubljana, Slovenia) Jon A. Wellner (University of  
Washington, Seattle)

Abstract: Functionals of Brownian bridge arise as limiting  
distributions in nonparametric statistics. In this paper we will give  
a derivation of distributions of extrema of the Brownian bridge based  
on excursion theory for Brownian motion. Only the Poisson character of  
the excursion process will be used. Particular cases of calculations  
include the distributions of the Kolmogorov-Smirnov statistic, the  
Kuiper statistic, and the ratio of the maximum positive ordinate to  
the minumum negative ordinate.

http://arxiv.org/abs/0904.3473


8427. Regularity of harmonic functions for a class of singular stable- 
like processes
Author(s): Richard F. Bass and Zhen-Qing Chen

Abstract: We consider the system of stochastic differential equations  
dX_t=A(X_{t-}) dZ_t, where Z_t^1, ..., Z^d_t are independent one- 
dimensional symmetric stable processes of order \alpha, and the matrix- 
valued function A is bounded, continuous and everywhere non- 
degenerate. We show that bounded harmonic functions associated with X  
are Holder continuous, but a Harnack inequality need not hold. The  
Levy measure associated with the vector-valued process Z is highly  
singular.

http://arxiv.org/abs/0904.3518


8428. A method for Hedging in continuous time
Author(s): Yoav Freund

Abstract: We present a method for hedging in continuous time.

http://arxiv.org/abs/0904.3356


8429. Application of the lent particle method to Poisson driven SDE's
Author(s): Nicolas Bouleau (CERMICS) and Laurent Denis (DP)

Abstract: We apply the Dirichlet forms version of Malliavin calculus  
to stochastic differential equations with jumps. As in the continuous  
case this weakens signi?cantly the assumptions on the coefficients of  
the SDE. In spite of the use of the Dirichlet forms theory, this  
approach brings also an important simpli?cation which was not  
available nor visible previously : an explicit formula giving the carr 
\'e du champ matrix, i.e. the Malliavin matrix. Following this formula  
a new procedure appears, called the lent particle method which  
shortens the computations both theoretically and in concrete examples.

http://arxiv.org/abs/0904.3613


8430. A spatially explicit Markovian individual-based model for  
terrestrial plant dynamics
Author(s): Fabien Campillo and Marc Joannides

Abstract: An individual-based model (IBM) of a spatiotemporal  
terrestrial ecological population is proposed. This model is spatially  
explicit and features the position of each individual together with  
another characteristic, such as the size of the individual, which  
evolves according to a given stochastic model. The population is  
locally regulated through an explicit competition kernel. The IBM is  
represented as a measure-valued branching/diffusing stochastic  
process. The approach allows (i) to describe the associated Monte  
Carlo simulation and (ii) to analyze the limit process under large  
initial population size asymptotic. The limit macroscopic model is a  
deterministic integro-differential equation.

http://arxiv.org/abs/0904.3632


8431. On adding a list of numbers (and other one-dependent  
determinantal processes)
Author(s): Alexei Borodin and Persi Diaconis and and Jason Fulman

Abstract: Adding a column of numbers produces "carries" along the way.  
We show that random digits produce a pattern of carries with a neat  
probabilistic description: the carries form a one-dependent  
determinantal point process. This makes it easy to answer natural  
questions: How many carries are typical? Where are they located? We  
show that many further examples, from combinatorics, algebra and group  
theory, have essentially the same neat formulae, and that any one- 
dependent point process on the integers is determinantal. The examples  
give a gentle introduction to the emerging fields of one-dependent and  
determinantal point processes.

http://arxiv.org/abs/0904.3740


8432. Levy solutions of a randomly forced Burgers equation
Author(s): Marie-Line Chabanol and Jean Duchon

Abstract: We consider the one dimensional Burgers equation forced by a  
brownian in space and white noise in time process $\partial_t u + u  
\partial_x u = f(x,t)$, with $2E(f(x,t)f(y,s)) = (|x|+|y|-|x-y|) 
\delta(t-s)$ and we show that there are Levy processes solutions, for  
which we give the evolution equation of the characteristic exponent.  
In particular we give the explicit solution in the case $u_0(x)=0$.

http://arxiv.org/abs/0904.3397


8433. New critical exponents for percolation and the random-cluster  
model
Author(s): Youjin Deng and Wei Zhang and Timothy M. Garoni and Alan D.  
Sokal and Andrea Sportiello

Abstract: We introduce several infinite families of new critical  
exponents for the random-cluster model, and give heuristic scaling  
arguments determining all but one of these exponents as a function of  
q in the two-dimensional case. We then give Monte Carlo simulations  
confirming these predictions. For the shortest-path fractal dimension  
we give the conjectured exact formula d_min = (g+2)(g+18)/(32g) where  
g is the Coulomb-gas coupling. Finally, we apply these exponents to  
provide a radically improved implementation of the Sweeny Monte Carlo  
algorithm.

http://arxiv.org/abs/0904.3448


8434. Remarks on Pickands theorem
Author(s): Zbigniew Michna

Abstract: In this article we present Pickands theorem and his double  
sum method. We follow Piterbarg's proof of this theorem. Since his  
proof relies on general lemmas we present a complete proof of Pickands  
theorem using Borell inequality and Slepian lemma. The original  
Pickands proof is rather complicated and is mixed with upcrossing  
probabilities for stationary Gaussian processes. We give a lower bound  
for Pickands constant.

http://arxiv.org/abs/0904.3832


8435. Matrix measures, random moments and Gaussian ensembles
Author(s): Jan Nagel and Holger Dette

Abstract: We consider the moment space $\mathcal{M}_n$ corresponding  
to $p \times p$ real or complex matrix measures defined on the  
interval $[0,1]$. The asymptotic properties of the first $k$  
components of a uniformly distributed vector $(S_{1,n}, ...,  
S_{n,n})^* \sim \mathcal{U} (\mathcal{M}_n)$ are studied if $n \to  
\infty$. In particular, it is shown that an appropriately centered and  
standardized version of the vector $(S_{1,n}, ..., S_{k,n})^*$  
converges weakly to a vector of $k$ independent $p \times p$ Gaussian  
ensembles. For the proof of our results we use some new relations  
between ordinary moments and canonical moments of matrix measures  
which are of own interest. In particular, it is shown that the first $k 
$ canonical moments corresponding to the uniform distribution on the  
real or complex moment space $\mathcal{M}_n$ are independent  
multivariate Beta distributed random variables and that each of these  
random variables converge in distribution (if the parameters converge  
to infinity) to the Gaussian orthogonal ensemble or to the Gaussian  
unitary ensemble, respectively.

http://arxiv.org/abs/0904.3847


8436. The Gapeev-K\"uhn stochastic game driven by a spectrally  
positive L\'evy process
Author(s): E.J. Baurdoux and A.E. Kyprianou and J.C. Pardo

Abstract: In Gapeev and K\"uhn (2005), the stochastic game  
corresponding to perpetual convertible bonds was considered when  
driven by a Brownian motion and a compound Poisson process with  
exponential jumps. We consider the same stochastic game but driven by  
a spectrally positive L\'evy process. We establish a complete solution  
to the game indicating four principle parameter regimes as well as  
characterizing the occurence of continuous and smooth fit. In Gapeev  
and K\"uhn (2005), the method of proof was mainly based on solving a  
free boundary value problem. In this paper, we instead use fluctuation  
theory and an auxiliary optimal stopping problem to find a solution to  
the game.

http://arxiv.org/abs/0904.3871


8437. On Marginal Markov Processes of Quantum Quadratic Stochastic  
Processes
Author(s): Farrukh Mukhamdov

Abstract: In the paper it is defined two marginal Markov processes on  
von Neumann algebras $\cm$ and $\cm\o\cm$, respectively, corresponding  
to given quantum quadratic stochastic process (q.q.s.p.). It is proved  
that such marginal processes uniquely determines the q.q.s.p.  
Moreover, certain ergodic relations between them are established as  
well.

http://arxiv.org/abs/0904.3790


8438. Metastable behavior for bootstrap percolation on regular trees
Author(s): Marek Biskup and Roberto H. Schonmann

Abstract: We examine bootstrap percolation on a regular (b+1)-ary tree  
with initial law given by Bernoulli(p). The sites are updated  
according to the usual rule: a vacant site becomes occupied if it has  
at least theta occupied neighbors, occupied sites remain occupied  
forever. It is known that, when b>theta>1, the limiting density q=q(p)  
of occupied sites exhibits a jump at some p_t=p_t(b,theta) in (0,1)  
from q_t:=q(p_t)<1 to q(p)=1 when p>p_t. We investigate the metastable  
behavior associated with this transition. Explicitly, we pick p=p_t+h  
with h>0 and show that, as h decreases to 0, the system lingers around  
the "critical" state for time order h^{-1/2} and then passes to fully  
occupied state in time O(1). The law of the entire configuration  
observed when the occupation density is q in (q_t,1) converges, as h  
tends to 0, to a well-defined measure.

http://arxiv.org/abs/0904.3965


8439. Some asymptotic properties of the spectrum of the Jacobi ensemble
Author(s): Holger Dette and Jan Nagel

Abstract: For the random eigenvalues with density corresponding to the  
Jacobi ensemble $$c \cdot \prod_{i < j} | \lambda_i - \lambda_j |^ 
\beta \prod^n_{i=1} (2 - \lambda_i)^a (2 + \lambda_i)^b I_{(-2,2)}  
(\lambda_i) $$ $(a, b > -1, \beta > 0) $ a strong uniform  
approximation by the roots of the Jacobi polynomials is derived if the  
parameters $a, b,$ $\beta$ depend on $n$ and $n \to \infty$. Roughly  
speaking, the eigenvalues can be uniformly approximated by roots of  
Jacobi polynomials with parameters $((2a+2)/\beta -1, (2b+2)/ 
\beta-1)$, where the error is of order $\{\log n/(a+b) \}^{1/4}$.  
These results are used to investigate the asymptotic properties of the  
corresponding spectral distribution if $n \to \infty$ and the  
parameters $a, b$ and $\beta$ vary with $n$. We also discuss further  
applications in the context of multivariate random $F$-matrices.

http://arxiv.org/abs/0904.4091


8440. Zero bias transformation and asymptotic expansions II : the  
Poisson case
Author(s): Ying Jiao (PMA)

Abstract: We apply a discrete version of the methodology in  
\cite{gauss} to obtain a recursive asymptotic expansion for $\esp[h(W)] 
$ in terms of Poisson expectations, where $W$ is a sum of independent  
integer-valued random variables and $h$ is a polynomially growing  
function. We also discuss the remainder estimations.

http://arxiv.org/abs/0904.4115


8441. Limiting Distributions for Sums of Independent Random Products
Author(s): Zakhar Kabluchko

Abstract: Let $\{X_{i,j}:(i,j)\in\mathbb N^2\}$ be a two-dimensional  
array of independent copies of a random variable $X$, and let $\{N_n 
\}_{n\in\mathbb N}$ be a sequence of natural numbers such that $\lim_{n 
\to\infty}e^{-cn}N_n=1$ for some $c>0$. Our main object of interest is  
the sum of independent random products $$Z_n=\sum_{i=1}^{N_n}  
\prod_{j=1}^{n}e^{X_{i,j}}.$$ It is shown that the limiting properties  
of $Z_n$, as $n\to\infty$, undergo phase transitions at two critical  
points $c=c_1$ and $c=c_2$. Namely, if $c>c_2$, then $Z_n$ satisfies  
the central limit theorem with the usual normalization, whereas for  
$cc_1$. If the random variable $X$ is Gaussian, we recover the results  
of Bovier, Kurkova, and L\"owe [Fluctuations of the free energy in the  
REM and the $p$-spin SK models. Ann. Probab. 30(2002), 605-651].

http://arxiv.org/abs/0904.4127


8442. A continuum-tree-valued Markov process
Author(s): Romain Abraham (MAPMO) and Jean-Francois Delmas (CERMICS)

Abstract: We present a construction of a L\'evy continuum random tree  
(CRT) associated with a super-critical continuous state branching  
process using the so-called exploration process and a Girsanov's  
theorem. We also extend the pruning procedure to this super-critical  
case. Let $\psi$ be a critical branching mechanism. We set $\psi_ 
\theta(\cdot)=\psi(\cdot+\theta)-\psi(\theta)$. Let $\Theta=(\theta_ 
\infty,+\infty)$ or $\Theta=[\theta_\infty,+\infty)$ be the set of  
values of $\theta$ for which $\psi_\theta$ is a branching mechanism.  
The pruning procedure allows to construct a decreasing L\'evy-CRT- 
valued Markov process $(\ct_\theta,\theta\in\Theta)$, such that $ 
\mathcal{T}_\theta$ has branching mechanism $\psi_\theta$. It is sub- 
critical if $\theta>0$ and super-critical if $\theta<0$. We then  
consider the explosion time $A$ of the CRT: the smaller (negative)  
time $\theta$ for which $\mathcal{T}_\theta$ has finite mass. We  
describe the law of $A$ as well as the distribution of the CRT just  
after this explosion time. The CRT just after explosion can be seen as  
a CRT conditioned not to be extinct which is pruned with an  
independent intensity related to $A$. We also study the evolution of  
the CRT-valued process after the explosion time. This extends results  
from Aldous and Pitman on Galton-Watson trees. For the particular case  
of the quadratic branching mechanism, we show that after explosion the  
total mass of the CRT behaves like the inverse of a stable  
subordinator with index 1/2. This result is related to the size of the  
tagged fragment for the fragmentation of Aldous' CRT.

http://arxiv.org/abs/0904.4175


8443. A uniqueness theorem for the martingale problem describing a  
diffusion in media with membranes
Author(s): Olga V. Aryasova and Mykola I. Portenko

Abstract: We formulate a martingale problem that describes a diffusion  
process in a multidimensional Euclidean space with a membrane located  
on a given smooth surface and having the properties of skewing and  
delaying. The theorem on the existence of no more than one solution to  
the problem is proved.

http://arxiv.org/abs/0904.4223


8444. Numerical Computation of First-Passage Times of Increasing Levy  
Processes
Author(s): Mark S. Veillette; Murad S. Taqqu

Abstract: Let $\{D(s), s \geq 0\}$ be a non-decreasing L\'evy process.  
The first-hitting time process $\{E(t) t \geq 0\}$ (which is sometimes  
referred to as an inverse subordinator) defined by $E(t) = \inf \{s:  
D(s) > t \}$ is a process which has arisen in many applications. Of  
particular interest is the mean first-hitting time $U(t)=\mathbb{E}E(t) 
$. This function characterizes all finite-dimensional distributions of  
the process $E$. The function $U$ can be calculated by inverting the  
Laplace transform of the function $\widetilde{U}(\lambda) = (\lambda  
\phi(\lambda))^{-1}$, where $\phi$ is the L\'evy exponent of the  
subordinator $D$. In this paper, we give two methods for computing  
numerically the inverse of this Laplace transform. The first is based  
on the Bromwich integral and the second is based on the Post-Widder  
inversion formula. The software written to support this work is  
available from the authors and we illustrate its use at the end of the  
paper.

http://arxiv.org/abs/0904.4232


8445. Exact maximum likelihood estimators for drift fractional  
Brownian motions
Author(s): Hu Yaozhong and Xiao Weilin and Zhang Weiguo

Abstract: This paper deals with the problems of consistence and strong  
consistence of the maximum likelihood estimators of the mean and  
variance of the drift fractional Brownian motions observed at discrete  
time instants. A central limit theorem for these estimators is also  
obtained by using the Malliavin calculus.

http://arxiv.org/abs/0904.4186


8446. On Limit theorems in $JW$- algebras
Author(s): Abdusalom Karimov and Farrukh Mukhamedov

Abstract: In the present paper, we study bundle convergence in $JW$-  
algebra and prove some ergodic theorems with respect to such  
convergence. Moreover, conditional expectations of $JW$-algebras are  
considered. Using such expectations, the convergence of  
supermartingales in $JW$- algebras is established.

http://arxiv.org/abs/0904.4070


8447. Large deviations of empirical zero point measures on Riemann  
surfaces,
Author(s): O. Zeitouni and S. Zelditch

Abstract: We prove an LDP for the empirical measure of complex zeros  
of a Gaussian random complex polynomial of degree N of one variable as  
N tends to infinity. The Gaussian measure is induced by an inner  
product defined by a smooth weight (Hermitian metric) $h$ and a  
Bernstein-Markov measure $\nu$. The speed is N^2 and the the unique  
minimizer of the rate function $I$ is the weighted equilibrium measure  
$\nu_{h, K}$ with respect to $h$ on the support $K$ of $\nu$.

http://arxiv.org/abs/0904.4271


8448. Continuous-time trading and the emergence of probability
Author(s): Vladimir Vovk

Abstract: This paper establishes a non-stochastic analogue of the  
celebrated result by Dubins and Schwarz about reduction of continuous  
martingales to Brownian motion via time change. We consider an  
idealized financial security with continuous price process, without  
making any stochastic assumptions. It is shown that almost all sample  
paths of the price process possess quadratic variation, where "almost  
all" is understood in the following game-theoretic sense: there exists  
a trading strategy that earns infinite capital without risking more  
than one monetary unit if the process of quadratic variation does not  
exist. Replacing time by the quadratic variation process, we show that  
the price process becomes Brownian motion. This is essentially the  
same conclusion as in the Dubins-Schwarz result, except that the  
probabilities (constituting the Wiener measure) emerge instead of  
being postulated. We also give an elegant statement, inspired by Peter  
McCullagh's unpublished work, of this result in terms of game- 
theoretic probability.

http://arxiv.org/abs/0904.4364


8449. Intrinsic ultracontractivity for Schrodinger operators based on  
fractional Laplacians
Author(s): Kamil Kaleta and Tadeusz Kulczycki

Abstract: We study the Feynman-Kac semigroup generated by the  
Schr{\"o}dinger operator based on the fractional Laplacian $-(- 
\Delta)^{\alpha/2} - q$ in $\Rd$, for $q \ge 0$, $\alpha \in (0,2)$.  
We obtain sharp estimates of the first eigenfunction $\phi_1$ of the  
Schr{\"o}dinger operator and conditions equivalent to intrinsic  
ultracontractivity of the Feynman-Kac semigroup. For potentials $q$  
such that $\lim_{|x| \to \infty} q(x) = \infty$ and comparable on unit  
balls we obtain that $\phi_1(x)$ is comparable to $(|x| + 1)^{-d -  
\alpha} (q(x) + 1)^{-1}$ and intrinsic ultracontractivity holds iff $ 
\lim_{|x| \to \infty} q(x)/\log|x| = \infty$. Proofs are based on  
uniform estimates of $q$-harmonic functions.

http://arxiv.org/abs/0904.4386


8450. Adaptive sampling for linear state estimation
Author(s): Maben Rabi and George V. Moustakides and John S. Baras

Abstract: State estimation under sampling rate constraints is  
important for Networked control. To obtain the lowest possible  
estimator distortion under such constraints, the samples must be  
chosen adaptively based on the trajectory of the signal being sampled,  
rather than deterministically. We treat the case of perfect  
observations at the sensor in which it measures a diffusion state  
process perfectly. The sensor has to choose causally, exactly N  
sampling times when it transmits samples to a supervisor which  
receives the samples without delay or distortion. Based on the causal  
sequence of samples it receives, the supervisor maintains a continuous  
MMSE estimate. In this paper we provide the optimal adaptive sampling  
rules to be used by the sensor that minimize the aggregate, finite- 
horizon, mean-square error distortion for scalar linear estimation. We  
also characterize the performance of the suboptimal class of Delta  
sampling schemes which uses fixed thresholds as sampling envelopes.  
The results of these calculations are surprising. Delta sampling  
performs worse than even the periodic sampling scheme, except possibly  
when the sample budget is quite small.

http://arxiv.org/abs/0904.4358


8451. Rotor Walks and Markov Chains
Author(s): Alexander E. Holroyd and James Propp

Abstract: The rotor walk is a derandomized version of the random walk  
on a graph. On successive visits to any given vertex, the walker is  
routed to each of the neighboring vertices in some fixed cyclic order,  
rather than to a random sequence of neighbors. The concept generalizes  
naturally to Markov chains on a countable state space. Subject to  
general conditions, we prove that many natural quantities associated  
with the rotor walk (including normalized hitting frequencies, hitting  
times and occupation frequencies) concentrate around their expected  
values for the random walk. Furthermore, the concentration is stronger  
than that associated with repeated runs of the random walk, with  
discrepancy at most C/n after n runs (for an explicit constant C),  
rather than constant/sqrt n.

http://arxiv.org/abs/0904.4507


8452. On the Representation Theorem of G-Expectations and Paths of G-- 
Brownian Motion
Author(s): Mingshang Hu and Shige Peng

Abstract: We give a very simple and elementary proof of the existence  
of a weakly compact family of probability measures $\{P_{\theta}: 
\theta \in \Theta \}$ to represent an important sublinear expectation-- 
G-expectation $\mathbb{E}[\cdot]$. We also give a concrete  
approximation of a bounded continuous function $X(\omega)$ by an  
increasing sequence of cylinder functions $L_{ip}(\Omega)$ in order to  
prove that $C_{b}(\Omega)$ belongs to the $\mathbb{E}[|\cdot|]$- 
completion of the $L_{ip}(\Omega)$.

http://arxiv.org/abs/0904.4519


8453. Metric properties of discrete time exclusion type processes in  
continuum
Author(s): Michael Blank

Abstract: A new class of exclusion type processes acting in continuum  
with synchronous updating is introduced and studied. Ergodic averages  
of particle velocities are obtained and their connections to other  
statistical quantities, in particular to the particle density (the so  
called Fundamental Diagram) is analyzed rigorously. The main technical  
tool is a "dynamical" coupling applied in a nonstandard fashion: we do  
not prove the existence of the successful coupling (which even might  
not hold) but instead use its presence/absence as an important  
diagnostic tool. Despite that this approach cannot be applied to  
lattice systems directly, it allows to obtain new results for the  
lattice systems embedding them to the systems in continuum.  
Applications to the traffic flows modelling are discussed as well.

http://arxiv.org/abs/0904.4585


8454. On random topological Markov chains with big images and preimages
Author(s): Manuel Stadlbauer

Abstract: We introduce a relative notion of the 'big images and  
preimages'-property for random topological Markov chains. This then  
implies that a relative version of the Ruelle-Perron-Frobenius theorem  
holds with respect to summable and locally Hoelder continuous  
potentials.

http://arxiv.org/abs/0904.4630


8455. VRRW on complete-like graphs: almost sure behavior
Author(s): Vlada Limic and Stanislav Volkov

Abstract: By a theorem of Volkov (2001) we know that on most graphs,  
with positive probability, the linearly vertex-reinforced random walk  
(VRRW) stays within a finite "trapping" subgraph at all large times.  
The question of whether this tail behavior occurs with probability one  
is open in general. R. Pemantle (1988) in his thesis proved, via a  
dynamical system approach, that for a VRRW on any complete graph the  
asymptotic frequency of visits is uniform over vertices. These  
techniques do not easily extend even to the setting of complete-like  
graphs, that is, complete graphs ornamented with finitely many leaves  
at each vertex. In this work we combine martingale and large deviation  
techniques to prove that almost surely the VRRW on any such graph  
spends positive (and equal) proportions of time on each of its non- 
leaf vertices. This behavior was previously shown to occur only up to  
event of positive probability, cf. Volkov (2001). We believe that our  
approach can be used as a building block in studying related questions  
on more general graphs. The same set of techniques is used to obtain  
explicit bounds on the speed of convergence of the empirical  
occupation measure.

http://arxiv.org/abs/0904.4722


8456. Restricted isometry property of matrices with independent  
columns and neighborly polytopes by random sampling
Author(s): Rados{\l}aw Adamczak and Alexander E. Litvak and Alain  
Pajor and Nicole Tomczak-Jaegermann

Abstract: This paper considers compressed sensing matrices and  
neighborliness of a centrally symmetric convex polytope generated by  
vectors $\pm X_1,...,\pm X_N\in\R^n$, ($N\ge n$). We introduce a class  
of random sampling matrices and show that they satisfy a restricted  
isometry property (RIP) with overwhelming probability. In particular,  
we prove that matrices with i.i.d. centered and variance 1 entries  
that satisfy uniformly a sub-exponential tail inequality possess this  
property RIP with overwhelming probability. We show that such  
"sensing" matrices are valid for the exact reconstruction process of $m 
$-sparse vectors via $\ell_1$ minimization with $m\le Cn/\log^2 (cN/n) 
$. The class of sampling matrices we study includes the case of  
matrices with columns that are independent isotropic vectors with log- 
concave densities. We deduce that if $K\subset \R^n$ is a convex body  
and $X_1,..., X_N\in K$ are i.i.d. random vectors uniformly  
distributed on $K$, then, with overwhelming probability, the symmetric  
convex hull of these points is an $m$-centrally-neighborly polytope  
with $m\sim n/\log^2 (cN/n)$.

http://arxiv.org/abs/0904.4723


8457. Current fluctuations of a system of one-dimensional random walks  
in random environment
Author(s): Jonathon Peterson and Timo Seppalainen

Abstract: We study the current of particles that move independently in  
a common static random environment on the one-dimensional integer  
lattice. A two-level fluctuation picture appears. On the central limit  
scale the quenched mean of the current process converges to a Brownian  
motion. On a smaller scale the current process centered at its  
quenched mean converges to a mixture of Gaussian process. These  
Gaussian processes are similar to those arising from classical random  
walks, but the environment makes itself felt through an additional  
Brownian random shift in the spatial argument of the limiting current  
process.

http://arxiv.org/abs/0904.4768


8458. Right Inverses of Levy processes
Author(s): R. Doney and M. Savov

Abstract: We call a right continuous increasing process K(x) a partial  
right inverse (PRI) of a given Levy process X if X(K{x))=x at least  
for all x in some random interval [0,c) of of positive length. In this  
paper we give a necessary and sufficient condition for the existence  
of a PRI in terms of the Levy triplet.

http://arxiv.org/abs/0904.4871


8459. Remarks on the fractional Brownian motion
Author(s): Denis Feyel and Arnaud De La Pradelle (Institut math jussieu)

Abstract: We study the fBm by use of convolution of the standard white  
noise with a certain distribution. This brings some simplifications  
and new results.

http://arxiv.org/abs/0904.4923


8460. A Supplement to the Paper Poisson Approximation in a Poisson  
Limit Theorem Inspired by Coupon Collecting
Author(s): Anna P\'osfai

Abstract: In this note we give a proof for the result stated as  
Theorem 4 in Poisson Approximation in a Poisson Limit Theorem Inspired  
by Coupon Collecting.

http://arxiv.org/abs/0904.4924


8461. Maximizing the probability of attaining a target prior to  
extinction
Author(s): Debasish Chatterjee and Eugenio Cinquemani and John Lygeros

Abstract: We present a dynamic programming-based solution to the  
problem of maximizing the probability of attaining a target set before  
hitting a cemetery set for a discrete-time Markov control process.  
Under mild hypotheses we establish that there exists a deterministic  
stationary policy that achieves the maximum value of this probability.  
We demonstrate how the maximization of this probability can be  
computed through the maximization of an expected total reward until  
the first hitting time to either the target or the cemetery set.  
Martingale characterizations of thrifty, equalizing, and optimal  
policies in the context of our problem are also established.

http://arxiv.org/abs/0904.4143


8462. Two speed TASEP
Author(s): Alexei Borodin (1) and Patrik L. Ferrari (2) and Tomohiro  
Sasamoto (3) ((1) Caltech, (2) Bonn University, (3) Chiba University  
and TU Munich)

Abstract: We consider the TASEP on Z with two blocks of particles  
having different jump rates. We study the large time behavior of  
particles' positions. It depends both on the jump rates and the region  
we focus on, and we determine the complete process diagram. In  
particular, we discover a new transition process in the region where  
the influence of the random and deterministic parts of the initial  
condition interact. Slow particles may create a shock, where the  
particle density is discontinuous and the distribution of a particle's  
position is asymptotically singular. We determine the diffusion  
coefficient of the shock without using second class particles. We also  
analyze the case where particles are effectively blocked by a wall  
moving with speed equal to their intrinsic jump rate.

http://arxiv.org/abs/0904.4655




-----------------------------------
Stefano M. Iacus
Department of Economics,
Business and Statistics
University of Milan
Via Conservatorio, 7
I-20123 Milan - Italy
Ph.: +39 02 50321 461
Fax: +39 02 50321 505
http://www.economia.unimi.it/iacus
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From pas at lists.imstat.org  Mon Jul 27 12:03:08 2009
From: pas at lists.imstat.org (Probability Abstract Service)
Date: Mon, 27 Jul 2009 19:03:08 +0200
Subject: [PAS] Probability Abstracts 110
Message-ID: <EFB7E415-6FB8-45C3-924B-BA450258E688@unimi.it>

Probability Abstracts 110
This document contains abstracts 8463-8724
from May-1-2009 to June-30-2009.
They have been mailed on July 27, 2009.

Apologizes for this delay. Next PAS letter will
be distributed regularly on early september.

This letter can be also found on line at
http://pas.imstat.org/Letters/letter_110.shtml


---------------------------------------------------------------

8463. POLYGONAL WEB REPRESENTATION FOR HIGHER ORDER CORRELATION  
FUNCTIONS OF  CONSISTENT POLYGONAL MARKOV FIELDS IN THE PLANE

Tomasz Schreiber

We consider polygonal Markov fields originally introduced by Arak and
Surgailis (1982,1989).
   Our attention is focused on fields with nodes of order two, which  
can be
regarded as continuum ensembles of non-intersecting contours in the  
plane,
sharing a number of salient features with the two-dimensional Ising  
model. The
purpose of this paper is to establish an explicit stochastic  
representation for
the higher-order correlation functions of polygonal Markov fields in  
their
consistency regime. The representation is given in terms of the so- 
called crop
functionals (defined by a Moebius-type formula) of polygonal webs  
which arise
in a graphical construction dual to that giving rise to polygonal  
fields. The
proof of our representation formula goes by constructing a martingale
interpolation between the correlation functions of polygonal fields  
and crop
functionals of polygonal webs.


  http://arxiv.org/abs/0905.0208

---------------------------------------------------------------

8464. LEVY'S ZERO-ONE LAW IN GAME-THEORETIC PROBABILITY

Glenn Shafer and  Vladimir Vovk and  and Akimichi Takemura

We prove a game-theoretic version of Levy's zero-one law, and deduce  
several
corollaries from it, including Kolmogorov's zero-one law, the  
ergodicity of
Bernoulli shifts, and a zero-one law for dependent trials. Our  
secondary goal
is to explore the basic definitions of game-theoretic probability  
theory, with
Levy's zero-one law serving a useful role.


  http://arxiv.org/abs/0905.0254

---------------------------------------------------------------

8465. A Q-ANALOGUE OF DE FINETTI'S THEOREM

Alexander Gnedin and Grigori Olshanski

A q-analogue of de Finetti's theorem is obtained in terms of a boundary
problem for the q-Pascal graph. For q a power of prime this leads to a
characterisation of random spaces over the Galois field F_q that are  
invariant
under the natural action of the infinite group of invertible matrices  
with
coefficients from F_q.


  http://arxiv.org/abs/0905.0367

---------------------------------------------------------------

8466. SUSCEPTIBILITY IN INHOMOGENEOUS RANDOM GRAPHS

Svante Janson and Oliver Riordan

We study the susceptibility, i.e., the mean size of the component  
containing
a random vertex, in a general model of inhomogeneous random graphs.  
This is one
of the fundamental quantities associated to (percolation) phase  
transitions; in
practice one of its main uses is that it often gives a way of  
determining the
critical point by solving certain linear equations. Here we relate the
susceptibility of suitable random graphs to a quantity associated to the
corresponding branching process, and study both quantities in various  
natural
examples.


  http://arxiv.org/abs/0905.0437

---------------------------------------------------------------

8467. ON ERGODIC TWO-ARMED BANDITS

Pierre Tarr\`es and Pierre Vandekerkhove

A device has two arms with unknown deterministic payoffs, and the aim  
is to
asymptotically identify the best one without spending too much time on  
the
other. The Narendra algorithm offers a stochastic procedure to this  
end. We
show under weak ergodic assumptions on these deterministic payoffs  
that the
procedure eventually chooses the best arm (i.e. with greatest Cesaro  
limit)
with probability one, for appropriate step sequences of the algorithm.  
In the
case of i.i.d. payoffs, this implies a "quenched" version of the  
"annealed"
result of Lamberton, Pages and Tarres in 2004 by the law of iterated  
logarithm,
thus generalizing it.
   More precisely, if $(\eta_{l,i})_{i\in\N}\in\{0,1\}^\N$, $l\in\{A,B 
\}$, are
the deterministic reward sequences we would get if we played at time $i 
$, we
obtain infallibility with the same assumption on nonincreasing step  
sequences
on the payoffs as in the result mentioned above, replacing the i.i.d.
assumption by the hypothesis that the empirical averages
$\sum_{i=1}^n\eta_{A,i}/n$ and$\sum_{i=1}^n\eta_{B,i}/n$ converge, as  
$n$ tends
to infinity, respectively to $\theta_A$ and $\theta_B$, with rate at  
least
$1/(\log n)^{1+\e}$, for some $\e>0$.


  http://arxiv.org/abs/0905.0463

---------------------------------------------------------------

8468. NON-GLOBALLY LIPSCHITZ COUNTEREXAMPLES FOR THE STOCHASTIC EULER  
SCHEME

Martin Hutzenthaler and Arnulf Jentzen

The stochastic Euler scheme is known to converge to the exact solution  
of a
stochastic differential equation with globally Lipschitz coefficients  
and even
with coefficients which grow at most linearly. For super-linearly  
growing
coefficients convergence in the strong and numerically weak sense  
remained an
open question. In this article we prove for many stochastic differential
equations with super-linearly growing coefficients that Euler's  
approximation
does not converge neither in the strong sense nor in the numerically  
weak sense
to the exact solution. Even worse, the difference of the exact  
solution and of
the numerical approximation diverges to infinity in the strong sense  
and in the
numerically weak sense.


  http://arxiv.org/abs/0905.0273

---------------------------------------------------------------

8469. MOD-POISSON CONVERGENCE IN PROBABILITY AND NUMBER THEORY

E. Kowalski and  A. Nikeghbali

Building on earlier work introducing the notion of "mod-Gaussian"  
convergence
of sequences of random variables, which arises naturally in Random  
Matrix
Theory and number theory, we discuss the analogue notion of "mod- 
Poisson"
convergence. We show in particular how it occurs naturally in analytic  
number
theory in the classical Erd\H{o}s-K\'ac Theorem. In fact, this case  
reveals
deep connections and analogies with conjectures concerning the  
distribution of
L-functions on the critical line, which belong to the mod-Gaussian  
framework,
and with analogues over finite fields, where it can be seen as a
zero-dimensional version of the Katz-Sarnak philosophy in the large  
conductor
limit.


  http://arxiv.org/abs/0905.0318

---------------------------------------------------------------

8470. DUALITY IN INHOMOGENEOUS RANDOM GRAPHS, AND THE CUT METRIC

Svante Janson and Oliver Riordan

The classical random graph model $G(n,\lambda/n)$ satisfies a `duality
principle', in that removing the giant component from a supercritical  
instance
of the model leaves (essentially) a subcritical instance. Such  
principles have
been proved for various models; they are useful since it is often much  
easier
to study the subcritical model than to directly study small components  
in the
supercritical model. Here we prove a duality principle of this type  
for a very
general class of random graphs with independence between the edges,  
defined by
convergence of the matrices of edge probabilities in the cut metric.


  http://arxiv.org/abs/0905.0434

---------------------------------------------------------------

8471. WHAT HAPPENS AFTER A DEFAULT: THE CONDITIONAL DENSITY APPROACH

Nicole El Karoui (PMA and  CMAP) and  Monique Jeanblanc (DP) and  Ying  
Jiao (PMA)

We present a general model for default time, making precise the role  
of the
intensity process, and showing that this process allows for a  
knowledge of the
conditional distribution of the default only "before the default".  
This lack of
information is crucial while working in a multi-default setting. In a  
single
default case, the knowledge of the intensity process does not allow to  
compute
the price of defaultable claims, except in the case where immersion  
property is
satisfied. We propose in this paper the density approach for default  
time. The
density process will give a full characterization of the links between  
the
default time and the reference filtration, in particular "after the  
default
time". We also investigate the description of martingales in the full
filtration in terms of martingales in the reference filtration, and  
the impact
of Girsanov transformation on the density and intensity processes, and  
also on
the immersion property.


  http://arxiv.org/abs/0905.0559

---------------------------------------------------------------

8472. LARGE CLIQUES IN A POWER-LAW RANDOM GRAPH

Svante Janson and  Tomasz {\L}uczak and  Ilkka Norros

We study the size of the largest clique $\omega(G(n,\alpha))$ in a  
random
graph $G(n,\alpha)$ on $n$ vertices which has power-law degree  
distribution
with exponent $\alpha$. We show that for `flat' degree sequences with
$\alpha>2$ whp the largest clique in $G(n,\alpha)$ is of a constant  
size, while
for the heavy tail distribution, when $0<\alpha<2$, $\omega(G(n, 
\alpha))$ grows
as a power of $n$. Moreover, we show that a natural simple algorithm  
whp finds
in $G(n,\alpha)$ a large clique of size $(1+o(1))\omega(G(n,\alpha))$ in
polynomial time.


  http://arxiv.org/abs/0905.0561

---------------------------------------------------------------

8473. COUNTING NONDECREASING INTEGER SEQUENCES THAT LIE BELOW A BARRIER

Robin Pemantle and Herbert S. Wilf

Given a barrier $0 \leq b_0 \leq b_1 \leq ...$, let $f(n)$ be the  
number of
nondecreasing integer sequences $0 \leq a_0 \leq a_1 \leq ... \leq a_n 
$ for
which $a_j \leq b_j$ for all $0 \leq j \leq n$. Known formul\ae for  
$f(n)$
include an $n \times n$ determinant whose entries are binomial  
coefficients
(Kreweras, 1965) and, in the special case of $b_j = rj+s$, a short  
explicit
formula (Proctor, 1988, p.320). A relatively easy bivariate recursion,
decomposing all sequences according to $n$ and $a_n$, leads to a  
bivariate
generating function, then a univariate generating function, then a  
linear
recursion for $\{f(n) \}$. Moreover, the coefficients of the bivariate
generating function have a probabilistic interpretation, leading to an  
analytic
inequality which is an identity for certain values of its argument.


  http://arxiv.org/abs/0905.0609

---------------------------------------------------------------

8474. RIGOROUS DERIVATION OF THE LANDAU EQUATION IN THE WEAK COUPLING  
LIMIT

Kay Kirkpatrick

We examine a family of microscopic models of plasmas, with a parameter
$\alpha$ comparing the typical distance between collisions to the  
strength of
the grazing collisions. These microscopic models converge in  
distribution, in
the weak coupling limit, to a velocity diffusion described by the  
linear Landau
equation (also known as the Fokker-Planck equation). The present work  
extends
and unifies previous results that handled the extremes of the parameter
$\alpha$, for the whole range (0, 1/2], by showing that clusters of  
overlapping
obstacles are negligible in the limit. Additionally, we study the  
diffusion
coefficient of the Landau equation and show it to be independent of the
parameter.


  http://arxiv.org/abs/0905.0649

---------------------------------------------------------------

8475. Q-DISTRIBUTIONS ON BOXED PLANE PARTITIONS

Alexei Borodin and  Vadim Gorin and  Eric M. Rains

We introduce elliptic weights of boxed plane partitions and prove that  
they
give rise to a generalization of MacMahon's product formula for the  
number of
plane partitions in a box. We then focus on the most general positive
degenerations of these weights that are related to orthogonal  
polynomials; they
form three two-dimensional families. For distributions from these  
families we
prove two types of results.
   First, we construct explicit Markov chains that preserve these  
distributions.
In particular, this leads to a relatively simple exact sampling  
algorithm.
   Second, we consider a limit when all dimensions of the box grow and  
plane
partitions become large, and prove that the local correlations  
converge to
those of ergodic translation invariant Gibbs measures. For fixed  
proportions of
the box, the slopes of the limiting Gibbs measures (that can also be  
viewed as
slopes of tangent planes to the hypothetical limit shape) are encoded  
by a
single quadratic polynomial.


  http://arxiv.org/abs/0905.0679

---------------------------------------------------------------

8476. MOMENT ESTIMATES FOR SOLUTIONS OF LINEAR STOCHASTIC  
DIFFERENTIAL  EQUATIONS DRIVEN BY ANALYTIC FRACTIONAL BROWNIAN MOTION

J\'er\'emie Unterberger (IECN)

As a general rule, differential equations driven by a multi-dimensional
irregular path $\Gamma$ are solved by constructing a rough path over $ 
\Gamma$.
The domain of definition ? and also estimates ? of the solutions  
depend on
upper bounds for the rough path; these general, deterministic  
estimates are too
crude to apply e.g. to the solutions of stochastic differential  
equations with
linear coefficients driven by a Gaussian process with H\"older  
regularity
$\alpha < 1/2$. We prove here (by showing convergence of Chen's  
series) that
linear stochastic differential equations driven by analytic fractional  
Brownian
motion [7, 8] with arbitrary Hurst index $\alpha \in (0, 1)$ may be  
solved on
the closed upper halfplane, and that the solutions have finite variance.


  http://arxiv.org/abs/0905.0782

---------------------------------------------------------------

8477. PATH REGULARITY AND EXPLICIT CONVERGENCE RATE FOR BSDE WITH  
TRUNCATED  QUADRATIC GROWTH

Peter Imkeller and Goncalo dos Reis

We consider backward stochastic differential equations with drivers of
quadratic growth (qgBSDE). We prove several statements concerning path
regularity and stochastic smoothness of the solution processes of the  
qgBSDE,
in particular we prove an extension of Zhang's path regularity theorem  
to the
quadratic growth setting. We give explicit convergence rates for the  
difference
between the solution of a qgBSDE and its truncation, filling an  
important gap
in numerics for qgBSDE. We give an alternative proof of second order  
Malliavin
differentiability for BSDE with drivers that are Lipschitz continuous  
(and
differentiable), and then derive the same result for qgBSDE.


  http://arxiv.org/abs/0905.0788

---------------------------------------------------------------

8478. RANDOM WALK ON THE INTEGERS WITH EQUIDISTANT MULTIPLE FUNCTION  
BARRIERS

Theo van Uem

We obtain expected number of arrivals, probability of arrival,  
absorption
probabilities and expected time before absorption for a discrete  
random walk on
the integers with an infinite set of equidistant multiple function  
barriers


  http://arxiv.org/abs/0905.0823

---------------------------------------------------------------

8479. STABILITY OF A GROWTH PROCESS GENERATED BY MONOMER FILLING WITH   
NEAREST-NEIGHBOR COOPERATIVE EFFECTS

Vadim Shcherbakov and  Stanislav Volkov

In this paper we study stability of a growth process generated by a
cooperative sequential adsorption model (CSA) on the lattice. The  
lattice CSA
can be regarded as a variant of Polya urn scheme with interaction and  
the
growth process is formed by the numbers of adsorbed (allocated)  
particles at
lattice sites, called heights. In our paper stability of the growth  
process,
loosely speaking, means that its components grow at approximately the  
same
rate. To assess stability quantitatively we study a stochastic process  
formed
by differences of heights.


  http://arxiv.org/abs/0905.0835

---------------------------------------------------------------

8480. UPPER TAILS FOR COUNTING OBJECTS IN RANDOMLY INDUCED  
SUBHYPERGRAPHS AND  ROOTED RANDOM GRAPHS

Svante Janson and Andrzej Rucinski

General upper tail estimates are given for counting edges in a random  
induced
subhypergraph of a fixed hypergraph H, with an easy proof by  
estimating the
moments. As an application we consider the numbers of arithmetic  
progressions
and Schur triples in random subsets of integers. In the second part of  
the
paper we return to the subgraph counts in random graphs and provide  
upper tail
estimates in the rooted case.


  http://arxiv.org/abs/0905.0972

---------------------------------------------------------------

8481. ON A STOCHASTIC WAVE EQUATION DRIVEN BY A NON-GAUSSIAN LEVY  
PROCESS

Lijun Bo (XIDIAN) and  Kehua Shi (NANKAI) and  Yongjin Wang (NANKAI)

This paper investigates a damped stochastic wave equation driven by a
non-Gaussian Levy noise. The weak solution is proved to exist and be  
unique.
Moreover we show the existence of a unique invariant measure  
associated with
the transition semigroup under mild conditions.


  http://arxiv.org/abs/0905.0992

---------------------------------------------------------------

8482. INTERMITTENCY AND AGING FOR THE SYMBIOTIC BRANCHING MODEL

Frank Aurzada and Leif D\"oring

For the symbiotic branching model introduced by Etheridge/Fleischmann  
(2004),
it is shown that aging and intermittency exhibit different behaviour for
negative, zero, and positive correlations. Our approach also provides an
alternative, elementary proof and refinements of classical results  
concerning
second moments of the parabolic Anderson model with Brownian  
potential. Some
refinements to more general (also infinite range) kernels of recent  
aging
results of Dembo/Deuschel (2007) for interacting diffusions are given.


  http://arxiv.org/abs/0905.1003

---------------------------------------------------------------

8483. STRONG MIXING PROPERTY FOR STIT TESSELLATION

Rapha\"el Lachi\`eze-Rey

The so-called STIT tessellations form the class of homogeneous  
(spatially
stationary) tessellations of $\mathbb{R}^d$ which are stable under the
nesting/iteration operation. In this paper, we establish the strong  
mixing
property for these tessellations and give the optimal form of the rate  
of decay
for the quantity $|\mathbb{P}({A}\cap Y=\emptyset,T_h B \cap
Y=\emptyset)-\mathbb{P}({A}\cap Y=\emptyset)\mathbb{P}({B}\cap Y= 
\emptyset)|$
when $A$ and $B$ are two compact sets, $h$ a vector of $\mathbb{R}^d$,  
$T_{h}$
the corresponding translation operator and $Y$ a STIT Tessellation.


  http://arxiv.org/abs/0905.1145

---------------------------------------------------------------

8484. $L^P$ BOUNDS FOR A COMBINATORIAL CENTRAL LIMIT THEOREM WITH  
INVOLUTIONS

Subhankar Ghosh

Let $E=((e_{ij}))_{n\times n}$ be a fixed array of real numbers such  
that
$e_{ij}=e_{ji}, e_{ii}=0$ for $1\le i,j \le n$. Let the symmetric  
group be
denoted by $S_n$ and the collection of involutions with no fixed  
points by
$\Pi_n$, that is, $\Pi_n=\{\pi\in S_n: \pi^2=id, \pi(i)\neq i \forall i 
\}$. For
$\pi$ uniformly chosen from $\Pi_n$, let $Y_E=\sum_{i=1}^n e_{i\pi(i)} 
$ and
$W=(Y_E-\mu_E)/\sigma_E$ where $\mu_E=E(Y_E)$ and $\sigma_E^2={Var} 
(Y_E)$.
Denoting by $F_W$ and $\Phi$ the distribution functions of $W$ and a
$\mathcal{N}(0,1)$ variate respectively, we bound $||F_W-\Phi||_p$ for  
$ 1\le
p\le \infty$ using Stein's method and zero bias transformations. The  
resulting
bound obtained is the product of a third moment type quantity  
multiplied by an
explicit constant, and in particular for $p=\infty$ is of the same  
form as the
one obtained by Bolthausen for Hoeffdings combinatorial central limit  
theorem
when $\pi$ is chosen uniformly from $S_n$. The approximation developed  
here for
involutions has applications in testing whether there is a significant  
degree
of similarity in certain matched pairs experiments.


  http://arxiv.org/abs/0905.1150

---------------------------------------------------------------

8485. TRANSITION PHENOMENA FOR LADDER EPOCHS OF RANDOM WALKS WITH  
SMALL  NEGATIVE DRIFT

Vitali Wachtel

For a family of random walks $\{S^{(a)}\}$ satisfying
$\mathbf{E}S_1^{(a)}=-a<0$ we consider ladder epochs $\tau^{(a)}=\min\ 
{k\geq1:
S_k^{(a)}<0\}$. We study the asymptotic, as $a\to0$, behaviour of
$\mathbf{P}(\tau^{(a)}>n)$ in the case when $n=n(a)\to\infty$. As a  
consequence
we obtain also the growth rates of the moments of $\tau^{(a)}$.


  http://arxiv.org/abs/0905.1186

---------------------------------------------------------------

8486. ERRATUM: PERCOLATION ON RANDOM JOHNSON-MEHL TESSELLATIONS AND  
RELATED  MODELS

Bela Bollobas and Oliver Riordan

We correct a simple error in Percolation on random Johnson-Mehl  
tessellations
and related models, Probability Theory and Related Fields 140 (2008),  
417-468.
(See also arXiv:math/0610716)


  http://arxiv.org/abs/0905.1275

---------------------------------------------------------------

8487. CENTRAL LIMIT THEOREMS FOR GROMOV HYPERBOLIC GROUPS

Michael Bjorklund

In this paper we study asymptotic properties of symmetric and non- 
degenerate
random walks on transient hyperbolic groups. We prove a central limit  
theorem
and a law of iterated logarithm for the drift of a random walk,  
extending
previous results by S. Sawyer and T. Steger and F. Ledrappier for  
certain CAT
minus one groups. The proofs use a result by A. Ancona on the  
identification of
the Martin boundary of a hyperbolic group with its Gromov boundary. We  
also
give a new interpretation, in terms of Hilbert metrics, of the Green  
metric,
first introduced by S. Brofferio and S. Blachere.


  http://arxiv.org/abs/0905.1297

---------------------------------------------------------------

8488. DIAMOND AGGREGATION

Wouter Kager and  Lionel Levine

Internal diffusion-limited aggregation is a growth model based on  
random walk
in Z^d. We study how the shape of the aggregate depends on the law of  
the
underlying walk, focusing on a family of walks in Z^2 for which the  
limiting
shape is a diamond. Certain of these walks -- those with a directional  
bias
toward the origin -- have at most logarithmic fluctuations around the  
limiting
shape. This contrasts with the simple random walk, where the limiting  
shape is
a disk and the best known bound on the fluctuations, due to Lawler, is  
a power
law. Our walks enjoy a uniform layering property which simplifies many  
of the
proofs.


  http://arxiv.org/abs/0905.1361

---------------------------------------------------------------

8489. TERM STRUCTURE MODELS DRIVEN BY WIENER PROCESS AND POISSON  
MEASURES:  EXISTENCE AND POSITIVITY

Damir Filipovic and  Stefan Tappe and  Josef Teichmann

In the spirit of Bj\"ork-DiMasi-Kabanov-Runggaldier, we investigate term
structure models driven by Wiener process and Poisson measures with  
forward
curve dependent volatilities. This includes a full existence and  
uniqueness
proof for the corresponding Heath--Jarrow--Morton type term structure  
equation.
Furthermore, we characterize positivity preserving models by means of  
the
characteristic coefficients, which was open for jump-diffusions.  
Additionally
we treat existence, uniqueness and positivity of the Brody-Hughston  
equation of
interest rate theory with jumps, an equation which we believe to be  
very useful
for applications. A key role in our investigation is played by the  
method of
the moving frame, which allows to transform the Heath--Jarrow--Morton-- 
Musiela
equation to a time-dependent SDE.


  http://arxiv.org/abs/0905.1413

---------------------------------------------------------------

8490. ESTIMATION OF THE DRIFT OF FRACTIONAL BROWNIAN MOTION

Es-Sebaiy Khalifa (SAMOS) and  Idir Ouassou and  Youssef Ouknine

We consider the problem of efficient estimation for the drift of  
fractional
Brownian motion $B^H:=(B^H_t)_{t\in[0,T]}$ with hurst parameter $H$  
less than
1/2. We also construct superefficient James-Stein type estimators which
dominate, under the usual quadratic risk, the natural maximum likelihood
estimator.


  http://arxiv.org/abs/0905.1419

---------------------------------------------------------------

8491. ON THE DOVBYSH-SUDAKOV REPRESENTATION RESULT

Dmitry Panchenko

We present a detailed proof of the Dovbysh-Sudakov representation for
symmetric positive definite weakly exchangeable infinite random  
arrays, called
Gram-de Finetti matrices, which is based on the representation result  
of Aldous
and Hoover for arbitrary (not necessarily positive definite) symmetric  
weakly
exchangeable arrays.


  http://arxiv.org/abs/0905.1524

---------------------------------------------------------------

8492. TOTAL VARIATION MIXING TIME OF KAC'S RANDOM WALK

Yunjiang Jiang

We show that the classical Kac's random walk on $S^{n-1}$ starting  
from the
point mass at $e_1$ mixes in $\mathcal{O}(n^5 \log n)$ steps in total  
variation
distance. This improves a previous bound by Diaconis and Saloff-Coste of
$\mathcal{O}(n^{2n})$.


  http://arxiv.org/abs/0905.1539

---------------------------------------------------------------

8493. ON SMALL BALLS PROBLEM FOR STABLE MEASURES IN A HILBERT SPACE

Vygantas Paulauskas

In the paper the old results on probabilities of small balls for stable
measures in a Hilbert space, obtained in 1977 and remaining  
unpublished, are
presented. Apart of historical value these results are interesting  
even now,
since they are comparable with recently obtained ones.


  http://arxiv.org/abs/0905.1658

---------------------------------------------------------------

8494. LOCALLY MOST POWERFUL SEQUENTIAL TESTS OF A SIMPLE HYPOTHESIS  
VS  ONE-SIDED ALTERNATIVES

Andrey Novikov and  Petr Novikov

Let $X_1,X_2,...$ be a discrete-time stochastic process with a  
distribution
$P_\theta$, $\theta\in\Theta$, where $\Theta$ is an open subset of the  
real
line. We consider the problem of testing a simple hypothesis $H_0:$
$\theta=\theta_0$ versus a composite alternative $H_1:$ $\theta> 
\theta_0$,
where $\theta_0\in\Theta$ is some fixed point. The main goal of this  
article is
to characterize the structure of locally most powerful sequential  
tests in this
problem.
   For any sequential test $(\psi,\phi)$ with a (randomized) stopping  
rule
$\psi$ and a (randomized) decision rule $\phi$ let $\alpha(\psi,\phi)$  
be the
type I error probability, $\dot \beta_0(\psi,\phi)$ the derivative, at
$\theta=\theta_0$, of the power function, and $\mathscr N(\psi)$ an  
average
sample number of the test $(\psi,\phi)$. Then we are concerned with  
the problem
of maximizing $\dot \beta_0(\psi,\phi)$ in the class of all sequential  
tests
such that $$ \alpha(\psi,\phi)\leq \alpha\quad{and}\quad \mathscr  
N(\psi)\leq
\mathscr N, $$ where $\alpha\in[0,1]$ and $\mathscr N\geq 1$ are some
restrictions. It is supposed that $\mathscr N(\psi)$ is calculated  
under some
fixed (not necessarily coinciding with one of $P_\theta$) distribution  
of the
process $X_1,X_2...$.
   The structure of optimal sequential tests is characterized.


  http://arxiv.org/abs/0905.1437

---------------------------------------------------------------

8495. INFORMATION RANKING AND POWER LAWS ON TREES

Predrag R. Jelenkovic and Mariana Olvera-Cravioto

We study the situations when the solution to a weighted stochastic  
recursion
has a power law tail. To this end, we develop two complementary  
approaches, the
first one extends Goldie's (1991) implicit renewal theorem to cover  
recursions
on trees; and the second one is based on a direct sample path large  
deviations
analysis of weighted recursive random sums. We believe that these  
methods may
be of independent interest in the analysis of more general weighted  
branching
processes as well as in the analysis of algorithms.


  http://arxiv.org/abs/0905.1738

---------------------------------------------------------------

8496. LARGE DEVIATION PRINCIPLE AND INVISCID SHELL MODELS

Hakima Bessaih and  Annie Millet (CES and  SAMOS and  PMA)

A LDP is proved for the inviscid shell model of turbulence. As the  
viscosity
coefficient converges to 0 and the noise intensity is multiplied by  
the square
root of the viscosity, we prove that some shell models of turbulence  
with a
multiplicative stochastic perturbation driven by a H-valued Brownian  
motion
satisfy a LDP in C([0,T],V) for the topology of uniform convergence on  
[0,T],
but where V is endowed with a topology weaker than the natural one.  
The initial
condition has to belong to V and the proof is based on the weak  
convergence of
a family of stochastic control equations. The rate function is  
described in
terms of the solution to the inviscid equation.


  http://arxiv.org/abs/0905.1854

---------------------------------------------------------------

8497. STOCHASTIC APPROXIMATIONS OF SET-VALUED DYNAMICAL SYSTEMS:  
CONVERGENCE  WITH POSITIVE PROBABILITY TO AN ATTRACTOR

Mathieu Faure (UNINE) and  Roth Gregory (UNINE)

A succesful method to describe the asymptotic behavior of a discrete  
time
stochastic process governed by some recursive formula is to relate it  
to the
limit sets of a well chosen mean differential equation. Under an  
attainability
condition, convergence to a given attractor of the flow induced by this
dynamical system was proved to occur with positive probability (Bena 
\"im, 1999)
for a class of Robbins Monro algorithms. Bena\"im et al. (2005)  
generalised
this approach for stochastic approximation algorithms whose average  
behavior is
related to a differential inclusion instead. We pursue the analogy by  
extending
to this setting the result of convergence with positive probability to  
an
attractor.


  http://arxiv.org/abs/0905.1858

---------------------------------------------------------------

8498. A PROBABILISTIC NUMERICAL METHOD FOR FULLY NONLINEAR PARABOLIC  
PDES

Arash Fahim and  Nizar Touzi and Xavier Warin

We consider the probabilistic numerical scheme for fully nonlinear PDEs
suggested in [10], and show that it can be introduced naturally as a
combination of Monte Carlo and finite differences scheme without  
appealing to
the theory of backward stochastic differential equations. Our first  
main result
provides the convergence of the discrete-time approximation and  
derives a bound
on the discretization error in terms of the time step. An explicit
implementable scheme requires to approximate the conditional expectation
operators involved in the discretization. This induces a further Monte  
Carlo
error. Our second main result is to prove the convergence of the latter
approximation scheme, and to derive an upper bound on the  
approximation error.
Numerical experiments are performed for the approximation of the  
solution of
the mean curvature flow equation in dimensions two and three, and for  
two and
five-dimensional (plus time) fully-nonlinear Hamilton-Jacobi-Bellman  
equations
arising in the theory of portfolio optimization in financial  
mathematics.


  http://arxiv.org/abs/0905.1863

---------------------------------------------------------------

8499. CONCENTRATION OF RANDOM DETERMINANTS AND PERMANENT ESTIMATORS

Kevin P. Costello and  Van Vu

We show that the absolute value of the determinant of a matrix with  
random
independent (but not necessarily iid) entries is strongly concentrated  
around
its mean. As an application, we show that the Godsil-Gutman and Barvinok
estimators for the permanent of a strictly positive matrix give sub- 
exponential
approximation ratios with high probability.


  http://arxiv.org/abs/0905.1909

---------------------------------------------------------------

8500. COERCIVE INEQUALITIES ON METRIC MEASURE SPACES

W. Hebisch and  B. Zegarlinski

We study coercive inequalities on finite dimensional metric spaces with
probability measures which do not have volume doubling property. This  
class of
inequalities includes Poincar\'e and Log-Sobolev inequality. Our main  
result is
proof of Log-Sobolev inequality on Heisenberg group equipped with  
either heat
kernel measure or "gaussian" density build from optimal control  
distance. As
intermediate results we prove so called U-bounds.


  http://arxiv.org/abs/0905.1713

---------------------------------------------------------------

8501. ON SCHROEDINGER'S EQUATION, 3-DIMENSIONAL BESSEL BRIDGES, AND  
PASSAGE  TIME PROBLEMS

Gerardo Hernandez-del-Valle

We obtain explicit solutions for the density $\varphi_T$ of the first- 
time
$T$ that a one-dimensional Brownian process $B$ reaches the twice,  
continuously
differentiable moving boundary $f$ and such that $f''(t)\geq 0$ for  
all $t\in
\mathbb{R}^+$. We do so by finding the expected value of some  
functionals of a
3-dimensional Bessel bridge $\tilde{X}$ and exploiting its  
relationship with
first-passage time problems as pointed out by Kardaras (2007). It  
turns out
that this problem is related to Schr\"odinger's equation with time- 
dependent
linear potential, see Feng (2001).


  http://arxiv.org/abs/0905.1971

---------------------------------------------------------------

8502. ON THE FIRST PASSAGE TIME DENSITY OF A CONTINUOUS MARTINGALE  
OVER A  MOVING BOUNDARY

Gerardo Hernandez-del-Valle

In this paper we derive the density $\varphi$ of the first time $T$  
that a
continuous martingale $M$ with non-random quadratic variation
$<M>_\cdot:=\int_0^\cdot h^2(u)du$ hits a moving boundary $f$ which is  
twice
continuously differentiable, and $f'/h\in\mathbb{C}^2[0,\infty)$.
   Thus, this work is an extension to case in which $M$ is in fact a
one-dimensional standard Brownian motion $B$, as studied in Hernandez- 
del-Valle
(2007).


  http://arxiv.org/abs/0905.1975

---------------------------------------------------------------

8503. NON-EXTINCTION OF A FLEMING-VIOT PARTICLE MODEL

Mariusz Bieniek and  Krzysztof Burdzy and  Sam Finch

We consider a branching particle model in which particles move inside a
Euclidean domain according to the following rules. The particles move as
independent Brownian motions until one of them hits the boundary. This  
particle
is killed but another randomly chosen particle branches into two  
particles, to
keep the population size constant. We prove that the particle  
population does
not approach the boundary simultaneously in a finite time in some  
Lipschitz
domains. This is used to prove a limit theorem for the empirical  
distribution
of the particle family.


  http://arxiv.org/abs/0905.1999

---------------------------------------------------------------

8504. $L_P$-THEORY FOR THE STOCHASTIC HEAT EQUATION WITH INFINITE- 
DIMENSIONAL  FRACTIONAL NOISE

Raluca Balan

In this article, we consider the stochastic heat equation $du=(\Delta
u+f(t,x))dt+ \sum_{k=1}^{\infty} g^{k}(t,x) \delta \beta_t^k, t \in  
[0,T]$,
with random coefficients $f$ and $g^k$, driven by a sequence $ 
(\beta^k)_k$ of
i.i.d. fractional Brownian motions of index $H>1/2$. Using the Malliavin
calculus techniques and a $p$-th moment maximal inequality for the  
infinite sum
of Skorohod integrals with respect to $(\beta^k)_k$, we prove that the  
equation
has a unique solution (in a Banach space of summability exponent $p  
\geq 2$),
and this solution is H\"older continuous in both time and space.


  http://arxiv.org/abs/0905.2150

---------------------------------------------------------------

8505. MARKOVIAN BRIDGES: WEAK CONTINUITY AND PATHWISE CONSTRUCTIONS

Lo\"ic Chaumont and  Ger\'onimo Uribe Bravo

A Markovian bridge is a probability measure taken from a  
disintegration of
the law of an initial part of the path of a Markov process given its  
terminal
value. As such, Markovian bridges admit a natural parameterization in  
terms of
the state space of the process. In the context of Feller processes with
continuous transition densities, we construct by weak convergence
considerations the only versions of Markovian bridges which are weakly
continuous with respect to their parameter. We use this weakly  
continuous
construction to provide an extension of the strong Markov property in  
which the
flow of time is reversed. In the context of self-similar Feller  
process, the
last result is shown to be useful in the construction of Markovian  
bridges out
of the trajectories of the original process.


  http://arxiv.org/abs/0905.2155

---------------------------------------------------------------

8506. NON-MARKOVIAN LIMITS OF ADDITIVE FUNCTIONALS OF MARKOV PROCESSES

Milton Jara and  Tomasz Komorowski

In this paper we consider an additive functional of an observable $V(x) 
$ of a
Markov jump process. We assume that the law of the expected jump time  
$t(x)$
under the invariant probability measure $\pi$ of the skeleton chain  
belongs to
the domain of attraction of a subordinator. Then, the scaled limit of  
the
functional is a Mittag-Leffler proces, provided that $\Psi(x):=V(x)t(x) 
$ is
square integrable w.r.t. $\pi$. When the law of $\Psi(x)$ belongs to a  
domain
of attraction of a stable law the resulting process can be described  
by a
composition of a stable process and the inverse of a subordinator and  
these
processes are not necessarily independent. On the other hand when the
singularities of $\Psi(x)$ and $t(x)$ do not overlap with large  
probability the
law of the resulting process has some scaling invariance property. We  
provide
an application of the results to a process that arises in quantum  
transport
theory.


  http://arxiv.org/abs/0905.2163

---------------------------------------------------------------

8507. AMENABILITY OF LINEAR-ACTIVITY AUTOMATON GROUPS

Gideon Amir and  Omer Angel and  Balint Virag

We prove that every linear-activity automaton group is amenable.
   The proof is based on showing that a sufficiently symmetric random  
walk on a
specially constructed degree 1 automaton group -- the mother group --  
has
asymptotic entropy 0.
   Our result answers an open question by Nekrashevich in the Kourovka  
notebook,
and gives a partial answer to a question of Sidki.


  http://arxiv.org/abs/0905.2007

---------------------------------------------------------------

8508. A GLOBAL VIEW OF BROWNIAN PENALISATIONS

Joseph Najnudel and  Bernard Roynette and  Marc Yor

In this monograph, we construct and study a sigma-finite measure on
continuous functions from R_+ to R, strongly related to many probability
measures obtained by penalisation of Brownian motion, i.e. as limits of
probabilities which are absolutely continuous with respect to Wiener  
measure.
This remarkable sigma-finite measure can be generalized in three other  
cases:
one can start from a two-dimensional Brownian motion, from a recurrent
diffusion with values in R_+, and from a discrete, recurrent Markov  
chain.


  http://arxiv.org/abs/0905.2220

---------------------------------------------------------------

8509. THE MEAN PERIMETER OF SOME RANDOM PLANE CONVEX SETS GENERATED BY  
A  BROWNIAN MOTION

Philippe Biane G\'erard Letac

If $C_1$ is the convex hull of the curve of the standard Brownian  
motion in
the complex plane watched from 0 to 1, we consider the convex hulls of  
$C_1$
and several rotations of it and we compute the mean of the length of  
their
perimeter by elementary calculations.


  http://arxiv.org/abs/0905.2256

---------------------------------------------------------------

8510. A SCALING ANALYSIS OF A CAT AND MOUSE MARKOV CHAIN

Nelly Litvak and  Philippe Robert (INRIA)

Motivated by an original on-line page-ranking algorithm, starting from  
an
arbitrary Markov chain (C_n) on a discrete state space S, a Markov chain
(C_n,M_n) on the product space S^2, the cat and mouse Markov chain, is
constructed. The first coordinate of this Markov chain behaves like the
original Markov chain and the second component changes only when both
coordinates are equal. The asymptotic properties of this Markov chain  
are
investigated. A representation of its invariant measure is in particular
obtained. When the state space is infinite it is shown that this  
Markov chain
is in fact null recurrent if the initial Markov chain (C_n) is positive
recurrent and reversible. In this context, the scaling properties of the
location of the second component, the mouse, are investigated in various
situations: simple random walks in Z and Z^2, reflected simple random  
walk in N
and also in a continuous time setting. For several of these processes,  
a time
scaling with rapid growth gives an interesting asymptotic behavior  
related to
limit results for occupation times and rare events of Markov processes.


  http://arxiv.org/abs/0905.2259

---------------------------------------------------------------

8511. PRODUCT FORMULA FOR JACOBI POLYNOMIALS, SPHERICAL HARMONICS AND   
GENERALIZED BESSEL FUNCTION OF DIHEDRAL TYPE

Nizar Demni

We work out the expression of the generalized Bessel function of type  
B in
the two-rank case. This is done using Dijskma and Koornwinder's  
product formula
for Jacobi polynomials and the obtained expression is given by multiple
integrals involving only a normalized modified Bessel function and two
symmetric Beta distributions. We think of that expression as the major  
step
toward the explicit expression of the Dunkl's intertwining V operator
reflections-invariant functions. Finally, we give in the same setting an
explicit formula for the action of V on a product of a power of the  
norm and a
spherical harmonic. The obtained formula extends to all dihedral  
systems and it
improves the one derived by Y.Xu.


  http://arxiv.org/abs/0905.2265

---------------------------------------------------------------

8512. PASSAGE TIME FROM FOUR TO TWO BLOCKS IN THE VOTER MODEL

Kilian Raschel

We consider a voter model in which there are two candidates and  
initially, in
the population $\mathbb{Z}$, four connected blocks of same opinions.  
We assume
that a citizen changes his mind at a rate proportional to the number  
of its
neighbors that disagree with him, and we study the passage from four  
to two
connected blocks of same opinions. More precisely we make explicit the
generating function of the probabilities to go from four to two blocks  
in time
$k$ and we find the asymptotic of these probabilities when $k$ goes to
infinity.


  http://arxiv.org/abs/0905.2310

---------------------------------------------------------------

8513. A STOCHASTIC MODEL FOR PHYLOGENETIC TREES

T. M. Liggett and R. B. Schinazi

We propose the following simple stochastic model for phylogenetic  
trees. New
types are born and die according to a birth and death chain. At each  
birth we
associate a fitness to the new type sampled from a fixed distribution.  
At each
death the type with the smallest fitness is killed. We show that if  
the birth
(i.e. mutation) rate is subcritical we get a phylogenetic tree  
consistent with
an influenza tree (few types at any given time and one dominating type  
lasting
a long time). When the birth rate is supercritical we get a  
phylogenetic tree
consistent with an HIV tree (many types at any given time, none  
lasting very
long).


  http://arxiv.org/abs/0905.2349

---------------------------------------------------------------

8514. EXTENDING THE SUPPORT THEOREM TO INFINITE DIMENSIONS

Jeremy J. Becnel

The Radon transform is one of the most useful and applicable tools in
functional analysis. First constructed by John Radon in 1917 it has  
now been
adapted to several settings. One of the principle theorems involving  
the Radon
transform is the Support Theorem. In this paper, we discuss how the  
Radon
transform can be constructed in the white noise setting. We also  
develop a
Support Theorem in this setting.


  http://arxiv.org/abs/0905.2372

---------------------------------------------------------------

8515. GLOBAL EXISTENCE FOR ROUGH DIFFERENTIAL EQUATIONS UNDER LINEAR  
GROWTH  CONDITIONS

Massimiliano Gubinelli (CEREMADE) and  Antoine Lejay (IECN and  INRIA  
Sophia  Antipolis / INRIA Lorraine / IECN)

We prove existence of global solutions for differential equations  
driven by a
geometric rough path under the condition that the vector fields have  
linear
growth. We show by an explicit counter-example that the linear growth  
condition
is not sufficient if the driving rough path is not geometric. This  
settle a
long-standing open question in the theory of rough paths. So in the  
geometric
setting we recover the usual sufficient condition for differential  
equation.
The proof rely on a simple mapping of the differential equation from the
Euclidean space to a manifold to obtain a rough differential equation  
with
bounded coefficients.


  http://arxiv.org/abs/0905.2399

---------------------------------------------------------------

8516. TAIL ASYMPTOTICS FOR EXPONENTIAL FUNCTIONALS OF LEVY PROCESSES:  
THE  CONVOLUTION EQUIVALENT CASE

V'\ictor Rivero

We determine the rate of decrease of the right tail distribution of the
exponential functional of a Levy process with a convolution equivalent  
Levy
measure. Our main result establishes that it decreases as the right  
tail of the
image under the exponential function of the Levy measure of the  
underlying Levy
process. The method of proof relies on fluctuation theory of Levy  
processes and
an explicit path-wise representation of the exponential functional as  
the
exponential functional of a bivariate subordinator. Our techniques  
allow us to
establish rather general estimates of the measure of the excursions  
out from
zero for the underlying Levy process reflected in its past infimum,  
whose area
under the exponential of the excursion path exceed a given value.


  http://arxiv.org/abs/0905.2401

---------------------------------------------------------------

8517. THE DISTRIBUTION ROUTE FROM ANCESTORS TO DESCENDANTS

Baruch Fischer and Moshe Zakai

We study the distribution of descendants of a known personality, or of
anybody else, as it propagates along generations from father or mother  
through
any of their children. We ask for the ratio of the descendants to the  
total
population and construct a model for the route of Distribution from  
Ancestors
to Descendants (DAD). The population ratio $r_n$ is found to be given  
by the
recursive equation $ r_{n+1} \approx (2-r_n) r_n ,$ that provides the
transition from the $n-$th to the $(n+1)$th generation. $ r_0 =1/N_0$  
and $N_0$
is the total relevant population at the first generation. The number of
generations it takes to make half the population descendants is $\log  
N_0/\log
2$ and additional $\sim 4$ generations make everyone a descendent  
(=the full
descendant spreading time). These results are independent of the  
population
growth factor even if it changes along generations. As a running  
example we
consider the offspring of King David. Assuming a population between  
$N_0 =
10^6$ and $5 \cdot 10^6$ of Israelites at King David's time ($\sim  
1000$ BC),
it took 24 to 26 generations (about 600-650 years, when taking 25  
years for a
generation) to make every Israelite a King David descendent. The  
conclusion is
that practically every Israelite living today (and in fact already at  
350-400
BC), and probably also many others beyond them, are descendants of  
King David.
We note that this work doesn't deal with any genetical aspect. We also  
didn't
take into account here any geo-social-demographic factor.  
Nevertheless, along
tens of generations, about 120 from King David's time till today, the  
DAD route
is likely to govern the distribution in communities that are not very  
isolated.


  http://arxiv.org/abs/0904.4792

---------------------------------------------------------------

8518. STATIONARY MAP COLORING

Omer Angel and  Itai Benjamini and  Ori Gurel-Gurevich and  Tom  
Meyerovitch and  Ron  Peled

We consider a planar Poisson process and its associated Voronoi map.  
We show
that there is a proper coloring with 6 colors of the map which is a
deterministic isometry-equivariant function of the Poisson process. As  
part of
the proof we show that the 6-core of the corresponding Delaunay  
triangulation
is empty.
   Generalizations, extensions and some open questions are discussed.


  http://arxiv.org/abs/0905.2563

---------------------------------------------------------------

8519. RANDOM QUANTUM CHANNELS I: GRAPHICAL CALCULUS AND THE BELL  
STATE  PHENOMENON

Beno\^it Collins (ICJ) and  Ion Nechita (ICJ)

This paper is the first of a series where we study quantum channels  
from the
random matrix point of view. We develop a graphical tool that allows  
us to
compute the expected moments of the output of a random quantum  
channel. As an
application, we study variations of random matrix models introduced by  
Hayden
\cite{hayden}, and show that their eigenvalues converge almost surely.  
In
particular we obtain for some models sharp improvements on the value  
of the
largest eigenvalue, and this is shown in a further work to have new
applications to minimal output entropy inequalities.


  http://arxiv.org/abs/0905.2313

---------------------------------------------------------------

8520. QUANTUM STOCHASTIC CONVOLUTION COCYCLES III

J. Martin Lindsay and Adam G. Skalski

The theory of quantum Levy processes on a compact quantum group, and  
more
generally quantum stochastic convolution cocycles on a C*-bialgebra, is
extended to locally compact quantum groups and multiplier C*- 
bialgebras. Strict
extension results obtained by Kustermans, and automatic strictness  
properties
developed here, are exploited to obtain existence and uniqueness for
coalgebraic quantum stochastic differential equations in this setting.  
Working
in the universal enveloping von Neumann bialgebra, the stochastic  
generators of
Markov-regular, completely positive, respectively *-homomorphic, quantum
stochastic convolution cocycles are characterised. Every Markov- 
regular quantum
Levy process on a multiplier C*-bialgebra is shown to be equivalent to  
one
governed by a quantum stochastic differential equation, and the  
generating
functionals of norm-continuous convolution semigroups on a multiplier
C*-bialgebra are characterised. Applying a recent result of Belton's,  
we give a
thorough treatment of the approximation of quantum stochastic  
convolution
cocycles by quantum random walks.


  http://arxiv.org/abs/0905.2410

---------------------------------------------------------------

8521. HEAT KERNEL ESTIMATES FOR THE FRACTIONAL LAPLACIAN

Krzysztof Bogdan and  Tomasz Grzywny and  Michal Ryznar

We give sharp estimates for the heat kernel of the fractional
   Laplacian with Dirichlet exterior condition for a general class of  
domains
including Lipschitz domains. The estimates are sharp and explicit for  
smooth
domains.


  http://arxiv.org/abs/0905.2626

---------------------------------------------------------------

8522. A NOTE ON A FENYMAN-KAC-TYPE FORMULA

Raluca Balan

In this article, we establish a probabilistic representation for the
second-order moment of the solution of stochastic heat equation in  
$[0,1]
\times \bR^d$, with multiplicative noise, which is fractional in time  
and
colored in space. This representation is similar to the one given in  
Dalang,
Mueller and Tribe (2008) in the case of an s.p.d.e. driven by a  
Gaussian noise,
which is white in time. Unlike the formula of Dalang, Mueller and  
Tribe (2008),
which is based on the usual Poisson process, our representation is  
based on the
planar Poisson process, due to the fractional component of the noise.


  http://arxiv.org/abs/0905.2698

---------------------------------------------------------------

8523. A STRONG LAW OF LARGE NUMBERS FOR MARTINGALE ARRAYS

Yves F. Atchade

We prove a martingale triangular array generalization of the
Chow-Birnbaum-Marshall's inequality. The result is used to derive a  
strong law
of large numbers for martingale triangular arrays whose rows are  
asymptotically
stable in a certain sense. To illustrate, we derive a simple proof,  
based on
martingale arguments, of the consistency of kernel regression with  
dependent
data. Another application can be found in \cite{atchadeetfort08} where  
the new
inequality is used to prove a strong law of large numbers for adaptive  
Markov
Chain Monte Carlo methods.


  http://arxiv.org/abs/0905.2761

---------------------------------------------------------------

8524. STATIONARY STOCHASTIC VISCOSITY SOLUTIONS OF SPDES

Qi Zhang

In this paper we aim to obtain the stationary stochastic viscosity  
solutions
of a parabolic type SPDEs through the infinite horizon backward doubly
stochastic differential equations (BDSDEs). For this, we study the  
existence,
uniqueness and regularity of solutions of infinite horizon BDSDEs as  
well as
the "perfection procedure" applied to the solutions of BDSDEs to  
derive the
"perfect" stationary stochastic viscosity solutions of SPDEs.


  http://arxiv.org/abs/0905.2806

---------------------------------------------------------------

8525. CYLINDRICAL LEVY PROCESSES IN BANACH SPACES

David Applebaum and  Markus Riedle

Cylindrical probability measures are finitely additive measures on  
Banach
spaces that have sigma-additive projections to Euclidean spaces of all
dimensions. They are naturally associated to notions of weak  
(cylindrical)
random variable and hence weak (cylindrical) stochastic processes. In  
this
paper we focus on cylindrical Levy processes. These have (weak) Levy-Ito
decompositions and an associated Levy-Khintchine formula. If the  
process is
weakly square integrable, its covariance operator can be used to  
construct a
reproducing kernel Hilbert space in which the process has a  
decomposition as an
infinite series built from a sequence of uncorrelated bona fide one- 
dimensional
Levy processes. This series is used to define cylindrical stochastic  
integrals
from which cylindrical Ornstein-Uhlenbeck processes may be constructed  
as
unique solutions of the associated Cauchy problem. We demonstrate that  
such
processes are cylindrical Markov processes and study their (cylindrical)
invariant measures.


  http://arxiv.org/abs/0905.2858

---------------------------------------------------------------

8526. A NOTE ON CORRELATIONS IN RANDOMLY ORIENTED GRAPHS

Svante Linusson

Two models are compared on a given graph G. On the one hand regular edge
percolation with probability 1/2 and on the other hand orienting G by  
giving
each edge a random direction. Two lemmas are presented proving that  
for a given
vertex v the probability distribution for the open cluster C_v in edge
percolation is equal to the distribution for the directed out-cluster
$\hpil{C}_v$ and then a similar statement for the joint distribution of
disjoint clusters around two different vertices.
   One application of the two lemmas is then to prove correlation  
inequalities
of the existence of directed paths. It is proven that for vertices  
a,b,s in G,
the events {s\to a} and {s\to b} are positively correlated. This is  
proven to
be true also if we first condition on that there does not exist a path  
from s
to t for any vertex t\neq s. With this conditioning it is also true  
that {s\to
b} and {a\to t} are negatively correlated.
   A concept of increasing events in random orientations is defined  
and a
general inequality corresponding to Harris inequality is given.
   The lemmas and applications are true also for another model of  
randomly
directed graphs.


  http://arxiv.org/abs/0905.2881

---------------------------------------------------------------

8527. SPECTRAL ANALYSIS OF 1D NEAREST-NEIGHBOR RANDOM WALKS WITH  
APPLICATIONS  TO SUBDIFFUSIVE RANDOM TRAP AND BARRIER MODELS

A. Faggionato

Given a family $X^{(n)}(t)$ of continuous--time nearest--neighbor random
walks on the one dimensional lattice $\bbZ$, parameterized by $n \in  
\bbN_+$,
we show that the spectral analysis of the Markov generator of $X^{(n)} 
$ with
Dirichlet conditions outside $(0,n)$ reduces to the analysis of the  
eigenvalues
and eigenfunctions of a suitable generalized second order differential  
operator
$-D_{m_n} D_x$ with Dirichlet conditions outside $(0,1)$. If in  
addition the
measures $dm_n$ weakly converge to some measure $dm$, similarly to  
Krein's
correspondence we prove a limit theorem of the eigenvalues and  
eigenfunctions
of $-D_{m_n}D_x$ to the corresponding spectral quantities of $-D_mD_x$.
Applying the above result together with the Dirichlet--Neumann  
bracketing, we
investigate the limiting behavior of the small eigenvalues of  
subdiffusive
random trap and barrier models and establish lower and upper bounds  
for the
asymptotic annealed eigenvalue counting functions.


  http://arxiv.org/abs/0905.2900

---------------------------------------------------------------

8528. EXPONENTIAL DEFICIENCY OF CONVOLUTIONS OF DENSITIES

Iosif Pinelis

If a probability density p(\x) (\x\in\R^k) is bounded and R(t) := \int
\exp(t\ell(\x)) \d\x < \infty for some linear functional \ell and all
t\in(0,1), then, for each t\in(0,1) and all large enough n, the n-fold
convolution of the t-tilted density p_t(\x) := \exp(t\ell(\x)) p(\x)/ 
R(t) is
bounded. This is a corollary of a general, "non-i.i.d." result, which  
is also
shown to enjoy a certain optimality property. Such results are useful  
for
saddle-point approximations.


  http://arxiv.org/abs/0905.2944

---------------------------------------------------------------

8529. USING THE SCHRAMM-LOEWNER EVOLUTION TO EXPLAIN CERTAIN NON- 
LOCAL  OBSERVABLES IN THE 2D CRITICAL ISING MODEL

Michael J. Kozdron (University of Regina)

We present a mathematical proof of theoretical predictions made by  
Arguin and
Saint-Aubin, as well as by Bauer, Bernard, and Kytola, about certain  
non-local
observables for the two-dimensional Ising model at criticality by  
combining
Smirnov's recent proof of the fact that the scaling limit of critical  
Ising
interfaces can be described by chordal SLE(3) with Kozdron and Lawler's
configurational measure on mutually avoiding chordal SLE paths. As an  
extension
of this result, we also compute the probability that an SLE(k) path, k  
in
(0,4], and a Brownian motion excursion do not intersect.


  http://arxiv.org/abs/0905.2430

---------------------------------------------------------------

8530. CURRENT FLUCTUATIONS FOR TASEP: A PROOF OF THE PR\"{A}HOFER- 
SPOHN  CONJECTURE

G. Ben Arous and  I. Corwin

We consider the family of two-sided initial conditions for TASEP  
which, as
the left and right densities (\rho_-,\rho_+) are varied, give rise to  
shock
waves and rarefaction fans -- the two phenomena which are typical to  
TASEP. We
provide a proof of Conjecture 7.1 of Pr\"{a}hofer and Spohn which  
characterizes
the order of and scaling functions for the fluctuations of the height  
function
of two-sided TASEP in terms of the two densities \rho_-,\rho_+ and the  
speed y
around which the height is observed. In proving this theorem for TASEP  
we also
prove a fluctuation theorem for a class of corner growth processes with
external sources, or equivalently for the last passage time in a  
directed last
passage percolation model with two-sided boundary conditions: \rho_- and
1-\rho_+. We provide a complete characterization of the order of and the
scaling functions for the fluctuations of this model's last passage  
time L(N,M)
as a function of three parameters: the two boundary/source rates  
\rho_- and
1-\rho_+, and the scaling ratio gamma^2=M/N. The proof of this theorem  
draws on
the results of P.L. Ferrari and Spohn and extensively on the work of  
Baik, Ben
Arous and P\'{e}ch\'{e} on finite rank perturbations of Wishart  
ensembles in
random matrix theory.


  http://arxiv.org/abs/0905.2993

---------------------------------------------------------------

8531. A POINCAR\'E INEQUALITY ON LOOP SPACES

Xin Chen and  Xue-Mei Li and  Bo Wu

We investigate properties of measures in infinite dimensional spaces  
in terms
of Poincar\'e inequalities.
   A Poincar\'e inequality states that the $L^2$ variance of an  
admissible
function is controlled by the homogeneous $H^1$ norm. In the case of  
Loop
spaces, it was observed by L. Gross that the homogeneous $H^1$ norm  
alone may
not control the $L^2$ norm and a potential term involving the end  
value of the
Brownian bridge is introduced. Aida, on the other hand, introduced a  
weight on
the Dirichlet form. We show that Aida's modified Logarithmic Sobolev  
inequality
implies weak Logarithmic Sobolev Inequalities and weak Poincar\'e  
inequalities
with precise estimates on the order of convergence. The order of  
convergence in
the weak Sobolev inequalities are related to weak $L^1$ estimates on  
the weight
function. This and a relation between Logarithmic Sobolev inequalities  
and weak
Poincar\'e inequalities lead to a Poincar\'e inequality on the loop  
space over
certain manifolds.


  http://arxiv.org/abs/0905.3007

---------------------------------------------------------------

8532. ON THE MIXING TIME OF THE 2D STOCHASTIC ISING MODEL WITH "PLUS"  
BOUNDARY  CONDITIONS AT LOW TEMPERATURE

F. Martinelli (Matematica and  Roma 3) and  F. Toninelli (CNRS and ENS  
Lyon)

We consider the Glauber dynamics for the 2D Ising model in a box of  
side L,
at inverse temperature $\beta$ and random boundary conditions $\tau$  
whose
distribution P either stochastically dominates the extremal plus phase  
(hence
the quotation marks in the title) or it is stochastically dominated by  
the
extremal minus phase. A particular case is when P is concentrated on the
homogeneous configuration identically equal to + (equal to -). For $ 
\beta$
large enough we show that for any $\epsilon$ there exists $c=c(\beta, 
\epsilon)$
such that the corresponding mixing time $T_{mix}$ satisfies
$\lim_{L\to\infty}P(T_{mix}> \exp({cL^\epsilon})) =0$. In the non- 
random case
$\tau\equiv +$ (or $\tau\equiv -$), this implies that $T_{mix}<
\exp({cL^\epsilon})$. The same bound holds when the boundary  
conditions are all
+ on three sides and all - on the remaining one. The result, although  
still
very far from the expected Lifshitz behaviour $T_{mix}=O(L^2)$,  
considerably
improves upon the previous known estimates of the form $T_{mix}\le  
\exp({c
L^{1/2 + \epsilon}})$. The techniques are based on induction over length
scales, combined with a judicious use of the so-called "censoring  
inequality"
of Y. Peres and P. Winkler, which in a sense allows us to guide the  
dynamics to
its equilibrium measure.


  http://arxiv.org/abs/0905.3040

---------------------------------------------------------------

8533. CENTRAL LIMIT THEOREM FOR THE HEAT KERNEL MEASURE ON THE UNITARY  
GROUP

Thierry L\'evy (DMA) and  Myl\`ene Ma\"ida (LM-Orsay)

We prove that for a finite collection of real-valued functions
$f_{1},...,f_{n}$ on the group of complex numbers of modulus 1 which are
derivable with Lipschitz continuous derivative, the distribution of $ 
(\tr
f_{1},...,\tr f_{n})$ under the properly scaled heat kernel measure at  
a given
time on the unitary group $\U(N)$ has Gaussian fluctuations as $N$  
tends to
infinity, with a covariance for which we give a formula and which is  
of order
$N^{-1}$. In the limit where the time tends to infinity, we prove that  
this
covariance converges to that obtained by P. Diaconis and S. Evans in a  
previous
work on uniformly distributed unitary matrices. Finally, we discuss some
combinatorial aspects of our results.


  http://arxiv.org/abs/0905.3282

---------------------------------------------------------------

8534. A FUNCTIONAL EQUATION WHOSE UNKNOWN IS P([0,1]) VALUED

Giacomo Aletti and  Caterina May and Piercesare Secchi

We study a functional equation whose unknown maps a euclidean space  
into the
space of probability distributions on [0,1]. We prove existence and  
uniqueness
of its solution under suitable regularity and boundary conditions and we
characterize solutions that are diffuse on [0,1]. A canonical solution  
is
obtained by means of a Randomly Reinforced Urn with different  
reinforcement
distributions having equal means.


  http://arxiv.org/abs/0905.3310

---------------------------------------------------------------

8535. PATH REGULARITY OF GAUSSIAN PROCESSES VIA SMALL DEVIATIONS

Frank Aurzada

We study the a.s. sample path regularity of Gaussian processes. To  
this end
we relate the path regularity directly to the theory of small  
deviations. In
particular, we show that if the process is $n$-times differentiable  
then the
exponential rate of decay of its small deviations is at most
$\varepsilon^{-1/n}$. We also show a similar result if $n$ is not an  
integer.


  http://arxiv.org/abs/0905.3358

---------------------------------------------------------------

8536. CUMULANTS AS ITERATED INTEGRALS

Franz Lehner

A formula expressing cumulants in terms of iterated integrals of the
distribution function is derived. It generalizes results of Jones and
Balakrishnan who computed expressions for cumulants up to order 4.


  http://arxiv.org/abs/0905.3375

---------------------------------------------------------------

8537. GLOBALLY OPTIMAL PARAMETER ESTIMATES FOR NON-LINEAR DIFFUSIONS

A. Mijatovi\'c and  P. Schneider

This paper studies an approximation method for the log likelihood  
function of
a non-linear diffusion process using the bridge of the diffusion. The  
main
result (Theorem 1) shows that this approximation converges uniformly  
to the
unknown likelihood function and can therefore be used efficiently with  
any
algorithm for sampling from the law of the bridge. We also introduce an
expected maximum likelihood (EML) algorithm for inferring the  
parameters of
discretely observed diffusion processes. The approach is applicable to a
subclass of non-linear SDEs with constant volatility and drift that is  
linear
in the model parameters. In this setting globally optimal parameters are
obtained in a single step by solving a square linear system whose  
dimension
equals the number of parameters in the model. Simulation studies to  
test the
EML algorithm show that it performs well when compared with algorithms  
based on
the exact maximum likelihood as well as closed-form likelihood  
expansions.


  http://arxiv.org/abs/0905.3321

---------------------------------------------------------------

8538. CONVERGENCE OF THE STRUCTURE FUNCTION OF A MULTIFRACTAL RANDOM  
WALK IN A  MIXED ASYMPTOTIC SETTING

Laurent Duvernet

Some asymptotic properties of a Brownian motion in multifractal time,  
also
called multifractal random walk, are established. We show the almost  
sure and
$L^1$ convergence of its structure function. This is an issue directly
connected to the scale invariance and multifractal property of the  
sample
paths. We place ourselves in a mixed asymptotic setting where both the
observation length and the sampling frequency may go together to  
infinity at
different rates. The results we obtain are similar to the ones that  
were given
by Ossiander and Waymire and Bacry \emph{et al.} in the simpler  
framework of
Mandelbrot cascades.


  http://arxiv.org/abs/0905.3405

---------------------------------------------------------------

8539. CUMULANTS OF A CONVOLUTION AND APPLICATIONS TO MONOTONE  
PROBABILITY  THEORY

Takahiro Hasebe

In non-commutative probability theory, cumulants and their generating
function are defined once a notion of independence is given. In this  
paper, we
give a definition of cumulants of a (non-commutative) convolution and  
prove the
uniqueness of cumulants. Then we define cumulants of monotone  
convolution and
prove limit theorems as applications.


  http://arxiv.org/abs/0905.3446

---------------------------------------------------------------

8540. A PHASE TRANSITION FOR THE HEIGHTS OF A FRAGMENTATION TREE

Adrien Joseph (PMA)

We provide information about the asymptotic regimes for a homogeneous
fragmentation of a finite set. We establish a phase transition for the
asymptotic behaviours of the shattering times, defined as the first  
instants
when all the blocks of the partition process have cardinality less  
than a fixed
integer. Our results may be applied to the study of certain random  
split trees.


  http://arxiv.org/abs/0905.3545

---------------------------------------------------------------

8541. A STOCHASTIC OPTIMAL CONTROL PROBLEM FOR THE HEAT EQUATION ON  
THE  HALFLINE WITH DIRICHLET BOUNDARY-NOISE AND BOUNDARY-CONTROL

Federica Masiero

We consider a controlled state equation of parabolic type on the  
halfline
$(0,+\infty)$ with boundary conditions of Dirichlet type in which the  
unknown
is equal to the sum of the control and of a white noise in time. We  
study
finite horizon and infinite horizon optimal control problem related by  
menas of
backward stochastic differential equations.


  http://arxiv.org/abs/0905.3628

---------------------------------------------------------------

8542. APPROXIMATION OF QUASI-STATIONARY DISTRIBUTIONS FOR 1- 
DIMENSIONAL KILLED  DIFFUSIONS WITH UNBOUNDED DRIFTS

Denis Villemonais (CMAP)

The long time behavior of an absorbed Markov process is well described  
by the
limiting distribution of the process conditioned to not be killed when  
it is
observed. Our aim is to give an approximation's method of this limit,  
when the
process is a 1-dimensional It\^o diffusion whose drift is allowed to  
explode at
the boundary. In a first step, we show how to restrict the study to  
the case of
a diffusion with values in a bounded interval and whose drift is  
bounded. In a
second step, we show an approximation method of the limiting conditional
distribution of such diffusions, based on a Fleming-Viot type  
interacting
particle system. We end the paper with two numerical applications : to  
the
logistic Feller diffusion and to the Wright-Fisher diffusion with  
values in
$]0,1[$ conditioned to be killed at 0.


  http://arxiv.org/abs/0905.3636

---------------------------------------------------------------

8543. ON A PROCESSOR SHARING QUEUE THAT MODELS BALKING

Qiang Zhen and  Johan S. H. van Leeuwaarden and  Charles Knessl

We consider the processor sharing $M/M/1$-PS queue which also models  
balking.
A customer that arrives and sees $n$ others in the system "balks" (i.e.,
decides not to enter) with probability $1-b_n$. If $b_n$ is inversely
proportional to $n+1$, we obtain explicit expressions for a tagged  
customer's
sojourn time distribution. We consider both the conditional  
distribution,
conditioned on the number of other customers present when the tagged  
customer
arrives, as well as the unconditional distribution. We then evaluate the
results in various asymptotic limits. These include large time (tail  
behavior)
and/or large $n$, lightly loaded systems where the arrival rate $ 
\lambda\to 0$,
and heavily loaded systems where $\lambda\to\infty$. We find that the
asymptotic structure for the problem with balking is much different  
from the
standard $M/M/1$-PS queue. We also discuss a perturbation method for  
deriving
the asymptotics, which should apply to more general balking functions.


  http://arxiv.org/abs/0905.3700

---------------------------------------------------------------

8544. ON THE MARTINGALE PROPERTY OF CERTAIN LOCAL MARTINGALES:  
CRITERIA AND  APPLICATIONS

Aleksandar Mijatovic and  Mikhail Urusov

The stochastic exponential $Z_t=\exp[M_t-M_0-(1/2)< M,M>_t]$ of a  
continuous
local martingale $M$ is itself a continuous local martingale. We give a
necessary and sufficient condition for the process $Z$ to be a true  
martingale
in the case where $M_t=\int_0^t b(Y_u) dW_u$ and $Y$ is a one- 
dimensional
diffusion driven by a Brownian motion $W$. Furthermore, we provide a  
necessary
and sufficient condition for $Z$ to be a uniformly integrable  
martingale in the
same setting. These conditions are deterministic and expressed only in  
terms of
the function $b$ and the drift and diffusion coefficients of $Y$. We  
also
classify, via deterministic necessary and sufficient conditions, when  
the
process $Z$ is a.s. strictly positive, when its limit $Z_\infty$ is a.s.
strictly positive, and when $Z_\infty$ is a.s. zero. This allows us to  
obtain a
deterministic necessary and sufficient condition in the one- 
dimensional setting
for a discounted stock price to be a true martingale under the risk- 
neutral
measure, and for it to be a uniformly integrable martingale. These  
results
enable us to ascertain the existence of financial bubbles in diffusion- 
based
models. Finally, we obtain a deterministic characterisation of the  
\emph{no
free lunch with vanishing risk}, the \emph{no generalised arbitrage},  
and the
\emph{no relative arbitrage} conditions in the one-dimensional setting  
and
examine how these notions of no-arbitrage relate to each other.


  http://arxiv.org/abs/0905.3701

---------------------------------------------------------------

8545. OPTIMAL STOPPING FOR NON-LINEAR EXPECTATIONS

Erhan Bayraktar and  Song Yao

We develop a theory for solving continuous time optimal stopping  
problems for
non-linear expectations. Our motivation is to consider problems in  
which the
stopper uses risk measures to evaluate future rewards.


  http://arxiv.org/abs/0905.3601

---------------------------------------------------------------

8546. A NOTE ON FURSTENBERG'S FILTERING PROBLEM

Rodolphe Garbit (LMJL)

This short note gives a positive answer to an elementary question in
probability theory that arose in Furstenberg's famous article  
"Disjointness in
Ergodic Theory". As a consequence, Furstenberg's filtering theorem holds
without any integrability assumption.


  http://arxiv.org/abs/0905.3879

---------------------------------------------------------------

8547. REVERSED DIRICHLET ENVIRONMENT AND DIRECTIONAL TRANSIENCE OF  
RANDOM  WALKS IN DIRICHLET RANDOM ENVIRONMENT

Christophe Sabot (ICJ) and  Laurent Tournier (ICJ)

We consider random walks in a random environment that is given by i.i.d.
Dirichlet distributions at each vertex of Z^d or, equivalently,  
oriented edge
reinforced random walks on Z^d. The parameters of the distribution are a
2d-uplet of positive real numbers indexed by the unit vectors of Z^d.  
We prove
that, as soon as these weights are nonsymmetric, the random walk in  
this random
environment is transient in a direction with positive probability. In  
dimension
2, this result can be strenghened to an almost sure directional  
transience
thanks to the 0-1 law from [ZM01]. Our proof relies on the property of
stability of Dirichlet environment by time reversal proved in [Sa09].  
In a
first part of this paper, we also give a probabilistic proof of this  
property
as an alternative to the change of variable computation used in that  
article.


  http://arxiv.org/abs/0905.3917

---------------------------------------------------------------

8548. ON QUADRATIC G-EVALUATIONS/EXPECTATIONS AND RELATED ANALYSIS

Jin Ma and  Song Yao

In this paper we extend the notion of g-evaluation, in particular
g-expectation, to the case where the generator g is allowed to have a  
quadratic
growth. We show that some important properties of the g-expectations,  
including
a representation theorem between the generator and the corresponding
g-expectation, and consequently the reverse comparison theorem of  
quadratic
BSDEs as well as the Jensen inequality, remain true in the quadratic  
case. Our
main results also include a Doob-Meyer type decomposition, the optional
sampling theorem, and the up-crossing inequality. The results of this  
paper are
important in the further development of the general quadratic nonlinear
expectations.


  http://arxiv.org/abs/0905.3941

---------------------------------------------------------------

8549. DISCRETE TIME SCALE INVARIANT MARKOV PROCESSES

N. Modarresi and S. Rezakhah

In this paper we consider a discrete scale invariant Markov process with
scale $l$ which by a scheme of sampling at discrete points we provide  
discrete
time scale invariant Markov(DT-SIM) process. We also define quasi  
Lamperti
transformation as a basic tool in relation with such sampling. We  
study the
properties of a DT-SIM process and find the covariance function of it  
which is
specified by the values of $\{R_{j}^H(1),R_{j}^H(0),j\in {\bf Z^+},  
0\leq j\leq
{T-1}\}$, where $R_j^H(k)$ is the covariance function $j$th and $(j+k) 
$th
observations of DT-SIM and $T$ is the number of observations in each  
scale. We
also define T-dimensional self-similar Markov process corresponding to  
DT-SIM
process and characterize its covariance matrix.


  http://arxiv.org/abs/0905.3959

---------------------------------------------------------------

8550. TAGGED PARTICLE PROCESSES AND THEIR NON-EXPLOSION CRITERIA

Hirofumi Osada

We give a derivation of tagged particle processes from unlabeled  
interacting
Brownian motions. We give a criteria of the non-explosion property of  
tagged
particle processes. We prove the quasi-regularity of Dirichlet forms  
describing
the environment seen from the tagged particle, which were used in  
previous
papers to prove the invariance principle of tagged particles of  
interacting
Brownian motions.


  http://arxiv.org/abs/0905.3973

---------------------------------------------------------------

8551. MAXIMUM OF DYSON BROWNIAN MOTION AND NON-COLLIDING SYSTEMS WITH  
A  BOUNDARY

Alexei Borodin and  Patrik L. Ferrari and  Michael Praehofer and   
Tomohiro  Sasamoto, Jon Warren

We prove an equality-in-law relating the maximum of GUE Dyson's Brownian
motion and the non-colliding systems with a wall. This generalizes the  
well
known relation between the maximum of a Brownian motion and a reflected
Brownian motion.


  http://arxiv.org/abs/0905.3989

---------------------------------------------------------------

8552. LIMITS OF RANDOMLY GROWN GRAPH SEQUENCES

C. Borgs and  J. Chayes and  L. Lov\'asz and  V.T. S\'os and  K.  
Vesztergombi

Motivated in part by various sequences of graphs growing under random  
rules
(like internet models), convergent sequences of dense graphs and their  
limits
were introduced by Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi and  
by
Lov\'asz and Szegedy. In this paper we use this framework to study one  
of the
motivating class of examples, namely randomly growing graphs. We prove  
the
(almost sure) convergence of several such randomly growing graph  
sequences, and
determine their limit. The analysis is not always straightforward: in  
some
cases the cut distance from a limit object can be directly estimated,  
in other
case densities of subgraphs can be shown to converge.


  http://arxiv.org/abs/0905.3806

---------------------------------------------------------------

8553. ERGODIC PROPERTIES OF MAX-INFINITELY DIVISIBLE PROCESSES

Zakhar Kabluchko and  Martin Schlather

We prove that a stationary max--infinitely divisible process is mixing
(ergodic) iff its dependence function converges to 0 (is Cesaro  
summable to 0).
These criteria are applied to some classes of max--infinitely divisible
processes.


  http://arxiv.org/abs/0905.4196

---------------------------------------------------------------

8554. ON CHARACTERIZATIONS BASED ON REGRESSION OF LINEAR COMBINATIONS  
OF  RECORDS

George P. Yanev and M. Ahsanullah

We characterize the exponential distribution in terms of the  
regression of a
record value with non-adjacent records as covariates. We also study
characterizations based on the regression of linear combinations of  
records.


  http://arxiv.org/abs/0905.4230

---------------------------------------------------------------

8555. ON ASYMPTOTIC EXPANSION IN THE RANDOM ALLOCATION OF PARTICLES BY  
SETS

Saidbek S.Mirakhmedov and Sherzod M.Mirakhmedov

We consider a scheme of equiprobable allocation of particles into  
cells by
sets. The Edgeworth type asymptotic expansion in the local central limit
theorem for a number of empty cells left after allocation of all sets of
particles is derived.


  http://arxiv.org/abs/0905.4247

---------------------------------------------------------------

8556. FIXED TRACE $\BETA$-HERMITE ENSEMBLES: ASYMPTOTIC EIGENVALUE  
DENSITY AND  THE EDGE OF THE DENSITY

Da-Sheng Zhou and  Dang-Zheng Liu and  Tao Qian

In the present paper, fixed trace $\beta$-Hermite ensembles  
generalizing the
fixed trace Gaussian Hermite ensemble are considered. For all $\beta$,  
we prove
the Wigner semicircle law for these ensembles by using two different  
methods:
one is the moment equivalence method with the help of the matrix model  
for
general $\beta$, the other is to use asymptotic analysis tools. At the  
edge of
the density, we prove that the edge scaling limit for $\beta$-HE  
implies the
same limit for fixed trace $\beta$-Hermite ensembles. Consequently,  
explicit
limit can be given for fixed trace GOE, GUE and GSE. Furthermore, for  
even
$\beta$, analogous to $\beta$-Hermite ensembles, a multiple integral  
of the
Konstevich type can be obtained.


  http://arxiv.org/abs/0905.4255

---------------------------------------------------------------

8557. BOUNDARY BEHAVIOUR OF HARMONIC FUNCTIONS ON HYPERBOLIC GROUPS

Camille Petit (IF)

We consider random walks with finite support on non-elementary Gromov
hyperbolic groups. For a given harmonic function on such a group, we  
prove that
asymptotic properties of non-tangential boundedness and non-tangential
convergence are almost everywhere equivalent. The proof is inspired  
from works
of F. Mouton in the cases of Riemannian manifolds of pinched negative  
curvature
and infinite trees. It involves geometric and probabilitistic methods.


  http://arxiv.org/abs/0905.4118

---------------------------------------------------------------

8558. ON THE INFLUENCES OF VARIABLES ON BOOLEAN FUNCTIONS IN PRODUCT  
SPACES

Nathan Keller

In this paper we consider the influences of variables on Boolean  
functions in
general product spaces. Unlike the case of functions on the discrete  
cube where
there is a clear definition of influence, in the general case at least  
three
definitions were presented in different papers. We propose a family of
definitions for the influence, that contains all the known  
definitions, as well
as other natural definitions, as special cases. We prove a  
generalization of
the BKKKL theorem, which is tight in terms of the definition of  
influence used
in the assertion, and use it to generalize several known results on  
influences
in general product spaces.


  http://arxiv.org/abs/0905.4216

---------------------------------------------------------------

8559. LDP APPLICATION FOR BILLINGSLEY'S EXAMPLE

R. Liptser

We consider a classical model discussed in Theorem 16.4 (Billingsley
\cite{Bil}) concerning to an empirical distribution function $$
F_n(t)=\frac{1}{n}\sum_{k=1}^nI_{\{\xi_k\le t\}}, $$ of $(\xi_k)_{i\ge  
1}$ -
i.i.d. sequence of random variables, supported on the interval  
$[0,1]$, with
$F(t)=\mathsf{P}(\xi_1\le t)$ the continuous distribution function. We  
give a
proof of Kolmogorov's exponential estimate
\mathsf{P}\Big(\sup_{t\in[0,1]}|F_n(t)-F(t)|\ge \varepsilon\Big) \le
2\exp\Big\{-n[\varepsilon/8\log(1+\varepsilon^2/32)-\varepsilon/8
+(4/\varepsilon)\log(1+\varepsilon^2/32\Big)]\} with the help of which  
jointly
with the large deviations technique, we establish a logarithmic  
asymptotic: for
any $T\in[F^{-1}({1/2}),1)$ and any $\alpha\in\big(0,{1/2}\big)$:
\lim_{n\to\infty}\frac{1}{n^{1-2\alpha}}
\log\mathsf{P}\bigg(\sup_{t\in[0,T]}n^\alpha\Big|F_n(t)-F(t)\Big|\ge
\varepsilon\bigg)=-2\varepsilon^2.


  http://arxiv.org/abs/0905.4334

---------------------------------------------------------------

8560. A DECOMPOSITION AND WEAK APPROXIMATION OF THE SUB-FRACTIONAL  
BROWNIAN  MOTION

Xavier Bardina and David Bascompte

We present a decomposition of the sub-fractional Brownian motion into  
the sum
of a fractional Brownian motion plus a stochastic process with  
absolutely
continuous trajectories. The first application we show of this  
decomposition is
the relation between the spaces of integrable functions with respect  
each one
of these three processes. A general result of weak convergence to  
integrals of
$L^2(\mathbb R^{+})$ functions with respect to standard Brownian  
motion is
proved, and this result permits us to obtain approximations in law of  
the
fractional Brownian motion and the sub-fractional Brownian motion with
parameter $H\in(0,1)$.


  http://arxiv.org/abs/0905.4360

---------------------------------------------------------------

8561. FROM THE LIFSHITZ TAIL TO THE QUENCHED SURVIVAL ASYMPTOTICS IN  
THE  TRAPPING PROBLEM

Ryoki Fukushima

The survival problem for a diffusing particle moving among random  
traps is
considered. We introduce a simple argument to derive the quenched  
asymptotics
of the survival probability from the Lifshitz tail effect for the  
associated
operator. In particular, the upper bound is proved in fairly general  
settings
and is shown to be sharp in the case of the Brownian motion in the  
Poissonian
obstacles. As an application, we derive the quenched asymptotics for the
Brownian motion in traps distributed according to a random  
perturbation of the
lattice.


  http://arxiv.org/abs/0905.4436

---------------------------------------------------------------

8562. EXTREME VALUE THEORY, POISSON-DIRICHLET DISTRIBUTIONS AND FPP ON  
RANDOM  NETWORKS

Shankar Bhamidi and  Remco van der Hofstad and  Gerard Hooghiemstra

We study first passage percolation on the configuration model (CM)  
having
power-law degrees with exponent $\tau\in [1,2)$. To this end we equip  
the edges
with exponential weights. We derive the distributional limit of the  
minimal
weight of a path between typical vertices in the network and the  
number of
edges on the minimal weight path, which can be computed in terms of the
Poisson-Dirichlet distribution. We explicitly describe these limits  
via the
construction of an infinite limiting object describing the FPP problem  
in the
densely connected core of the network. We consider two separate cases,  
namely,
the {\it original CM}, in which each edge, regardless of its  
multiplicity,
receives an independent exponential weight, as well as the {\it erased  
CM}, for
which there is an independent exponential weight between any pair of  
direct
neighbors. While the results are qualitatively similar, surprisingly the
limiting random variables are quite different.
   Our results imply that the flow carrying properties of the network  
are
markedly different from either the mean-field setting or the locally  
tree-like
setting, which occurs as $\tau>2$, and for which the hopcount between  
typical
vertices scales as $\log{n}$. In our setting the hopcount is tight and  
has an
explicit limiting distribution, showing that one can transfer  
information
remarkably quickly between different vertices in the network. This  
efficiency
has a down side in that such networks are remarkably fragile to directed
attacks. These results continue a general program by the authors to  
obtain a
complete picture of how random disorder changes the inherent geometry of
various random network models.


  http://arxiv.org/abs/0905.4438

---------------------------------------------------------------

8563. DEDUCING VERTEX WEIGHTS FROM EMPIRICAL OCCUPATION TIMES

Joshua N. Cooper

We consider the following problem arising from the study of human  
problem
solving: Let $G$ be a vertex-weighted graph with marked "in" and "out"
vertices. Suppose a random walker begins at the in-vertex, steps to  
neighbors
of vertices with probability proportional to their weights, and stops  
upon
reaching the out-vertex. Could one deduce the weights from the paths  
that many
such walkers take? We analyze an iterative numerical solution to this
reconstruction problem, in particular, given the empirical mean  
occupation
times of the walkers. In the process, a result concerning the  
differentiation
of a matrix pseudoinverse is given, which may be of independent  
interest. We
then consider the existence of a choice of weights for the given  
occupation
times, formulating a natural conjecture to the effect that -- barring  
obvious
obstructions -- a solution always exists. It is shown that the  
conjecture holds
for a class of graphs that includes all trees and complete graphs.  
Several open
problems are discussed.


  http://arxiv.org/abs/0905.4391

---------------------------------------------------------------

8564. L^P ESTIMATES FOR FEYNMAN-KAC PROPAGATORS WITH TIME-DEPENDENT  
REFERENCE  MEASURES

Andreas Eberle and Carlo Marinelli

We introduce a class of time-inhomogeneous transition operators of
Feynman-Kac type that can be considered as a generalization of  
symmetric Markov
semigroups to the case of a time-dependent reference measure. Applying  
weighted
Poincar\'e and logarithmic Sobolev inequalities, we derive L^p-L^p and  
L^p-L^q
estimates for the transition operators. Since the operators are not  
Markovian,
the estimates depend crucially on the value of p. Our studies are  
motivated by
applications to sequential Markov Chain Monte Carlo methods.


  http://arxiv.org/abs/0905.4411

---------------------------------------------------------------

8565. QUANTUM PROBABILITY, RENORMALIZATION AND INFINITE-DIMENSIONAL *- 
LIE  ALGEBRAS

Luigi Accardi and Andreas Boukas

The present paper reviews some intriguing connections which link  
together a
new renormalization technique, the theory of *-representations of  
infinite
dimensional *-Lie algebras, quantum probability, white noise and  
stochastic
calculus and the theory of classical and quantum infinitely divisible
processes.


  http://arxiv.org/abs/0905.4491

---------------------------------------------------------------

8566. OMNIBUS SEQUENCES, COUPON COLLECTION, AND MISSING WORD COUNTS

Sunil Abraham and  Greg Brockman and  Stephanie Sapp and  Anant P.  
Godbole

An {\it Omnibus Sequence} of length $n$ is one that has each possible
"message" of length $k$ embedded in it as a subsequence. We study  
various
properties of Omnibus Sequences in this paper, making connections,  
whenever
possible, to the classical coupon collector problem.


  http://arxiv.org/abs/0905.4517

---------------------------------------------------------------

8567. PERCOLATION OF WORDS ON $\Z^D$ WITH LONG RANGE CONNECTIONS

Bernardo N. B. de Lima and  Remy Sanchis and  Roger W. C. Silva

Consider an independent site percolation model on $\Z^d$, with  
parameter $p
\in (0,1)$, where all long range connections in the axes directions are
allowed. In this work we show that given any parameter $p$, there  
exists and
integer $K(p)$ such that all binary sequences (words) $\xi \in  
\{0,1\}^{\N}$
can be seen simultaneously, almost surely, even if all connections  
whose length
is bigger than $K(p)$ are suppressed. We also show some results  
concerning the
question how $K(p)$ should scale with $p$ when $p$ goes to zero. Related
results are also obtained for the question of whether or not almost  
all words
are seen.


  http://arxiv.org/abs/0905.4615

---------------------------------------------------------------

8568. HAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS

Jozsef Balogh and  Bela Bollobas and  Mark Walters

We prove that, in the Gilbert model for a random geometric graph, almost
every graph becomes Hamiltonian exactly when it first becomes 2- 
connected. This
proves a conjecture of Penrose.
   We also show that in the $k$-nearest neighbour model, there is a  
constant
$\kappa$ such that almost every $\kappa$-connected graph has a  
Hamilton cycle.


  http://arxiv.org/abs/0905.4650

---------------------------------------------------------------

8569. BULK UNIVERSALITY FOR WIGNER MATRICES

Laszlo Erdos and  Jose A. Ramirez and  Benjamin Schlein and  Horng- 
Tzer Yau

We consider $N\times N$ Hermitian Wigner random matrices $H$ where the
probability density for each matrix element is given by the density $ 
\nu(x)=
e^{- U(x)}$. We prove that the eigenvalue statistics in the bulk is  
given by
Dyson sine kernel provided that $U \in C^6(\RR)$ with at most  
polynomially
growing derivatives and $\nu(x) \le C e^{- C |x|}$ for $x$ large. The  
proof is
based upon an approximate time reversal of the Dyson Brownian motion  
combined
with the convergence of the eigenvalue density to the Wigner  
semicircle law on
short scales.


  http://arxiv.org/abs/0905.4176

---------------------------------------------------------------

8570. CHARACTERIZING PREDICTABLE CLASSES OF PROCESSES

Daniil Ryabko (INRIA Futurs and  LIFL and  INRIA Lille - Nord Europe)

The problem is sequence prediction in the following setting. A sequence
$x_1,...,x_n,...$ of discrete-valued observations is generated  
according to
some unknown probabilistic law (measure) $\mu$. After observing each  
outcome,
it is required to give the conditional probabilities of the next  
observation.
The measure $\mu$ belongs to an arbitrary class $\C$ of stochastic  
processes.
We are interested in predictors $\rho$ whose conditional probabilities  
converge
to the "true" $\mu$-conditional probabilities if any $\mu\in\C$ is  
chosen to
generate the data. We show that if such a predictor exists, then a  
predictor
can also be obtained as a convex combination of a countably many  
elements of
$\C$. In other words, it can be obtained as a Bayesian predictor whose  
prior is
concentrated on a countable set. This result is established for two very
different measures of performance of prediction, one of which is very  
strong,
namely, total variation, and the other is very weak, namely,  
prediction in
expected average Kullback-Leibler divergence.


  http://arxiv.org/abs/0905.4341

---------------------------------------------------------------

8571. RIFFLE SHUFFLES OF A DECK WITH REPEATED CARDS

Sami Assaf and  Persi Diaconis and  K. Soundararajan

We study the Gilbert-Shannon-Reeds model for riffle shuffles and ask
   'How many times must a deck of cards be shuffled for the deck to be  
in close
to random order?'. In 1992, Bayer and Diaconis gave a solution which  
gives
exact and asymptotic results for all decks of practical interest, e.g.  
a deck
of 52 cards. But what if one only cares about the colors of the cards or
disregards the suits focusing solely on the ranks? More generally, how  
does the
rate of convergence of a Markov chain change if we are interested in  
only
certain features? Our exploration of this problem takes us through  
random walks
on groups and their cosets, discovering along the way exact formulas  
leading to
interesting combinatorics, an 'amazing matrix', and new analytic  
methods which
produce a completely general asymptotic solution that is remarkable  
accurate.


  http://arxiv.org/abs/0905.4698

---------------------------------------------------------------

8572. ASYMPTOTICS OF THE VISIBILITY FUNCTION IN THE BOOLEAN MODEL

Pierre Calka (MAP5) and  Julien Michel (UMPA-ENSL) and  Sylvain Porret- 
Blanc  (UMPA-ENSL)

The aim of this paper is to give a precise estimate on the tail  
probability
of the visibility function in a germ-grain model: this function is  
defined as
the length of the longest ray starting at the origin that does not  
intersect an
obstacle in a Boolean model. We proceed in two or more dimensions using
coverage techniques. Moreover, convergence results involving a type I  
extreme
value distribution are shown in the two particular cases of small  
obstacles or
a large obstacle-free region.


  http://arxiv.org/abs/0905.4874

---------------------------------------------------------------

8573. A CRITERION FOR HYPOTHESIS TESTING FOR STATIONARY PROCESSES

Daniil Ryabko (INRIA Futurs and  LIFL and  INRIA Lille - Nord Europe)

Given a discrete-valued sample X_1... X_n we wish to test whether it was
generated by a process belonging to a family H_0, or it was generated  
by a
process outside H_0. All process distributions are assumed stationary  
ergodic,
and no further probabilistic or parametric assumptions are made. We  
require the
Type I error of the test to be uniformly bounded, while the  
probability of Type
II error has to tend to zero as the sample size increases. For this  
notion of
consistency we provide necessary and sufficient conditions on the  
family H_0
for the existence of a consistent test. This criterion is illustrated  
with
applications to testing for a membership to parametric families,  
generalizing
some existing results.


  http://arxiv.org/abs/0905.4937

---------------------------------------------------------------

8574. LONG AND SHORT PATHS IN UNIFORM RANDOM RECURSIVE DAGS

Luc Devroye and Svante Janson

In a uniform random recursive k-dag, there is a root, 0, and each node  
in
turn, from 1 to n, chooses k uniform random parents from among the  
nodes of
smaller index. If S_n is the shortest path distance from node n to the  
root,
then we determine the constant \sigma such that S_n/log(n) tends to  
\sigma in
probability as n tends to infinity. We also show that max_{1 \le i \le  
n}
S_i/log(n) tends to \sigma in probability.


  http://arxiv.org/abs/0906.0152

---------------------------------------------------------------

8575. COMPUTATIONAL METHODS FOR STOCHASTIC RELATIONS AND MARKOVIAN  
COUPLINGS

Lasse Leskel\"a (Helsinki University of Technology)

Order-preserving couplings are elegant tools for obtaining robust  
estimates
of the time-dependent and stationary distributions of Markov processes  
that are
too complex to be analyzed exactly. The starting point of this paper  
is to
study stochastic relations, which may be viewed as natural  
generalizations of
stochastic orders. This generalization is motivated by the observation  
that for
the stochastic ordering of two Markov processes, it suffices that the
generators of the processes preserve some, not necessarily reflexive or
transitive, subrelation of the order relation. The main contributions  
of the
paper are an algorithmic characterization of stochastic relations  
between
finite spaces, and a truncation approach for comparing infinite-state  
Markov
processes. The methods are illustrated with applications to loss  
networks and
parallel queues.


  http://arxiv.org/abs/0906.0153

---------------------------------------------------------------

8576. QUASI-MARTINGALES WITH A LINEARLY ORDERED INDEX SET

Gianluca Cassese

We prove a version of Rao decomposition for quasi-martingales indexed  
by a
linearly ordered set.


  http://arxiv.org/abs/0906.0183

---------------------------------------------------------------

8577. A COUNTER-INTUITIVE CORRELATION IN A RANDOM TOURNAMENT

Sven Erick Alm and  Svante Linusson

Consider a randomly oriented graph $G=(V,E)$ and let $a$, $s$ and $b$ be
three distinct vertices in $V$. We study the correlation between the  
events
$\{a\to s\}$ and $\{s\to b\}$. We show that, when $G$ is the complete  
graph
$K_n$, the correlation is negative for $n=3$, zero for $n=4$, and that,
counter-intuitively, it is positive for $n\ge 5$. We also show that the
correlation is always negative when $G$ is a cycle, $C_n$, and  
negative or zero
when $G$ is a tree (or a forest).


  http://arxiv.org/abs/0906.0240

---------------------------------------------------------------

8578. APPROXIMATING A DIFFUSION BY A HIDDEN MARKOV MODEL

Ioannis Kontoyiannis and  Sean P. Meyn

For a wide class of continuous-time Markov processes, including all
irreducible hypoelliptic diffusions evolving on an open, connected  
subset of
$\RL^d$, the following are shown to be equivalent: (i) The process  
satisfies (a
slightly weaker version of) the classical Donsker-Varadhan conditions;  
(ii) The
transition semigroup of the process can be approximated by a finite- 
state
hidden Markov model, in a strong sense in terms of an associated  
operator norm;
(iii) The resolvent kernel of the process is `$v$-separable', that is,  
it can
be approximated arbitrarily well in operator norm by finite-rank  
kernels. Under
any (hence all) of the above conditions, the Markov process is shown  
to have a
purely discrete spectrum on a naturally associated weighted $L_\infty$  
space.


  http://arxiv.org/abs/0906.0259

---------------------------------------------------------------

8579. BRANCHING BROWNIAN MOTION: ALMOST SURE GROWTH ALONG SCALED PATHS

Simon Harris and  Matthew Roberts

We give a proof of a result on the growth of the number of particles  
along
chosen paths in a branching Brownian motion. The work follows the  
approach of
classical large deviations results, in which paths in $C[0,1]$ are  
rescaled
onto $C[0,T]$ for large $T$. The methods used are probabilistic and take
advantage of modern spine techniques.


  http://arxiv.org/abs/0906.0291

---------------------------------------------------------------

8580. BERRY-ESSEEN BOUNDS FOR GENERAL NONLINEAR STATISTICS, WITH  
APPLICATIONS  TO PEARSON'S AND NON-CENTRAL STUDENT'S AND HOTELLING'S

Iosif Pinelis and Raymond Molzon

Recently Chen and Shao developed a Stein-type method to obtain bounds  
on the
closeness of the distribution of a general nonlinear statistic to that  
of a
linear approximation. We generalize these results so as to allow one  
to use
lesser moment restrictions when applied to nonlinear statistics  
expressed as
smooth enough functions of sums of independent random vectors. Our main
innovation in the method is the use of a Cramer-type of tilt  
transform. Other
techniques used to obtain improvements include exponential and  
Rosenthal-type
inequalities for sums of random vectors established by Pinelis and  
Sakhanenko.
As applications, Berry-Esseen type bounds are obtained for concrete  
nonlinear
statistics such as the Pearson correlation coefficient and the non- 
central
Student and Hotelling statistics.


  http://arxiv.org/abs/0906.0177

---------------------------------------------------------------

8581. HAMILTONICITY OF THE RANDOM GEOMETRIC GRAPH

Michael Krivelevich and Tobias Muller

Let $X_1,X_2,...$ be independent, uniformly random points from  
$[0,1]^2$. For
$r\geq 0$ the {\em random geometric graph} $G(n,r)$ has vertex set  
$V_n :=
\{X_1,...,X_n\}$ and an edge $X_iX_j \in E_n$ iff. $\norm{X_i-X_j}  
\leq r$. The
"hitting radius" $\rho_n(\Pcal)$ of an increasing graph property $\Pcal 
$ is the
least $r$ such that $G(n,r)$ satisfies $\Pcal$, i.e. $\rho_n(\Pcal) :=  
\inf\{r
\geq 0 : G(n,r) \text{satisfies} \Pcal \}$. Here we prove that $ 
{\mathcal P}[
\rho_n(\text{min. degree}\geq 2) = \rho_n(\text{Hamiltonian}) ] \to 1$  
as $n
\to \infty$. This answers an open question of Penrose in the  
affirmative and
provides an analogue for the random geometric graph of a celebrated  
result of
Ajtai, Koml{\'o}s and Szemer{\'e}di on the Erd\H{o}s-R\'enyi random  
graph. The
proof generalises to uniform random points on the $d$-dimensional  
hypercube
with $\norm{.}$ any $l_p$-norm.


  http://arxiv.org/abs/0906.0071

---------------------------------------------------------------

8582. INTRINSIC VOLUMES OF INSCRIBED RANDOM POLYTOPES IN SMOOTH CONVEX  
BODIES

Imre B\'ar\'any and  Ferenc Fodor and  Viktor V\'igh

Let $K$ be a $d$ dimensional convex body with a twice continuously
differentiable boundary and everywhere positive Gauss-Kronecker  
curvature.
Denote by $K_n$ the convex hull of $n$ points chosen randomly and  
independently
from $K$ according to the uniform distribution. Matching lower and  
upper bounds
are obtained for the orders of magnitude of the variances of the $s$-th
intrinsic volumes $V_s(K_n)$ of $K_n$ for $s\in\{1, ..., d\}$.  
Furthermore,
strong laws of large numbers are proved for the intrinsic volumes of  
$K_n$. The
essential tools are the Economic Cap Covering Theorem of B\'ar\'any  
and Larman,
and the Efron-Stein jackknife inequality.


  http://arxiv.org/abs/0906.0309

---------------------------------------------------------------

8583. LARGE DEVIATIONS OF U-EMPIRICAL KOLMOGOROV-SMIRNOV TESTS, AND  
THEIR  EFFICIENCY

Yakov Nikitin

Non-degenerate U-empirical Kolmogorov-Smirnov tests are studied and  
their
large deviation asymptotics under the null-hypothesis is described.  
Several
examples of such statistics used for testing goodness-of-fit and  
symmetry are
considered. It is shown how to calculate their local Bahadur efficiency.


  http://arxiv.org/abs/0906.0428

---------------------------------------------------------------

8584. RANDOM MATRICES: UNIVERSALITY OF LOCAL EIGENVALUE STATISTICS

Terence Tao and  Van Vu

In this paper, we consider the universality of the local eigenvalue
statistics of random matrices. Our main result shows that these  
statistics are
determined by the first four moments of the distribution of the  
entries. As a
consequence, we derive the universality of eigenvalue gap distribution  
and
$k$-point correlation and many other statistics (under some mild  
assumptions)
for both Wigner Hermitian matrices and Wigner real symmetric matrices.


  http://arxiv.org/abs/0906.0510

---------------------------------------------------------------

8585. RATE OF CONVERGENCE OF STOCHASTIC PROCESSES WITH VALUES IN  $ 
\MATHBB{R}$-TREES AND HADAMARD MANIFOLDS

Kei Funano

Under K.-T. Sturm's formulation, we obtain a Gaussian upper bound for  
tail
probability of mean value of independent, identically distributed random
variables with values in $\mathbb{R}$-trees and Hadamard manifolds.


  http://arxiv.org/abs/0906.0649

---------------------------------------------------------------

8586. CORRELATIONS FOR PATHS IN RANDOM ORIENTATIONS OF G(N,P)

Sven Erick Alm and  Svante Linusson

We study the random graph G(n,p) with a random orientation. For three  
fixed
vertices s,a,b in G(n,p) we study the correlation of the events {a\to  
s} and
{s\to b}. We prove that for a fixed p<1/2 the correlation is negative  
for large
enough n and for p>1/2 the correlation is positive for large enough n.  
We
present exact recursions to compute P(a\to s) and P(a\to s, s\to b). We
conjecture that for a fixed n>26 the correlation changes sign three  
times for
three critical values of p.


  http://arxiv.org/abs/0906.0720

---------------------------------------------------------------

8587. ON THE UNIQUENESS OF SOLUTIONS TO QUADRATIC BSDES WITH CONVEX  
GENERATORS  AND UNBOUNDED TERMINAL CONDITIONS

Freddy Delbaen (Department of Mathematics) and  Ying Hu (IRMAR) and   
Adrien  Richou (IRMAR)

In a previous work, P. Briand and Y. Hu proved the uniqueness among the
solutions which admit every exponential moments. In this paper, we  
prove that
uniqueness holds among solutions which admit some given exponential  
moments.
These exponential moments are natural as they are given by the existence
theorem. Thanks to this uniqueness result we can strengthen the  
nonlinear
Feynman-Kac formula proved by P. Briand and Y. Hu.


  http://arxiv.org/abs/0906.0752

---------------------------------------------------------------

8588. EXPONENTIAL AND GAUSSIAN CONCENTRATION OF 1-LIPSCHITZ MAPS

Kei Funano

In this paper, we prove an exponential and Ganssian concentration  
inequality
for 1-Lipschitz maps from mm-spaces to Hadamard manifolds. In  
particular, we
give a complete answer to a question by M. Gromov.


  http://arxiv.org/abs/0906.0648

---------------------------------------------------------------

8589. THINNING, ENTROPY AND THE LAW OF THIN NUMBERS

Peter Harremoes and  Oliver Johnson and  Ioannis Kontoyiannis

Renyi's "thinning" operation on a discrete random variable is a natural
discrete analog of the scaling operation for continuous random  
variables. The
properties of thinning are investigated in an information-theoretic  
context,
especially in connection with information-theoretic inequalities  
related to
Poisson approximation results. The classical Binomial-to-Poisson  
convergence
(sometimes referred to as the "law of small numbers" is seen to be a  
special
case of a thinning limit theorem for convolutions of discrete  
distributions. A
rate of convergence is provided for this limit, and nonasymptotic  
bounds are
also established. This development parallels, in part, the development  
of
Gaussian inequalities leading to the information-theoretic version of  
the
central limit theorem. In particular, a "thinning Markov chain" is  
introduced,
and it is shown to play a role analogous to that of the Ornstein- 
Uhlenbeck
process in connection to the entropy power inequality.


  http://arxiv.org/abs/0906.0690

---------------------------------------------------------------

8590. A MARKOVIAN SLOT MACHINE AND PARRONDO'S PARADOX

S. N. Ethier and Jiyeon Lee

The antique Mills Futurity slot machine has two unusual features.  
First, if a
player loses 10 times in a row, the 10 lost coins are returned.  
Second, the
payout distribution varies from coup to coup in a manner that is  
nonrandom and
periodic with period 10. It follows that the machine is driven by a  
100-state
irreducible period-10 Markov chain. Here we evaluate the stationary
distribution of the Markov chain, and this leads to a strong law of  
large
numbers and a central limit theorem for the sequence of payouts.  
Following a
suggestion of Pyke (2003), we address the question of whether there  
exists a
two-armed version of this "one-armed bandit" that obeys Parrondo's  
paradox.
More precisely, is there such a machine with the property that the  
casino can
honestly advertise that both arms are fair, yet when players alternate  
arms in
certain random or nonrandom ways, the casino makes money in the long  
run? The
answer is a qualified yes. Although this "history-dependent" game is
conceptually simpler than the original such games of Parrondo, Harmer,  
and
Abbott (2000), it is nearly as complicated analytically, and open  
problems
remain.


  http://arxiv.org/abs/0906.0792

---------------------------------------------------------------

8591. BANDIT PROBLEMS WITH LEVY PAYOFF PROCESSES

Asaf Cohen and Eilon Solan

We study two-armed Levy bandits in continuous-time, which have one  
safe arm
that yields a constant payoff s, and one risky arm that can be either  
of type
High or Low; both types yield stochastic payoffs generated by a Levy  
process.
The expectation of the Levy process when the arm is High is greater  
than s, and
lower than s if the arm is Low.
   The decision maker (DM) has to choose, at any given time t, the  
fraction of
resource to be allocated to each arm over the time interval [t,t+dt).  
We show
that under proper conditions on the Levy processes, there is a unique  
optimal
strategy, which is a cut-off strategy, and we provide an explicit  
formula for
the cut-off and the optimal payoff, as a function of the data of the  
problem.
We also examine the case where the DM has incorrect prior over the  
type of the
risky arm, and we calculate the expected payoff gained by a DM who  
plays the
optimal strategy that corresponds to the incorrect prior.
   In addition, we study two applications of the results: (a) we show  
how to
price information in two-armed Levy bandit problem, and (b) we  
investigate who
fares better in two-armed bandit problems: an optimist who assigns to  
High a
probability higher than the true probability, or a pessimist who  
assigns to
High a probability lower than the true probability.


  http://arxiv.org/abs/0906.0835

---------------------------------------------------------------

8592. FUNCTIONAL INTEGRAL REPRESENTATIONS FOR SELF-AVOIDING WALK

David C. Brydges and  John Z. Imbrie and  Gordon Slade

We give a survey and unified treatment of functional integral  
representations
for both simple random walk and some self-avoiding walk models,  
including
models with strict self-avoidance, with weak self-avoidance, and a  
model of
walks and loops. Our representation for the strictly self-avoiding  
walk is new.
The representations have recently been used as the point of departure  
for
rigorous renormalization group analyses of self-avoiding walk models in
dimension 4. For the models without loops, the integral  
representations involve
fermions, and we also provide an introduction to fermionic integrals.  
The
fermionic integrals are in terms of anti-commuting Grassmann  
variables, which
can be conveniently interpreted as differential forms.


  http://arxiv.org/abs/0906.0922

---------------------------------------------------------------

8593. DIRECTED POLYMERS ON HIERARCHICAL LATTICES WITH SITE DISORDER

Hubert Lacoin (PMA) and  Gregorio Moreno Flores (PMA)

We study a polymer model on hierarchical lattices very close to the one
introduced and studied in \cite{DGr, CD}. For this model, we prove the
existence of free energy and derive the necessary and sufficient  
condition for
which very strong disorder holds for all $\gb$, and give some accurate  
results
on the behavior of the free energy at high-temperature. We obtain  
these results
by using a combination of fractional moment method and change of  
measure over
the environment to obtain an upper bound, and second moment method to  
get a
lower bound. We also get lower bounds on the fluctuation exponent of $ 
\log
Z_n$, and study the infinite polymer measure in the weak disorder phase.


  http://arxiv.org/abs/0906.0992

---------------------------------------------------------------

8594. APPLICATIONS OF STEIN'S METHOD FOR CONCENTRATION INEQUALITIES

Sourav Chatterjee and Partha S. Dey

Stein's method for concentration inequalities was introduced to prove
concentration of measure in problems involving complex dependencies  
such as
random permutations and Gibbs measures. In this paper, we provide some
extensions of the theory and three applications: (1) We obtain a  
concentration
inequality for the magnetization in the Curie-Weiss model at critical
temperature (where it obeys a non-standard normalization and super- 
Gaussian
concentration). (2) We derive exact large deviation asymptotics for  
the number
of triangles in the Erdos-Renyi random graph G(n,p) when p \ge 0.31.  
Similar
results are derived also for general subgraph counts. (3) We obtain some
interesting concentration inequalities for the Ising model on lattices  
that
hold at all temperatures.


  http://arxiv.org/abs/0906.1034

---------------------------------------------------------------

8595. ON ASYMPTOTIC EFFICIENCY OF MULTIVARIATE VERSION OF SPEARMAN'S RHO

Alexander Nazarov and  Natalia Stepanova

A multivariate version of Spearman's rho for testing independence is
considered. Its asymptotic efficiency is calculated under a general
distribution model specified by the dependence function. The efficiency
comparison study that involves other multivariate Spearman-type test  
statistics
is made. Conditions for Pitman optimality of the test are established.  
Examples
that illustrate the quality of the multivariate Spearman's test are  
included.


  http://arxiv.org/abs/0906.1059

---------------------------------------------------------------

8596. A SYMBOLIC COMPUTATIONAL APPROACH TO A PROBLEM INVOLVING  
MULTIVARIATE  POISSON DISTRIBUTIONS

Eduardo Sontag and  Doron Zeilberger

Multivariate Poisson random variables subject to linear integer  
constraints
arise in several application areas, such as queuing and biomolecular  
networks.
This note shows how to compute conditional statistics in this context,  
by
employing WF Theory and associated algorithms. A symbolic computation  
package
has been developed and is made freely available. A discussion of  
motivating
biomolecular problems is also provided.


  http://arxiv.org/abs/0906.1141

---------------------------------------------------------------

8597. CONFORMAL INVARIANCE AND UNIVERSAL CRITICAL EXPONENTS IN THE   
TWO-DIMENSIONAL PERCOLATION MODEL

Yu Zhang

For most two-dimensional critical percolation models, we show the  
existence
of a scaling limit for the crossing probabilities in an isosceles right
triangle. Furthermore, by justifying the lattice, the scaling limit is a
conformal invariance satisfying Cardy's formula in Carleson's form.  
Together
with the standard results of the $SLE_6$ process, we show that most  
critical
exponents exist in the sense of universality predicted by physics on  
most
two-dimensional lattices.


  http://arxiv.org/abs/0906.1203

---------------------------------------------------------------

8598. A NOTE ON WIENER-HOPF FACTORIZATION FOR MARKOV ADDITIVE PROCESSES

Przemyslaw Klusik and Zbigniew Palmowski

We prove the Wiener-Hopf factorization for Markov Additive processes. We
derive also Spitzer-Rogozin theorem for this class of processes which  
serves
for obtaining Kendall's formula and Fristedt representation of the  
cumulant
matrix of the ladder epoch process. Finally, we also obtain the so- 
called
ballot theorem.


  http://arxiv.org/abs/0906.1223

---------------------------------------------------------------

8599. GLOBAL HEAT KERNEL ESTIMATES FOR FRACTIONAL LAPLACIANS IN  
UNBOUNDED OPEN  SETS

Zhen-Qing Chen and Joshua Tokle

In this paper, we derive global sharp heat kernel estimates for  
symmetric
alpha-stable processes (or equivalently, for the fractional Laplacian  
with zero
exterior condition) in two classes of unbounded C^{1,1} open sets in  
R^d:
half-space-like open sets and exterior open sets. These open sets can be
disconnected. We focus in particular on explicit estimates for  
p_D(t,x,y) for
all t>0 and x, y\in D. Our approach is based on the idea that for x  
and y in
$D$ far from the boundary and t sufficiently large, we can compare  
p_D(t,x,y)
to the heat kernel in a well understood open set: either a half-space  
or R^d;
while for the general case we can reduce them to the above case by  
pushing $x$
and $y$ inside away from the boundary. As a consequence, sharp Green  
functions
estimates are obtained for the Dirichlet fractional Laplacian in these  
two
types of open sets. Global sharp heat kernel estimates and Green  
function
estimates are also obtained for censored stable processes (or  
equivalently, for
regional fractional Laplacian) in exterior open sets.


  http://arxiv.org/abs/0906.1234

---------------------------------------------------------------

8600. A RECURSIVE APPROACH FOR ALDOUS' SPECTRAL GAP CONJECTURE

Pietro Caputo and  Thomas M. Liggett and  Thomas Richthammer

Aldous' spectral gap conjecture asserts that on any graph the random  
walk
process and the random transposition (or interchange) process have the  
same
spectral gap. We describe a recursive strategy which could potentially  
settle
the conjecture for arbitrary weighted graphs. The approach is a natural
extension of the recursive method already used to prove the validity  
of the
conjecture on trees. The novelty is an idea based on electric network  
reduction
which reduces the proof of the conjecture to the proof of an explicit
inequality for a random transposition operator involving both positive  
and
negative rates. At present we are able to check the latter inequality  
only in
some cases. This already allows us to prove the conjecture for the  
class of
weighted graphs that can be reduced to a single edge by composing one- 
vertex
network reductions where at each step at most 3 edges are removed. In
particular, this includes all weighted trees, cycles and many more  
graphs with
small degree.


  http://arxiv.org/abs/0906.1238

---------------------------------------------------------------

8601. COMPUTING EXPECTATIONS WITH CONTINUOUS P-BOXES: UNIVARIATE CASE

L. Utkin and S. Destercke

Given an imprecise probabilistic model over a continuous space,  
computing
lower/upper expectations is often computationally hard to achieve,  
even in
simple cases. Because expectations are essential in decision making  
and risk
analysis, tractable methods to compute them are crucial in many  
applications
involving imprecise probabilistic models. We concentrate on p-boxes (a  
simple
and popular model), and on the computation of lower expectations of
non-monotone functions. This paper is devoted to the univariate case,  
that is
where only one variable has uncertainty. We propose and compare two  
approaches
: the first using general linear programming, and the second using the  
fact
that p-boxes are special cases of random sets. We underline the  
complementarity
of both approaches, as well as the differences.


  http://arxiv.org/abs/0906.1260

---------------------------------------------------------------

8602. A L\'EVY AREA BY FOURIER NORMAL ORDERING FOR MULTIDIMENSIONAL  
FRACTIONAL  BROWNIAN MOTION WITH SMALL HURST INDEX

Jeremie Unterberger (IECN)

The main tool for stochastic calculus with respect to a multidimensional
process $B$ with small H\"older regularity index is rough path theory.  
Once $B$
has been lifted to a rough path, a stochastic calculus -- as well as  
solutions
to stochastic differential equations driven by $B$ -- follow by standard
arguments. Although such a lift has been proved to exist by abstract  
arguments
\cite{LyoVic07}, a first general, explicit construction has been  
proposed in
\cite{Unt09,Unt09bis} under the name of Fourier normal ordering. The  
purpose of
this short note is to convey the main ideas of the Fourier normal  
ordering
method in the particular case of the iterated integrals of lowest  
order of
fractional Brownian motion with arbitrary Hurst index.


  http://arxiv.org/abs/0906.1416

---------------------------------------------------------------

8603. TREND TO EQUILIBRIUM AND PARTICLE APPROXIMATION FOR A WEAKLY   
SELFCONSISTENT VLASOV-FOKKER-PLANCK EQUATION

Francois Bolley (CEREMADE) and  Arnaud Guillin and  Florent Malrieu  
(IRMAR)

We consider a Vlasov-Fokker-Planck equation governing the evolution of  
the
density of interacting and diffusive matter in the space of positions  
and
velocities. We use a probabilistic interpretation to obtain  
convergence towards
equilibrium in Wasserstein distance with an explicit exponential rate.  
We also
prove a propagation of chaos property for an associated particle  
system, and
give rates on the approximation of the solution by the particle system.
Finally, a transportation inequality for the distribution of the  
particle
system leads to quantitative deviation bounds on the approximation of  
the
equilibrium solution of the equation by an empirical mean of the  
particles at
given time.


  http://arxiv.org/abs/0906.1417

---------------------------------------------------------------

8604. A WORLD RECORD AT AN ATLANTIC CITY CASINO AND THE DISTRIBUTION  
OF THE  LENGTH OF THE CRAPSHOOTER'S HAND

S. N. Ethier and Fred M. Hoppe

It was widely reported in the media that, on 23 May 2009, at the Borgata
Hotel Casino & Spa in Atlantic City, Patricia DeMauro, playing craps  
for only
the second time, rolled the dice for four hours and 18 minutes, finally
sevening out at the 154th roll, a world record. Initial estimates of the
probability of this event were erroneous, but consensus was reached  
within
days: one chance in 5.6 billion. More generally, what is P(L \ge n),  
where the
random variable L denotes the length of the crapshooter's hand (154 in  
Ms.
DeMauro's case) and n is a positive integer? It is well known that these
probabilities can be derived recursively or by Markov chain methods.  
Our aim
here is to give an explicit closed-form expression for them, showing  
that the
distribution of L is a linear combination (not a convex combination)  
of four
geometric distributions.


  http://arxiv.org/abs/0906.1545

---------------------------------------------------------------

8605. A $Q$-ANALOGUE OF THE FKG INEQUALITY AND SOME APPLICATIONS

Anders Bj\"orner

Let $L$ be a finite distributive lattice and $\mu : L \to {\mathbb  
R}^{+}$ a
log-supermodular function. For functions $k: L \to {\mathbb R}^{+}$ let
$$E_{\mu} (k; q) \defeq \sum_{x\in L} k(x) \mu (x) q^{{\mathrm rank} 
(x)} \in
{\mathbb R}^{+}[q].$$ We prove for any pair $g,h: L\to {\mathbb R}^{+} 
$ of
monotonely increasing functions, that $$E_{\mu} (g; q)\cdot E_{\mu}  
(h; q) \ll
   E_{\mu} (1; q)\cdot E_{\mu} (gh; q), $$ where ``$ \ll $'' denotes
coefficientwise inequality of real polynomials. The FKG inequality of  
Fortuin,
Kasteleyn and Ginibre (1971) is the real number inequality obtained by
specializing to $q=1$.
   The polynomial FKG inequality has applications to $f$-vectors of  
joins and
intersections of simplicial complexes, to Betti numbers of  
intersections of
certain Schubert varieties, and to the following kind of correlation  
inequality
for power series weighted by Young tableaux. Let $Y$ be the set of all  
integer
partitions. Given functions $k, \mu: Y \rarr \R^+$, define the formal  
power
series $$F_{\mu}(k ; z) \defeq \sum_{\la\in Y} k(\la) \mu(\la) f_{\la}
\frac{z^{|\la|}}{|\la| !} \in \R^+ [[z]],$$ where $f_{\la}$ is the  
number of
standard Young tableaux of shape $\la$. Assume that $\mu: Y\rarr \R^+$  
is
log-supermodular, and that $g, h: Y \rarr \R^+$ are monotonely  
increasing with
respect to containment order of partition shapes. Then $$F_{\mu}(g;z)  
\cdot
F_{\mu}(h;z) \ll F_{\mu}(1;z) \cdot F_{\mu}(gh;z). $$


  http://arxiv.org/abs/0906.1389

---------------------------------------------------------------

8606. DYNAMIC RISK DIVERSIFICATION AND INSURANCE PREMIUM PRINCIPLES

Kei Fukuda and  Akihiko Inoue and Yumiharu Nakano

We present an approach to the dynamic valuation of exposure risks in the
multi-period setting, which incorporates a dynamic and multiple  
diversification
of risks in Pareto optimal sense. This approach extends classical  
indifference
premium principles and can be applied for the valuation of insurance  
risks. In
particular, our method produces explicit computation formulas for the  
dynamic
version of the exponential premium principles. Moreover, we show limit  
theorems
asserting that the risk loading for our valuation decreases to zero  
when the
number of divisions of a risk goes to infinity.


  http://arxiv.org/abs/0906.1632

---------------------------------------------------------------

8607. WEIGHTED POINCAR\'{E}-TYPE INEQUALITIES FOR CAUCHY AND OTHER  
CONVEX  MEASURES

Sergey G. Bobkov and  Michel Ledoux

Brascamp--Lieb-type, weighted Poincar\'{e}-type and related analytic
inequalities are studied for multidimensional Cauchy distributions and  
more
general $\kappa$-concave probability measures (in the hierarchy of  
convex
measures). In analogy with the limiting (infinite-dimensional log- 
concave)
Gaussian model, the weighted inequalities fully describe the measure
concentration and large deviation properties of this family of measures.
Cheeger-type isoperimetric inequalities are investigated similarly,  
giving rise
to a common weight in the class of concave probability measures under
consideration.


  http://arxiv.org/abs/0906.1651

---------------------------------------------------------------

8608. INFINITE PATHS IN RANDOM SHIFT GRAPHS

Matteo Novaga and  Pietro Majer

We determine the probability tresholds for the existence of infinite  
paths in
random shift graphs.


  http://arxiv.org/abs/0906.1689

---------------------------------------------------------------

8609. ON SIMULTANEOUS HITTING OF MEMBRANES BY TWO SKEW BROWNIAN MOTIONS

Olga Aryasova and  Andrey Pilipenko

We consider two depending Wiener processes which have membranes at  
zero with
different permeability coefficients. Starting from different points, the
processes almost surely do not meet at any fixed point except that where
membranes are situated. The necessary and sufficient conditions for  
the meeting
of the processes are found. It is shown that the probability of  
meeting is
equal to zero or one.


  http://arxiv.org/abs/0906.1695

---------------------------------------------------------------

8610. AN EXTENSION OF THE YAMADA-WATANABE CONDITION FOR PATHWISE  
UNIQUENESS TO  STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS

Reinhard Hoepfner

We extend the Yamada-Watanabe condition for pathwise uniqueness to  
stochastic
differential equations with jumps, in the special case where small  
jumps are
summable.


  http://arxiv.org/abs/0906.1699

---------------------------------------------------------------

8611. INTERLACINGS FOR RANDOM WALKS ON WEIGHTED GRAPHS AND THE  
INTERCHANGE  PROCESS

A. B. Dieker

We study Aldous' conjecture that the spectral gap of the interchange  
process
on a weighted undirected graph equals the spectral gap of the random  
walk on
this graph. We present a conjecture in the form of an inequality, and  
prove
that this inequality implies Aldous' conjecture by combining an  
interlacing
result for Laplacians of random walks on weighted graphs with  
representation
theory. We prove the conjectured inequality for several important  
instances. As
an application of the developed theory, we prove Aldous' conjecture  
for a large
class of weighted graphs, which includes all wheel graphs, all graphs  
with four
vertices, certain nonplanar graphs, certain graphs with several  
weighted cycles
of arbitrary length, as well as all trees.


  http://arxiv.org/abs/0906.1716

---------------------------------------------------------------

8612. CLARK--OCONE FORMULA AND VARIATIONAL REPRESENTATION FOR POISSON   
FUNCTIONALS

Xicheng Zhang

In this paper we first prove a Clark--Ocone formula for any bounded
measurable functional on Poisson space. Then using this formula, under  
some
conditions on the intensity measure of Poisson random measure, we  
prove a
variational representation formula for the Laplace transform of  
bounded Poisson
functionals, which has been conjectured by Dupuis and Ellis [A Weak  
Convergence
Approach to the Theory of Large Deviations (1997) Wiley], p. 122.


  http://arxiv.org/abs/0906.1721

---------------------------------------------------------------

8613. ON SOME UNIVERSAL SIGMA FINITE MEASURES AND SOME EXTENSIONS OF  
DOOB'S  OPTIONAL STOPPING THEOREM

Joseph Najnudel and Ashkan Nikeghbali

In this paper, we associate, to any submartingale of class $(\Sigma)$,
defined on a filtered probability space $(\Omega, \mathcal{F},  
\mathbb{P},
(\mathcal{F}_t)_{t \geq 0})$, which satisfies some technical  
conditions, a
$\sigma$-finite measure $\mathcal{Q}$ on $(\Omega, \mathcal{F})$, such  
that for
all $t \geq 0$, and for all events $\Lambda_t
   \in \mathcal{F}_t$: $$ \mathcal{Q} [\Lambda_t, g\leq t] =
\mathbb{E}_{\mathbb{P}} [\mathds{1}_{\Lambda_t} X_t]$$ where $g$ is  
the last
hitting time of zero of the process $X$. This measure $\mathcal{Q}$  
has already
been defined in several particular cases, some of them are involved in  
the
study of Brownian penalisation, and others are related with problems in
mathematical finance. More precisely, the existence of $\mathcal{Q}$  
in the
general case solves a problem stated by D. Madan, B. Roynette and M.  
Yor, in a
paper studying the link between Black-Scholes formula and last passage  
times of
certain submartingales. Moreover, the equality defining $\mathcal{Q}$  
remains
true if one replaces the fixed time $t$ by any bounded stopping time.  
This
generalization can be viewed as an extension of Doob's optional stopping
theorem.


  http://arxiv.org/abs/0906.1782

---------------------------------------------------------------

8614. UNIQUE DECOMPOSITIONS, FACES, AND AUTOMORPHISMS OF SEPARABLE  
STATES

Erik Alfsen and Fred Shultz

We show that the set of separable states of length at most max(m,n) on  
B(C^m
otimes C^n) admits an open dense set of states with unique  
decomposition as a
convex combination of pure product states, and we describe all  
possible convex
decompositions for a larger set of separable states. In both cases we  
describe
the associated faces of the space of separable states, which in the  
first case
are simplexes, and in the second case are direct convex sums of faces  
that are
isomorphic to state spaces of full matrix algebras. As an application  
of these
results, we characterize all affine automorphisms of the convex set of
separable states, and all automorphisms of the state space of B(C^m  
otimes C^n)
that preserve entanglement and separability.


  http://arxiv.org/abs/0906.1761

---------------------------------------------------------------

8615. A MODEL FOR SEXED COAGULATION

Raoul Normand

We consider in this work a model for aggregation, where the coalescing
particles initially have a certain number of potential links (called  
arms)
which are used to perform coagulations. This model is sexed, is the  
sense that
there are male and female arms: two particles may coagulate only if  
one has an
available male arm, and the other has an available female arm. After a
coagulation, the used arms are no longer available. We are interested  
in the
concentrations of the different types of particles, which are governed  
by a
modification of Smoluchowski's coagulation equation -- that is, an  
infinite
system of nonlinear differential equations. Using generating functions  
and
solving a nonlinear PDE, we show that, up to some critical time, there  
is a
unique solution to this equation. The Lagrange Inversion Formula  
allows in some
cases to obtain explicit solutions, and to relate our model to two  
recent
models for limited aggregation. We also show that, whenever the  
critical time
is infinite, the concentrations converge to a state where all arms have
disappeared, and the distribution of the masses is related to the law  
of the
size of some two-type Galton-Watson tree. Finally, we consider a  
microscopic
model for coagulation: we construct a sequence of Marcus-Luschnikov  
processes,
and show that it converges, before the critical time, to the solution  
of our
modified Smoluchowski's equation.


  http://arxiv.org/abs/0906.1773

---------------------------------------------------------------

8616. A TIME INHOMOGENEOUS COX-INGERSOLL-ROSS DIFFUSION WITH JUMPS

Reinhard Hoepfner

We consider a time inhomogeneous Cox-Ingersoll-Ross diffusion with  
positive
jumps. We exploit a branching property to prove existence of a unique  
strong
solution under a restrictive condition on the jump measure. We give  
Laplace
transforms for the transition probabilities, with an interpretation in  
terms of
limits of mixtures over Gamma laws.


  http://arxiv.org/abs/0906.1856

---------------------------------------------------------------

8617. RANDOM QUANTUM CHANNELS II: ENTANGLEMENT OF RANDOM SUBSPACES,  
RENYI  ENTROPY ESTIMATES AND ADDITIVITY PROBLEMS

Beno\^it Collins (ICJ) and  Ion Nechita (ICJ)

In this paper we obtain new bounds for the minimum output entropies of  
random
quantum channels. These bounds rely on random matrix techniques  
arising from
free probability theory. We then revisit the counterexamples developed  
by
Hayden and Winter to get violations of the additivity equalities for  
minimum
output R\'enyi entropies. We show that random channels obtained by  
randomly
coupling the input to a qubit violate the additivity of the $p$-R\'enyi
entropy. For some sequences of random quantum channels, we compute  
almost
surely the limit of their Schatten $S_1 \to S_p$ norms.


  http://arxiv.org/abs/0906.1877

---------------------------------------------------------------

8618. RIESZ EXPONENTIAL FAMILIES ON HOMOGENEOUS CONES

Imen Boutouria and  Abdelhamid Hassairi

In this paper, we introduce, for a multiplier $\chi$, a notion of  
generalized
power function $x\mapsto \Delta_{\chi}(x),$ defined on the homogeneous  
cone
${\mathcal{P}}$ of a Vinberg algebra ${\mathcal{A}}$. We then extend to
${\mathcal{A}}$ the famous Gindikin result, that is we determine the  
set of
multipliers $\chi$ such that the map $\theta \mapsto \Delta_{\chi} 
(\theta
^{-1})$, defined on ${\mathcal{P}}^{\ast}$, is the Laplace transform  
of a
positive measure $R_{\chi}$. We also determine the set of $\chi $ such  
that
$R_{\chi}$ generates an exponential family, and we calculate the  
variance
function of this family


  http://arxiv.org/abs/0906.1892

---------------------------------------------------------------

8619. THE REAL ZEROS OF A RANDOM POLYNOMIAL WITH DEPENDENT COEFFICIENTS

Jeffrey Matayoshi

Mark Kac gave one of the first results analyzing random polynomial  
zeros. He
considered the case of independent standard normal coefficients and  
was able to
show that the expected number of real zeros for a degree n polynomial  
is on the
order of (2/pi)log(n), as n goes to infinity. Several years later,  
Sambandham
considered two cases with some dependence assumed among the  
coefficients. The
first case looked at coefficients with an exponentially decaying  
covariance
function, while the second assumed a constant covariance. He showed  
that the
expectation of the number of real zeros for an exponentially decaying
covariance matches the independent case, while having a constant  
covariance
reduces the expected number of zeros in half. In this paper we will  
apply
techniques similar to Sambandham's and extend his results to a wider  
class of
covariance functions. Under certain restrictions on the spectral  
density, we
will show that the order of the expected number of real zeros remains  
the same
as in the independent case.


  http://arxiv.org/abs/0906.1996

---------------------------------------------------------------

8620. BESSEL PROCESS AND CONFORMAL QUANTUM MECHANICS

M. A. Rajabpour

Different aspects of the connection between the Bessel process and the
conformal quantum mechanics (CQM) are discussed. The meaning of the  
possible
generalizations of both models is investigated with respect to the  
other model,
including self adjoint extension of the CQM. Some other  
generalizations such as
the Bessel process in the wide sense and radial Ornstein- Uhlenbeck  
process are
discussed with respect to the underlying conformal group structure.


  http://arxiv.org/abs/0906.1728

---------------------------------------------------------------

8621. ANATOMY OF A YOUNG GIANT COMPONENT IN THE RANDOM GRAPH

Jian Ding and  Jeong Han Kim and  Eyal Lubetzky and  Yuval Peres

We provide a complete description of the giant component of the
Erd\H{o}s-R\'enyi random graph $G(n,p)$ as soon as it emerges from the  
scaling
window, i.e., for $p = (1+\epsilon)/n$ where $\epsilon^3 n \to \infty$  
and
$\epsilon=o(1)$.
   Our description is particularly simple for $\epsilon = o(n^{-1/4}) 
$, where
the giant component $C_1$ is contiguous with the following model  
(i.e., every
graph property that holds with high probability for this model also  
holds
w.h.p. for $C_1$). Let $Z$ be normal with mean $\frac23 \epsilon^3 n$  
and
variance $\epsilon^3 n$, and let $K$ be a random 3-regular graph on  
$2\lfloor
Z\rfloor$ vertices. Replace each edge of $K$ by a path, where the path  
lengths
are i.i.d. geometric with mean $1/\epsilon$. Finally, attach an  
independent
Poisson($1-\epsilon$)-Galton-Watson tree to each vertex.
   A similar picture is obtained for larger $\epsilon=o(1)$, in which  
case the
random 3-regular graph is replaced by a random graph with $N_k$  
vertices of
degree $k$ for $k\geq 3$, where $N_k$ has mean and variance of order
$\epsilon^k n$.
   This description enables us to determine fundamental  
characteristics of the
supercritical random graph. Namely, we can infer the asymptotics of the
diameter of the giant component for any rate of decay of $\epsilon$,  
as well as
the mixing time of the random walk on $C_1$.


  http://arxiv.org/abs/0906.1839

---------------------------------------------------------------

8622. DIAMETERS IN SUPERCRITICAL RANDOM GRAPHS VIA FIRST PASSAGE  
PERCOLATION

Jian Ding and  Jeong Han Kim and  Eyal Lubetzky and  Yuval Peres

We study the diameter of $C_1$, the largest component of the
Erd\H{o}s-R\'enyi random graph $\cG(n,p)$ emerging from the critical  
window,
i.e., for $p = \frac{1+\epsilon}n$ where $\epsilon^3 n \to \infty$ and
$\epsilon=o(1)$. This parameter was extensively studied for fixed $ 
\epsilon >
0$, yet results for $\epsilon=o(1)$ outside the critical window were  
only
obtained very recently: Riordan and Wormald gave precise estimates on  
the
diameter, however these do not cover the entire supercritical regime  
(namely,
when $\epsilon^3 n\to\infty$ arbitrarily slowly); {\L}uczak and  
Seierstad
estimated its order throughout this regime, yet their upper and lower  
bounds
differ by a factor of $\frac{1000}7$.
   We show that for any $\epsilon=o(1)$ with $\epsilon^3n\to\infty$, the
diameter of $C_1$ is with high probability asymptotic to $D
(\epsilon,n)=(3/\epsilon)\log(\epsilon^3 n)$. We also prove that, in  
this
regime, the diameter of the 2-core of $C_1$ is w.h.p. asymptotic to $ 
\frac23
D(\epsilon,n)$, and the maximal distance in it between any pair of  
kernel
vertices is w.h.p. asymptotic to $\frac{5}9D(\epsilon,n)$.
   The proofs rely on a recent structure result for the supercritical  
giant
component, which reduces the problem of estimating distances between its
vertices to the study of passage times in first-passage percolation.


  http://arxiv.org/abs/0906.1840

---------------------------------------------------------------

8623. VECTOR MEASURES OF BOUNDED GAMMA-VARIATION AND STOCHASTIC  
INTEGRALS

Jan van Neerven and Lutz Weis

We introduce the class of vector measures of bounded $\gamma$- 
variation and
study its relationship with vector-valued stochastic integrals with  
respect to
Brownian motions.


  http://arxiv.org/abs/0906.1883

---------------------------------------------------------------

8624. THE SUBELLIPTIC HEAT KERNEL ON SL(2,R): AN INTEGRAL  
REPRESENTATION AND  SOME FUNCTIONAL INEQUALITIES

Michel Bonnefont

In this paper, we study a subelliptic heat kernel on the Lie group  
SL(2,R).
The subelliptic structure on SL(2,R) comes from the fibration $SO(2) - 
 > SL(2,R)
-> H^2$. First, we derive an integral representation for this heat  
kernel. This
expression allows us to obtain some asymptotics in small time of this  
heat
kernel and gives a way to compute the subriemannian distance. Then, we
establish some gradient estimates and some functional inequalities  
like a
Li-Yau type estimate and a reverse Poincar\'e inequality.


  http://arxiv.org/abs/0906.1977

---------------------------------------------------------------

8625. INVARIANT TRANSPORTS OF STATIONARY RANDOM MEASURES AND MASS- 
STATIONARITY

G\"unter Last and  Hermann Thorisson

We introduce and study invariant (weighted) transport-kernels balancing
stationary random measures on a locally compact Abelian group. The  
first main
result is an associated fundamental invariance property of Palm  
measures,
derived from a generalization of Neveu's exchange formula. The second  
main
result is a simple sufficient and necessary criterion for the  
existence of
balancing invariant transport-kernels. We then introduce (in a  
nonstationary
setting) the concept of mass-stationarity with respect to a random  
measure,
formalizing the intuitive idea that the origin is a typical location  
in the
mass. The third main result of the paper is that a measure is a Palm  
measure if
and only if it is mass-stationary.


  http://arxiv.org/abs/0906.2062

---------------------------------------------------------------

8626. RATIONAL BEHAVIOUR IN THE PRESENCE OF STOCHASTIC PERTURBATIONS

Panayotis Mertikopoulos and  Aris L. Moustakas

We study repeated games where players employ an exponential learning  
scheme
in order to adapt to an ever-changing environment. If the game's  
payoffs are
subject to random perturbations, this scheme leads to a new stochastic  
version
of the replicator dynamics that is quite different from the "aggregate  
shocks"
approach of evolutionary game theory. Irrespective of the perturbations'
magnitude, we find that strategies which are dominated (even  
iteratively)
eventually become extinct and that the game's strict Nash equilibria are
stochastically asymptotically stable. We complement our analysis by
illustrating these results in the case of congestion games.


  http://arxiv.org/abs/0906.2094

---------------------------------------------------------------

8627. DE FINETTI'S DIVIDEND PROBLEM AND IMPULSE CONTROL FOR A TWO- 
DIMENSIONAL  INSURANCE RISK PROCESS

Irmina Czarna and Zbigniew Palmowski

Consider two insurance companies (or two branches of the same company)  
that
have the same claims and they divide premia in some specified  
proportions. We
model the occurrence of claims according to a Poisson process. The  
ruin is
achieved if the corresponding two-dimensional risk process first leave  
the
positive quadrant. We consider different kinds of linear barriers. We  
will
consider two scenarios of controlled process. In first one when two- 
dimensional
risk process hits the barrier the minimal amount of dividends is payed  
out to
keep the risk process within the region bounded by the barrier. In the  
second
scenario whenever process hits horizontal line, the risk process is  
reduced by
paying dividend to some fixed point in the positive quadrant and waits  
there
for the first claim to arrive. In both models we calculate discounted
cumulative dividend payments until the ruin time.


  http://arxiv.org/abs/0906.2100

---------------------------------------------------------------

8628. VARIATIONAL CHARACTERISATION OF GIBBS MEASURES WITH DELAUNAY  
TRIANGLE  INTERACTION

David Dereudre and  Hans-Otto Georgii

This paper deals with stationary Gibbsian point processes on the plane  
with
an interaction that depends on the tiles of the Delaunay triangulation  
of
points via a bounded triangle potential. It is shown that the class of  
these
Gibbs processes includes all minimisers of the associated free energy  
density
and is therefore nonempty. Conversely, each such Gibbs process  
minimises the
free energy density, provided the potential satisfies a weak long-range
assumption.


  http://arxiv.org/abs/0906.2153

---------------------------------------------------------------

8629. INFINITESIMAL NON-CROSSING CUMULANTS AND FREE PROBABILITY OF  
TYPE B

Maxime Fevrier and  Alexandru Nica

Free probabilistic considerations of type B first appeared in a paper by
Biane, Goodman and Nica in 2003. Recently, connections between type B  
and
infinitesimal free probability were put into evidence by Belinschi and
Shlyakhtenko (arXiv:0903.2721). The interplay between "type B" and
"infinitesimal" is also the object of the present paper. We study  
infinitesimal
freeness for a family of unital subalgebras A_1, ..., A_k in an  
infinitesimal
noncommutative probability space (A, phi, phi'), and we introduce a  
concept of
infinitesimal non-crossing cumulant functionals for (A, phi, phi'),  
obtained by
taking a formal derivative in the formula for usual non-crossing  
cumulants. We
prove that the infinitesimal freeness of A_1, ... A_k is equivalent to a
vanishing condition for mixed cumulants; this gives the infinitesimal
counterpart for a theorem of Speicher from "usual" free probability.  
We show
that the lattices of non-crossing partitions of type B appear in the
combinatorial study of (A, phi, phi'), in the formulas for infinitesimal
cumulants and when describing alternating products of infinitesimally  
free
random variables. As an application of alternating free products, we  
observe
the infinitesimal analogue for the well-known fact that freeness is  
preserved
under compression with a free projection. As another application, we  
observe
the infinitesimal analogue for a well-known procedure used to  
construct free
families of free Poisson elements. Finally, we discuss situations when  
the
freeness of A_1, ..., A_k in (A, phi) can be naturally upgraded to
infinitesimal freeness in (A, phi, phi'), for a suitable choice of a  
"companion
functional" phi'.


  http://arxiv.org/abs/0906.2017

---------------------------------------------------------------

8630. CONVERGENCE RATES OF THE SPLITTING SCHEME FOR PARABOLIC LINEAR   
STOCHASTIC CAUCHY PROBLEMS

Sonja Cox and Jan van Neerven

We study the splitting scheme associated with the linear stochastic  
Cauchy
problem dU(t) = AU(t) dt + dW(t), where A is the generator of an  
analytic
C_0-semigroup S={S(t)} on a Banach space E and W={W(t)} is a Brownian  
motion
with values in a fractional domain space E_\b associated with A. We  
prove that
if \a,\b,\g,\th \ge 0 are such that \g + \th < 1 and max[0,(\a-\b+ 
\th)] + \g <
1/2, then the approximate solutions U_n (where n is the number of time  
steps)
converge to the solution U in the Holder space C^\g([0,T];E_\a), both in
L^p-means and almost surely, with rate 1/n^\th.


  http://arxiv.org/abs/0906.2129

---------------------------------------------------------------

8631. ASYMPTOTIC RESULTS FOR THE TWO-PARAMETER POISSON-DIRICHLET  
DISTRIBUTION

Shui Feng and Fuqing Gao

The two-parameter Poisson-Dirichlet distribution is the law of a  
sequence of
decreasing nonnegative random variables with total sum one. It can be
constructed from stable and Gamma subordinators with the two-parameters,
$\alpha$ and $\theta$, corresponding to the stable component and Gamma
component respectively. The moderate deviation principles are  
established for
the two-parameter Poisson-Dirichlet distribution and the corresponding
homozygosity when $\theta$ approaches infinity, and the large deviation
principle is established for the two-parameter Poisson-Dirichlet  
distribution
when both $\alpha$ and $\theta$ approach zero.


  http://arxiv.org/abs/0906.2217

---------------------------------------------------------------

8632. ON THE EXPECTATIONS OF MAXIMA OF SETS OF INDEPENDENT RANDOM  
VARIABLES

D. V. Tokarev and  K. A. Borovkov

Let $X^1, ..., X^k$ and $Y^1, ..., Y^m$ be jointly independent copies of
random variables $X$ and $Y$, respectively. For a fixed total number $n 
$ of
random variables, we aim at maximising $M(k,m):= E \max \{X^1, ...,  
X^k, Y^1,
 >..., Y^{m} \}$ in $k = n-m\ge 0$, which corresponds to maximising  
the expected
lifetime of an $n$-component parallel system whose components can be  
chosen
from two different types. We show that the lattice $\{M(k,m): k, m\ge  
0\}$ is
concave, give sufficient conditions on $X$ and $Y$ for M(n,0) to be  
always or
ultimately maximal and derive a bound on the number of sign changes in  
the
sequence $M(n,0)-M(0,n)$, $n\ge 1$. The results are applied to a mixed
population of Bienayme-Galton-Watson processes, with the objective to  
derive
the optimal initial composition to maximise the expected time to  
extinction.


  http://arxiv.org/abs/0906.2270

---------------------------------------------------------------

8633. COEXISTENCE IN STOCHASTIC SPATIAL MODELS

Rick Durrett

In this paper I will review twenty years of work on the question: When  
is
there coexistence in stochastic spatial models? The answer, announced in
Durrett and Levin [Theor. Pop. Biol. 46 (1994) 363--394], and that we  
explain
in this paper is that this can be determined by examining the mean- 
field ODE.
There are a number of rigorous results in support of this picture, but  
we will
state nine challenging and important open problems, most of which date  
from the
1990's.


  http://arxiv.org/abs/0906.2293

---------------------------------------------------------------

8634. CRITICALLY LOADED QUEUEING MODELS THAT ARE THROUGHPUT SUBOPTIMAL

Rami Atar and  Gennady Shaikhet

This paper introduces and analyzes the notion of throughput  
suboptimality for
many-server queueing systems in heavy traffic. The queueing model under
consideration has multiple customer classes, indexed by a finite set
$\mathcal{I}$, and heterogenous, exponential servers. Servers are  
dynamically
chosen to serve customers, and buffers are available for customers  
waiting to
be served. The arrival rates and the number of servers are scaled up  
in such a
way that the processes representing the number of class-$i$ customers  
in the
system, $i\in\mathcal{I}$, fluctuate about a static fluid model, that is
assumed to be critically loaded in a standard sense. At the same time,  
the
fluid model is assumed to be throughput suboptimal. Roughly, this  
means that
the servers can be allocated so as to achieve a total processing rate  
that is
greater than the total arrival rate. We show that there exists a dynamic
control policy for the queueing model that is efficient in the  
following strong
sense: Under this policy, for every finite $T$, the measure of the set  
of times
prior to $T$, at which at least one customer is in the buffer,  
converges to
zero in probability as the arrival rates and number of servers go to  
infinity.
On the way to prove our main result, we provide a characterization of
throughput suboptimality in terms of properties of the buffer-station  
graph.


  http://arxiv.org/abs/0906.2305

---------------------------------------------------------------

8635. NO ARBITRAGE WITHOUT SEMIMARTINGALES

Robert A. Jarrow and  Philip Protter and  Hasanjan Sayit

We show that with suitable restrictions on allowable trading  
strategies, one
has no arbitrage in settings where the traditional theory would admit  
arbitrage
possibilities. In particular, price processes that are not  
semimartingales are
possible in our setting, for example, fractional Brownian motion.


  http://arxiv.org/abs/0906.2318

---------------------------------------------------------------

8636. PORTFOLIO CHOICE WITH JUMPS: A CLOSED-FORM SOLUTION

Yacine A\"it-Sahalia and  Julio Cacho-Diaz and  T. R. Hurd

We analyze the consumption-portfolio selection problem of an investor  
facing
both Brownian and jump risks. We bring new tools, in the form of  
orthogonal
decompositions, to bear on the problem in order to determine the optimal
portfolio in closed form. We show that the optimal policy is for the  
investor
to focus on controlling his exposure to the jump risk, while exploiting
differences in the Brownian risk of the asset returns that lies in the
orthogonal space.


  http://arxiv.org/abs/0906.2324

---------------------------------------------------------------

8637. THE ASYMPTOTIC DISTRIBUTION OF A CLUSTER-INDEX FOR I.I.D. NORMAL  
RANDOM  VARIABLES

Yannis G. Yatracos

In a sample variance decomposition, with components functions of the  
sample's
spacings, the largest component $\tilde{I}_n$ is used in cluster  
detection. It
is shown for normal samples that the asymptotic distribution of $ 
\tilde{I}_n$
is the Gumbel distribution.


  http://arxiv.org/abs/0906.2334

---------------------------------------------------------------

8638. CONDITIONS FOR RAPID MIXING OF PARALLEL AND SIMULATED TEMPERING  
ON  MULTIMODAL DISTRIBUTIONS

Dawn B. Woodard and  Scott C. Schmidler and  Mark Huber

We give conditions under which a Markov chain constructed via parallel  
or
simulated tempering is guaranteed to be rapidly mixing, which are  
applicable to
a wide range of multimodal distributions arising in Bayesian statistical
inference and statistical mechanics. We provide lower bounds on the  
spectral
gaps of parallel and simulated tempering. These bounds imply a single  
set of
sufficient conditions for rapid mixing of both techniques. A direct  
consequence
of our results is rapid mixing of parallel and simulated tempering for  
several
normal mixture models, and for the mean-field Ising model.


  http://arxiv.org/abs/0906.2341

---------------------------------------------------------------

8639. ERROR BOUNDS FOR COMPUTING THE EXPECTATION BY MARKOV CHAIN MONTE  
CARLO

Daniel Rudolf

We study the error of reversible Markov chain Monte Carlo methods for
approximating the expectation of a function. Explicit error bounds  
with respect
to different norms of the function are proven. By the estimation the  
well known
asymptotical limit of the error is attained, i.e. there is no gap  
between the
estimate and the asymptotical behavior. We discuss the dependence of  
the error
on a burn-in of the Markov chain. Furthermore we suggest and justify a  
specific
burn-in for optimizing the algorithm.


  http://arxiv.org/abs/0906.2359

---------------------------------------------------------------

8640. OPTIMAL PORTFOLIO LIQUIDATION WITH EXECUTION COST AND RISK

Idris Kharroubi (PMA and  Crest) and  Huyen Pham (PMA and  Crest)

We study the optimal portfolio liquidation problem over a finite  
horizon in a
limit order book with bid-ask spread and temporary market price impact
penalizing speedy execution trades. We use a continuous-time modeling
framework, but in contrast with previous related papers (see e.g. [24]  
and
[25]), we do not assume continuous-time trading strategies. We  
consider instead
real trading that occur in discrete-time, and this is formulated as an  
impulse
control problem under a solvency constraint, including the lag variable
tracking the time interval between trades. A first important result of  
our
paper is to show that nearly optimal execution strategies in this  
context lead
actually to a finite number of trading times, and this holds true  
without
assuming ad hoc any fixed transaction fee. Next, we derive the dynamic
programming quasi-variational inequality satisfied by the value  
function in the
sense of constrained viscosity solutions. We also introduce a family  
of value
functions converging to our value function, and which is characterized  
as the
unique constrained viscosity solutions of an approximation of our  
dynamic
programming equation. This convergence result is useful for numerical  
purpose,
postponed in a further study.


  http://arxiv.org/abs/0906.2565

---------------------------------------------------------------

8641. A FLUCTUATION LIMIT THEOREM OF BRANCHING PROCESSES WITH  
IMMIGRATION AND  STATISTICAL APPLICATIONS

Chunhua Ma

We prove a general fluctuation limit theorem for Galton-Watson branching
processes with immigration. The limit is a time-inhomogeneous OU type  
process
driven by a spectrally positive Levy process. As applications of this  
result,
we obtain some asymptotic estimates for the conditional least-squares  
estimator
of the offspring mean.


  http://arxiv.org/abs/0906.2586

---------------------------------------------------------------

8642. CHARACTERISTIC POLYNOMIALS OF SAMPLE COVARIANCE MATRICES

Holger K\"osters

We investigate the second-order correlation function of the  
characteristic
polynomial of a sample covariance matrix. Starting from an explicit  
formula for
the generating function, we re-obtain several well-known kernels from  
random
matrix theory.


  http://arxiv.org/abs/0906.2763

---------------------------------------------------------------

8643. HOT SCATTERERS AND TRACERS FOR THE TRANSFER OF HEAT IN  
COLLISIONAL  DYNAMICS

Raphael Lefevere and  Lorenzo Zambotti

We introduce stochastic models for the transport of heat in systems  
described
by local collisional dynamics. The dynamics consists of tracer  
particles moving
through an array of hot scatterers describing the effect of heat baths  
at fixed
temperatures. Those models have the structure of Markov renewal  
processes. We
study their ergodic properties in details and provide a useful formula  
for the
cumulant generating function of the time integrated energy current. We  
observe
that out of thermal equilibrium, the generating function is not  
analytic. When
the set of temperatures of the scatterers is fixed by the condition  
that in
average no energy is exchanged between the scatterers and the system,  
different
behaviours may arise. When the tracer particles are allowed to travel  
freely
through the whole array of scatterers, the temperature profile is  
linear. If
the particles are locked in between scatterers, the temperature  
profile becomes
nonlinear. In both cases, the thermal conductivity is interpreted as a
frequency of collision between tracers and scatterers.


  http://arxiv.org/abs/0904.0020

---------------------------------------------------------------

8644. ALGORITHMIC INFORMATION THEORY AND MARTINGALES

Laurent Bienvenu and  Alexander Shen

The notion of an individual random sequence goes back to von Mises. We
describe the evolution of this notion, especially the use of martingales
(suggested by Ville), and the development of algorithmic information  
theory in
1960s and 1970s (Solomonov, Kolmogorov, Martin-Lof, Levin, Chaitin,  
Schnorr and
others). We conclude with some remarks about the use of the algorithmic
information theory in the foundations of probability theory.


  http://arxiv.org/abs/0906.2614

---------------------------------------------------------------

8645. RESISTANCE BOUNDARIES OF INFINITE NETWORKS

Palle E. T. Jorgensen and Erin P. J. Pearse

A resistance network is a connected graph $(G,c)$ with edges (and edge
weights) determined by the conductance function $c_{xy}$. The  
Dirichlet energy
form $\mathcal E$ produces a Hilbert space structure (which we call  
the energy
space ${\mathcal H}_{\mathcal E}$) on the space of functions of finite  
energy.
In a previous paper, we constructed a reproducing kernel $\{v_x\}$ for  
this
Hilbert space and used it to prove a discrete Gauss-Green identity \ 
[{\mathcal
E}(u,v) = \sum_{G} u \Delta v + \sum_{\operatorname{bd}G} u
\frac{\partial}{\partial \mathbf{n}} v,\] where the latter sum is  
understood in
a limiting sense. Applying this formula to a harmonic function $u \in  
{\mathcal
H}_{\mathcal E}$ gives a boundary representation \[u(x) =
\sum_{\operatorname{bd}G} u \frac{\partial}{\partial \mathbf{n}} v +  
u(o),\]
where $o$ is a fixed reference vertex.
   In this paper, we use techniques from stochastic integration to  
make the
boundary $\operatorname{bd}G$ precise as a measure space, and replace  
the
latter formula with a boundary integral representation (in a sense  
analogous to
that of Poisson or Martin boundary theory). We construct a Gel'fand  
triple $S
\ci {\mathcal H}_{\mathcal E} \ci S'$ and obtain a probability measure
$\mathbb{P}$ and an isometric embedding of ${\mathcal H}_{\mathcal E}$  
into
$L^2(S',\mathbb{P})$. This gives a concrete representation of the  
boundary as a
certain subset of $S'$.


  http://arxiv.org/abs/0906.2745

---------------------------------------------------------------

8646. A SOLVABLE MODEL FOR HOMOPOLYMERS AND THE CRITICAL PHENOMENA

M. Cranston and  L. Koralov and  S. Molchanov and  B. Vainberg

We consider a model for the distribution of a long homopolymer with a
zero-range potential at the origin in $\mathbb{R}^3$. The distribution  
can be
obtained as a limit of Gibbs distributions corresponding to properly  
normalized
potentials concentrated in small neighborhoods of the origin as the  
size of the
neighborhoods tends to zero. The distribution depends on the length $T 
$ of the
polymer and a parameter $\gamma$ that corresponds, roughly speaking,  
to the
difference between the inverse temperature in our model and the  
critical value
of the inverse temperature.
   At the critical point $\gamma_{cr} = 0$ the transition occurs from  
the
globular phase (positive recurrent behavior of the polymer, $\gamma >  
0$) to
the extended phase (Brownian type behavior, $\gamma < 0$). The main  
result of
the paper is a detailed analysis of the behavior of the polymer when $ 
\gamma$
is near $\gamma_{cr}$.
   Our approach is based on analyzing the semigroups generated by the
self-adjoint extensions $\mathcal{L}_\gamma$ of the Laplacian on
$C_0^\infty(\mathbb{R}^3 \setminus \{0\})$ parametrized by $\gamma$,  
which are
related to the distribution of the polymer. The main technical tool of  
the
paper is the explicit formula for the resolvent of the operator
$\mathcal{L}_\gamma$.


  http://arxiv.org/abs/0906.2816

---------------------------------------------------------------

8647. A THREE-PARAMETER BINOMIAL APPROXIMATION

Vydas \v{C}ekanavi\v{c}ius and  Erol A. Pek\"oz and  Adrian R\"ollin  
and  Michael  Shwartz

We approximate the distribution of the sum of independent but not  
necessarily
identically distributed Bernoulli random variables using a shifted  
binomial
distribution where the three parameters (the number of trials, the  
probability
of success, and the shift amount) are chosen to match up the first three
moments of the two distributions. We give a bound on the approximation  
error in
terms of the total variation metric using Stein's method. A numerical  
study is
discussed that shows shifted binomial approximations typically are more
accurate than Poisson or standard binomial approximations. The  
application of
the approximation to solving a problem arising in Bayesian hierarchical
modeling is also discussed.


  http://arxiv.org/abs/0906.2855

---------------------------------------------------------------

8648. INFINITE VARIANCE STABLE LIMITS FOR SUMS OF DEPENDENT RANDOM  
VARIABLES

Katarzyna Bartkiewicz and  Adam Jakubowski and  Thomas Mikosch and   
Olivier  Wintenberger (CEREMADE)

The aim of this paper is to provide conditions which ensure that the  
affinely
transformed partial sums of a strictly stationary process converge in
distribution to an in?nite variance stable distribution. Conditions  
for this
convergence to hold are known in the literature. However, most of  
these results
are qualitative in the sense that the parameters of the limit  
distribution are
expressed in terms of some limiting point process. In this paper we  
will be
able to determine the parameters of the limiting stable distribution  
in terms
of some tail characteristics of the underlying stationary sequence. We  
will
apply our results to some standard time series models, including the  
GARCH(1,
1) process and its squares, the stochastic volatility models and  
solutions to
stochastic recurrence equations.


  http://www.arxiv.org

---------------------------------------------------------------

8649. MATHEMATICAL ANALYSIS OF STOCHASTIC MODELS FOR TUMOR-IMMUNE  
SYSTEMS

O. Chis and  D. Opris

In this paper we investigate some stochastic models for tumor-immune  
systems.
To describe these models we used a Wiener process, as the noise has a
stabilization effect. Their dynamics are studied in terms of stochastic
stability in the equilibrium points, by constructing the Lyapunov  
exponent,
depending on the parameters that describe the model. We have studied and
analyzed a Kuznetsov-Taylor like stochastic model and a Bell  
stochastic model
for tumor-immune systems. These stochastic models are studied from  
stability
point of view and they were represented using the Euler second order  
scheme.


  http://arxiv.org/abs/0906.2794

---------------------------------------------------------------

8650. REGULAR SETS AND COUNTING IN FREE GROUPS

Elizaveta Frenkel and  Alexei G. Myasnikov and  Vladimir N.  
Remeslennikov

In this paper we study asymptotic behavior of regular subsets in a  
free group
F of finite rank, compare their sizes at infinity, and develop  
techniques to
compute the probabilities of sets relative to distributions on F that  
come
naturally from no-return random walks on the Cayley graph of F. We  
apply these
techniques to study cosets, double cosets, and Schreier  
representatives of
finitely generated subgroups of F and also to analyze relative sizes  
of regular
prefixed-closed subsets in F.


  http://arxiv.org/abs/0906.2850

---------------------------------------------------------------

8651. A FINITE DIFFERENCE APPROACH TO THE INFINITY LAPLACE EQUATION  
AND  TUG-OF-WAR GAMES

Scott N. Armstrong and Charles K. Smart

We present a modified version of the $\epsilon$-step tug-of-war game  
recently
introduced by Peres, Schramm, Sheffield, and Wilson. This new tug-of- 
war game
is identical to the original except near the boundary of the domain $ 
\partial
\Omega$, but its associated value functions are more regular. Using  
the dynamic
programming principle, we show that the value functions satisfy a  
certain
finite difference equation. By studying solutions of this difference  
equation
directly, we are able to adapt techniques from viscosity solution  
theory to
prove a number of new results.
   The finite difference equation has unique maximal and minimal  
solutions,
which are identified as the value functions for the two tug-of-war  
players. We
demonstrate uniqueness, and hence the existence of a value for the  
game, in the
case that the running payoff function is nonnegative. In the limit $ 
\epsilon
\to 0$, we obtain the convergence of the value functions to a viscosity
solution of the normalized infinity Laplace equation.
   We also obtain several new results for the normalized infinity  
Laplace
equation $-\Delta_\infty u = f$. In particular, we demonstrate the  
existence of
solutions to the Dirichlet problem for any $f \in C(\Omega) \cap
L^\infty(\Omega)$ and continuous boundary data, as well as the  
uniqueness of
solutions to this problem in the generic case. We also give a new  
elementary
proof of Jensen's theorem on the uniqueness of infinity harmonic  
functions.


  http://arxiv.org/abs/0906.2871

---------------------------------------------------------------

8652. UNIQUELY ERGODIC MINIMAL TILING SPACES WITH POSITIVE ENTROPY

Ian Palmer and Jean Bellissard

Strictly ergodic spaces of tilings with positive entropy are constructed
using tools from information and probability theory. Statistical  
estimates are
made to create a one-dimensional subshift with these dynamical  
properties,
yielding a space of repetitive tilings of R^D wit finite local  
complexity that
is also equivalent to a symbolic dynamical system with a Z^D action.


  http://arxiv.org/abs/0906.2997

---------------------------------------------------------------

8653. ON THE DENSITY OF THE SUM OF TWO INDEPENDENT STUDENT T-RANDOM  
VECTORS

C. Berg and C. Vignat

In this paper, we find an expression for the density of the sum of two
independent $d-$dimensional Student $t-$random vectors $\mathbf{X}$ and
$\mathbf{Y}$ with arbitrary degrees of freedom. As a byproduct we also  
obtain
an expression for the density of the sum $\mathbf{N}+\mathbf{X}$, where
$\mathbf{N}$ is normal and $\mathbf{X}$ is an independent Student $t- 
$vector.
In both cases the density is given as an infinite series \ 
[ \sum_{n=0}^{\infty}
c_{n}f_{n} \] where $f_{n}$ is a sequence of probability densities on
$\mathbb{R}^{d}$ and $(c_{n} )$ is a sequence of positive numbers of  
sum 1,
i.e. the distribution of a non-negative integer-valued random variable  
$C$,
which turns out to be infinitely divisible for $d=1$ and $d=2.$ When  
$d=1$ and
the degrees of freedom of the Student variables are equal, we recover  
an old
result of Ruben.


  http://arxiv.org/abs/0906.3037

---------------------------------------------------------------

8654. FEYNMAN-KAC FORMULA FOR HEAT EQUATION DRIVEN BY FRACTIONAL WHITE  
NOISE

Yaozhong Hu and  David Nualart and  Jian Song

In this paper we establish a version of the Feynman-Kac formula for the
stochastic heat equation with a multiplicative fractional Brownian  
sheet. We
prove the smoothness of the density of the solution, and the H\"older
regularity in the space and time variables.


  http://arxiv.org/abs/0906.3076

---------------------------------------------------------------

8655. ZERO DISSIPATION LIMIT IN THE ABELIAN SANDPILE MODEL

Antal A.Jarai and  F. Redig and  E. Saada

We study the abelian avalanche model, an analogue of the abelian  
sandpile
model with continuous heights, which allows for arbitrary small values  
of
dissipation. We prove that for non-zero dissipation, the infinite  
volume limit
of the stationary measures of the abelian avalanche model exists and  
can be
obtained via a weighted spanning tree measure. Moreover we obtain  
exponential
decay of spatial covariances of local observables in the non-zero  
dissipation
regime. We then study the zero dissipation limit and prove that the
self-organized critical model is recovered, both for the stationary  
measures
and for the dynamics.


  http://arxiv.org/abs/0906.3128

---------------------------------------------------------------

8656. FIXED POINTS OF THE MIN-TRANSFORMATION

Matthias Meiners and Gerold Alsmeyer

In this paper, the stochastic fixed-point equation X \stackrel{d}{=}  
\inf_{i
\geq 1: T_i > 0} X_i/T_i is studied, where $\stackrel{d}{=}$ means  
equality in
law, $(X_{n})_{n\ge 1}$ and $T = (T_i)_{i \geq 1}$ denote independent  
sequences
of non-negative random variables, and $X_1, X_2, ...$ are independent  
copies of
the random variable $X$. Under suitable conditions on the given weight  
sequence
$T$, most notably that $1<\E N<\infty$ for $N=\sum_{i\ge 1}\1_{\ 
{T_{i}>0\}}$
and that $T$ has a characteristic exponent, defined as the minimal $ 
\alpha>0$
such that $\sum_{i\ge 1}\E T_{i}^{\alpha}=1$, we determine the set of  
all
solutions to the equation, viz. all distributions of $X$ such that the
distributional identity holds true. These turn out to be certain  
mixtures of
Weibull distributions (or periodic variants). This extends earlier  
results by
Alsmeyer and R\"osler for the case of deterministic $T$ and by the  
present
authors who showed (under weaker assumptions on $T$) that these  
distributions
are the only ones within a certain subclass of distributions on  
$[0,\infty)$.
It further extends results by Durrett and Liggett and by Liu on the  
related
equation $X \stackrel{d}{=} \sum_{i \geq 1} T_i X_i$ after the  
observation that
any Laplace transform of a solution to this latter equation may also  
be viewed
as the survival function of a solution to the above min-type equation.


  http://arxiv.org/abs/0906.3133

---------------------------------------------------------------

8657. FIXATION FOR DISTRIBUTED CLUSTERING PROCESSES

Marcelo R. Hilario and  Oren Louidor and  Charles M. Newman and   
Leonardo T.  Rolla, Scott Sheffield, Vladas Sidoravicius

We study a discrete-time resource flow in $Z^d$, where wealthier  
vertices
attract the resources of their less rich neighbors. For any
translation-invariant probability distribution of initial resource  
quantities,
we prove that the flow at each vertex terminates after finitely many  
steps.
This answers (a generalized version of) a question posed by Van den  
Berg and
Meester in 1991. The proof uses the mass-transport principle and  
extends to
other graphs.


  http://arxiv.org/abs/0906.3154

---------------------------------------------------------------

8658. ESTIMATION FOR THE CHANGE POINT OF THE VOLATILITY IN A  
STOCHASTIC  DIFFERENTIAL EQUATION

Stefano M. Iacus and  Nakahiro Yoshida

We consider a multidimensional It\^o process $Y=(Y_t)_{t\in[0,T]}$  
with some
unknown drift coefficient process $b_t$ and volatility coefficient
$\sigma(X_t,\theta)$ with covariate process $X=(X_t)_{t\in[0,T]}$, the  
function
$\sigma(x,\theta)$ being known up to $\theta\in\Theta$. For this model  
we
consider a change point problem for the parameter $\theta$ in the  
volatility
component. The change is supposed to occur at some point $t^*\in (0,T) 
$. Given
discrete time observations from the process $(X,Y)$, we propose quasi- 
maximum
likelihood estimation of the change point. We present the rate of  
convergence
of the change point estimator and the limit thereoms of aymptotically  
mixed
type.


  http://arxiv.org/abs/0906.3108

---------------------------------------------------------------

8659. SPECTRAL PROPERTIES OF THE CAUCHY PROCESS

Tadeusz Kulczycki and  Mateusz Kwa\'snicki and  Jacek Ma{\l}ecki and   
Andrzej Stos

We study the spectral properties of the transition semigroup of the  
killed
one-dimensional Cauchy process on the half-line (0,infty) and the  
interval
(-1,1). This process is related to the square root of one-dimensional  
Laplacian
A = -sqrt(-d^2/dx^2) with a Dirichlet exterior condition (on a  
complement of a
domain), and to a mixed Steklov problem in the half-plane. For the  
half-line,
an explicit formula for generalized eigenfunctions psi_lambda of A is  
derived,
and then used to construct spectral representation of A. Explicit  
formulas for
the transition density of the killed Cauchy process in the half-line  
(or the
heat kernel of A in (0,infty)), and for the distribution of the first  
exit time
from the half-line follow. The formula for psi_lambda is also used to  
construct
approximations to eigenfunctions of A in the interval. For the  
eigenvalues
lambda_n of A in the interval the asymptotic formula lambda_n = n pi/2  
- pi/8 +
O(1/n) is derived, and all eigenvalues lambda_n are proved to be simple.
Finally, efficient numerical methods of estimation of eigenvalues  
lambda_n are
applied to obtain lower and upper numerical bounds for the first few
eigenvalues up to 9th decimal point.


  http://arxiv.org/abs/0906.3113

---------------------------------------------------------------

8660. AFFINE PROCESSES ARE REGULAR

Martin Keller-Ressel and  Walter Schachermayer and  Josef Teichmann

We show that stochastically continuous, time-homogeneous affine  
processes on
the canonical state space $\mathbb{R}_{\geq 0}^m \times \mathbb{R}^n$  
are
always regular. In the paper of Duffie-Filipovi\'c-Schachermayer (2003)
regularity was used as a crucial basic assumption. However, a  
counterexample of
a non-regular, stochastically continuous affine process was neither  
known nor
expected. We now show that the regularity assumption is indeed  
superfluous,
since regularity follows from stochastic continuity and the  
exponentially
affine behavior of the characteristic function. For the proof we combine
classic results on the differentiability of transformation semigroups  
with the
method of the moving frame which has been recently found to be useful  
in the
theory of SPDEs.


  http://arxiv.org/abs/0906.3392

---------------------------------------------------------------

8661. AROUND TSIRELSON'S EQUATION, OR: THE EVOLUTION PROCESS MAY NOT  
EXPLAIN  EVERYTHING

Kouji Yano and Marc Yor

We present a synthesis of a number of developments which have been made
around the celebrated Tsirelson's equation (1975), conveniently  
modified in the
framework of a Markov chain taking values in a compact group $ G $,  
and indexed
by negative time. To illustrate, we discuss in detail the case of the
one-dimensional torus $ G=\bT $.


  http://arxiv.org/abs/0906.3442

---------------------------------------------------------------

8662. RATE OF CONVERGENCE FOR NUMERICAL SOLUTIONS TO SFDES WITH JUMPS

Jianhai Bao and  Xuerong Mao and Chenggui Yuan

In this paper, we are interested in the numerical solutions of  
stochastic
functional differential equations (SFDEs) with {\it jumps}. Under the  
global
Lipschitz condition, we show that the $p$th moment convergence of the
Euler-Maruyama (EM) numerical solutions to SFDEs with jumps has order  
$1/p$ for
any $p\ge 2$. This is significantly different from the case of SFDEs  
without
jumps where the order is 1/2 for any $p\ge 2$. It is therefore best to  
use the
mean-square convergence for SFDEs with jumps. Consequently, under the  
local
Lipschitz condition, we reveal that the order of the mean-square  
convergence is
close to 1/2, provided that the local Lipschitz constants, valid on  
balls of
radius $j$, do not grow faster than $\log j$.


  http://arxiv.org/abs/0906.3455

---------------------------------------------------------------

8663. A WEAK TRAPEZOIDAL METHOD FOR A CLASS OF STOCHASTIC  
DIFFERENTIAL  EQUATIONS

David F. Anderson and  Jonathan C. Mattingly

We present a numerical method for the approximation of solutions for the
class of stochastic differential equations driven by Brownian motions  
which
induce stochastic variation in fixed directions. This class of  
equations arises
naturally in the study of population processes and chemical reaction  
kinetics.
We show that the method constructs paths that are second order  
accurate in the
weak sense. The method is simpler than many second order methods in  
that it
neither requires the construction of iterated Ito integrals nor the  
evaluation
of any derivatives. The method consists of two steps. In the first an  
explicit
Euler step is used to take a fractional step. This fractional point is  
then
combined with the initial point to obtain a higher order, trapezoidal  
like,
approximation. The higher order of accuracy stems from the fact that  
both the
drift and the quadratic variation of the underlying SDE are  
approximated to
second order.


  http://arxiv.org/abs/0906.3475

---------------------------------------------------------------

8664. BROWNIAN MOTION IN A BALL IN THE PRESENCE OF SPHERICAL OBSTACLES

Julie O'Donovan

We study the problem of when a Brownian motion in the unit ball has a
positive probability of avoiding a countable collection of spherical  
obstacles.
We give a necessary and sufficient integral condition for such a  
collection to
be avoidable.


  http://arxiv.org/abs/0906.3481

---------------------------------------------------------------

8665. CRITICAL HOMOGENIZATION OF LEVY PROCESS DRIVEN SDES IN RANDOM  
MEDIUM

R\'emi Rhodes (CEREMADE) and  Bamba A. Sow (LERSTAD)

We are concerned with homogenization of stochastic differential  
equations
(SDE) with stationary coefficients driven by Poisson random measures and
Brownian motions in the critical case, that is when the limiting  
equation
admits both a Brownian part as well as a pure jump part. We state an  
annealed
convergence theorem. This problem is deeply connected with  
homogenization of
integral partial differential equations


  http://arxiv.org/abs/0906.3569

---------------------------------------------------------------

8666. ON MONOCHROMATIC ARM EXPONENTS FOR 2D CRITICAL PERCOLATION

Vincent Beffara (UMPA-ENSL) and  Pierre Nolin (CIMS)

We investigate the so-called monochromatic arm exponents for critical
percolation in two dimensions. These exponents, describing the  
probability of
observing j disjoint macroscopic paths, are shown to exist and to form a
different family from the (now well-understood) polychromatic exponents.


  http://arxiv.org/abs/0906.3570

---------------------------------------------------------------

8667. CENTRAL LIMIT THEOREM FOR COLOURED HARD-DIMERS

Maria Simonetta Bernabei and  Horst Thaler

Using an averaged generating function for coloured hard-dimers, some  
random
variables of interest are studied. The main result lies in the fact  
that all
their probability distributions obey a central limit theorem.


  http://arxiv.org/abs/0906.3652

---------------------------------------------------------------

8668. SHARP THRESHOLD FOR PERCOLATION ON EXPANDERS

Itai Benjamini and  Stephane Boucheron and  Gabor Lugosi and  Raphael  
Rossignol

We study the appearance of the giant component in random subgraphs of  
a given
finite graph G=(V,E) in which each edge is present independently with
probability p. We show that if G is an expander with vertices of bounded
degree, then for any c in ]0,1[, the property that the random subgraph  
contains
a giant component of size c|V| has a sharp threshold.


  http://arxiv.org/abs/0906.3657

---------------------------------------------------------------

8669. ZEROS OF AIRY FUNCTION AND RELAXATION PROCESS

Makoto Katori and  Hideki Tanemura

One-dimensional system of Brownian motions called Dyson's model is the
particle system with long-range repulsive forces acting between any  
pair of
particles, where the strength of force is $\beta/2$ times the inverse of
particle distance. When $\beta=2$, it is realized as the Brownian  
motions in
one dimension conditioned never to collide with each other. For any  
initial
configuration, it is proved that Dyson's model with $\beta=2$ and $N$
particles, $\X(t)=(X_1(t), ..., X_N(t)), t \in [0,\infty), 2 \leq N <  
\infty$,
is determinantal in the sense that any multitime correlation function  
is given
by a determinant with a continuous kernel. The Airy function $\Ai(z)$  
is an
entire function with zeros all located on the negative part of the  
real axis
$\R$. We consider Dyson's model with $\beta=2$ starting from the first  
$N$
zeros of $\Ai(z)$, $0 > a_1 > ... > a_N$, $N \geq 2$. In order to  
properly
control the effect of such initial confinement of particles in the  
negative
region of $\R$, we put the drift term to each Brownian motion, which  
increases
in time as a parabolic function :
$Y_j(t)=X_j(t)+t^2/4+\{d_1+\sum_{\ell=1}^{N}(1/a_{\ell})\}t, 1 \leq j  
\leq N$,
where $d_1=\Ai'(0)/\Ai(0)$. We show that, as the $N \to \infty$ limit of
$\Y(t)=(Y_1(t), ..., Y_N(t)), t \in [0, \infty)$, we obtain an infinite
particle system, which is the relaxation process from the  
configuration, in
which every zero of $\Ai(z)$ on the negative $\R$ is occupied by one  
particle,
to the stationary state $\mu_{\Ai}$. The stationary state $\mu_{\Ai}$  
is the
determinantal point process with the Airy kernel, which is spatially
inhomogeneous on $\R$ and in which the Tracy-Widom distribution  
describes the
rightmost particle position.


  http://arxiv.org/abs/0906.3666

---------------------------------------------------------------

8670. THE NATURAL PARAMETRIZATION FOR THE SCHRAMM-LOEWNER EVOLUTION

Gregory F. Lawler and  Scott Sheffield

The Schramm-Loewner evolution (SLE_\kappa) is a candidate for the  
scaling
limit of random curves arising in two-dimensional critical phenomena.  
When
\kappa < 8, an instance of SLE_\kappa is a random planar curve with  
almost sure
Hausdorff dimension d = 1 + \kappa/8 < 2. This curve is conventionally
parametrized by its half plane capacity, rather than by any measure of  
its
d-dimensional volume. For \kappa < 8, we use a Doob-Meyer  
decomposition to
construct the unique (under mild assumptions) Markovian  
parametrization of
SLE_\kappa that transforms like a d-dimensional volume measure under  
conformal
maps. We prove that this parametrization is non-trivial (i.e., the  
curve is not
entirely traversed in zero time) for \kappa < 4(7 - \sqrt{33}) =  
5.021 ....


  http://arxiv.org/abs/0906.3804

---------------------------------------------------------------

8671. MARKOV CHAINS CONDITIONED NEVER TO WAIT TOO LONG AT THE ORIGIN

Saul Jacka

Motivated by Feller's coin-tossing problem, we consider the problem of
conditioning an irreducible Markov chain never to wait too long at 0.  
Denoting
by $\tau$ the first time that the chain, $X$, waits for at least one  
unit of
time at the origin, we consider conditioning the chain on the event $ 
(\tau>T)$.
We show there is a weak limit as $T\to \infty$ in the cases where  
either the
statespace is finite or $X$ is transient. We give sufficient  
conditions for the
existence of a weak limit in other cases and show that we have vague
convergence to a defective limit if the time to hit zero has a lighter  
tail
than $\tau$ and $\tau$ is subexponential.


  http://arxiv.org/abs/0906.3876

---------------------------------------------------------------

8672. CONCENTRATION OF MEASURES VIA SIZE BIASED COUPLINGS

Subhankar Ghosh and  Larry Goldstein

Let $Y$ be a nonnegative random variable with mean $\mu$ and finite  
positive
variance $\sigma^2$, and let $Y^s$, defined on the same space as $Y$,  
have the
$Y$ size biased distribution, that is, the distribution characterized by
E[Yf(Y)]=\mu E f(Y^s) for all functions $f$ for which these  
expectations exist.
Under a variety of conditions on the coupling of Y and $Y^s$, including
combinations of boundedness and monotonicity, concentration of measure
inequalities hold. Examples include the number of relatively ordered
subsequences of a random permutation, sliding window statistics  
including the
number of m-runs in a sequence of coin tosses, the number of local  
maximum of a
random function on a lattice, the number of urns containing exactly  
one ball in
an urn allocation model, the volume covered by the union of $n$ balls  
placed
uniformly over a volume n subset of d diml Euclidean space, the number  
of bulbs
switched on at the terminal time in the so called lightbulb process,  
the number
of isolated vertices in the Erdos-Renyi random graph model, and the  
infinitely
divisible and compound Poisson distributions that satisfy a bounded  
moment
generating function condition.


  http://arxiv.org/abs/0906.3886

---------------------------------------------------------------

8673. SURVIVAL AND GROWTH OF A BRANCHING RANDOM WALK IN RANDOM  
ENVIRONMENT

Christian Bartsch and  Nina Gantert and  Michael Kochler

We consider a particular Branching Random Walk in Random Environment  
(BRWRE)
on $N_0$ started with one particle at the origin. Particles reproduce  
according
to an offspring distribution (which depends on the location) and move  
either
one step to the right (with a probability in $(0,1]$ which may also  
depend on
the location) or stay in the same place. We give criteria for local  
and global
survival and show that global survival is equivalent to exponential  
growth of
the moments.


  http://arxiv.org/abs/0906.4033

---------------------------------------------------------------

8674. ONE-SIDED CAUCHY-STIELTJES KERNEL FAMILIES

Wlodzimierz Bryc and Abdelhamid Hassairi

This paper continues the study of a kernel family which uses the
Cauchy-Stieltjes kernel in place of the celebrated exponential kernel  
of the
exponential families theory. We extend the theory to cover generating  
measures
with support that is unbounded on one side. We illustrate the need for  
such an
extension by showing that cubic pseudo-variance functions correspond to
free-infinitely divisible laws without the first moment. We also  
determine the
domain of means, advancing the understanding of
   Cauchy-Stieltjes kernel families also for compactly supported  
generating
measures.


  http://arxiv.org/abs/0906.4073

---------------------------------------------------------------

8675. IMPROVEMENT OF TWO HUNGARIAN BIVARIATE THEOREMS

Nathalie Castelle (LM-Orsay)

We introduce a new technique to establish Hungarian multivariate  
theorems. In
this article we apply this technique to the strong approximation  
bivariate
theorems of the uniform empirical process. It improves the Komlos,  
Major and
Tusn\'ady (1975) result, as well as our own (1998). More precisely, we  
show
that the error in the approximation of the uniform bivariate $n$- 
empirical
process by a bivariate Brownian bridge is of order $n^{-1/2}(log  
(nab))^{3/2}$
on the rectangle $[0,a]x[0,b]$, $0 <a, b <1$, and that the error in the
approximation of the uniform univariate $n$-empirical process by a  
Kiefer
process is of order $n^{-1/2}(log (na))^{3/2}$ on the interval $[0,a] 
$, $0 < a
< 1$. In both cases, the global error bound is therefore of order  
$n^{-1/2}(log
(n))^{3/2}$. Previously, from the 1975 article of Komlos, Major and  
Tusn\'ady,
the global error bound was of order $n^{-1/2}(log (n))^{2}$, and from  
our 1998
article, the local error bounds were of order $n^{-1/2}(log  
(nab))^{2}$ or
$n^{-1/2}(log (na))^{2}$. We think that, in the d-variate case, the  
global
error bound between the uniform d-variate $n$-empirical process and the
associated Gaussian process is of order $n^{-1/2}(log (n))^{(d+1)/2}$,  
and that
this result is optimal. The new feature of this article is to identify
martingales in the error terms and to apply to them an exponential  
inequality.
The idea is to bound of the compensator of the error term, instead of  
bounding
of the error term itself.


  http://arxiv.org/abs/0906.3952

---------------------------------------------------------------

8676. THE SPECTRAL EDGE OF SOME RANDOM BAND MATRICES

Sasha Sodin

We study the asymptotic distribution of the eigenvalues of random  
Hermitian
periodic band matrices, focusing on the spectral edges. The  
eigenvalues close
to the edges converge in distribution to the Airy point process if  
(and only
if) the band is sufficiently wide (W >> N^{5/6}.) Otherwise, a different
limiting distribution appears.


  http://arxiv.org/abs/0906.4047

---------------------------------------------------------------

8677. A CENTRAL LIMIT THEOREM VIA DIFFERENTIAL EQUATIONS

Taral Guldahl Seierstad

In a paper from 1995, Wormald gave general criteria for certain  
parameters in
a family of discrete random processes to converge to the solution of a  
system
of differential equations. Based on this method, we show that if some  
further
conditions are satisfied, the parameters converge to a multivariate  
normal
distribution.


  http://arxiv.org/abs/0906.4202

---------------------------------------------------------------

8678. POISSON--VORONOI APPROXIMATION

Matthias Heveling and  Matthias Reitzner

Let $X$ be a Poisson point process and $K\subset\mathbb{R}^d$ a  
measurable
set. Construct the Voronoi cells of all points $x\in X$ with respect  
to $X$,
and denote by $v_X(K)$ the union of all Voronoi cells with nucleus in  
$K$. For
$K$ a compact convex set the expectation of the volume difference
$V(v_X(K))-V(K)$ and the symmetric difference $V(v_X(K)\triangle K)$ is
computed. Precise estimates for the variance of both quantities are  
obtained
which follow from a new jackknife inequality for the variance of  
functionals of
a Poisson point process. Concentration inequalities for both  
quantities are
proved using Azuma's inequality.


  http://arxiv.org/abs/0906.4238

---------------------------------------------------------------

8679. RATES OF CONVERGENCE OF SOME MULTIVARIATE MARKOV CHAINS WITH  
POLYNOMIAL  EIGENFUNCTIONS

Kshitij Khare and  Hua Zhou

We provide a sharp nonasymptotic analysis of the rates of convergence  
for
some standard multivariate Markov chains using spectral techniques.  
All chains
under consideration have multivariate orthogonal polynomial as  
eigenfunctions.
Our examples include the Moran model in population genetics and its  
variants in
community ecology, the Dirichlet-multinomial Gibbs sampler, a class of
generalized Bernoulli--Laplace processes, a generalized Ehrenfest urn  
model and
the multivariate normal autoregressive process.


  http://arxiv.org/abs/0906.4242

---------------------------------------------------------------

8680. TREE BASED FUNCTIONAL EXPANSIONS FOR FEYNMAN--KAC PARTICLE MODELS

Pierre Del Moral and  Fr\'ed\'eric Patras and  Sylvain Rubenthaler

We design exact polynomial expansions of a class of Feynman--Kac  
particle
distributions. These expansions are finite and are parametrized by  
coalescent
trees and other related combinatorial quantities. The accuracy of the
expansions at any order is related naturally to the number of  
coalescences of
the trees. Our results include an extension of the Wick product  
formula to
interacting particle systems. They also provide refined nonasymptotic
propagation of chaos-type properties, as well as sharp $\mathbb{L}_p$- 
mean
error bounds, and laws of large numbers for $U$-statistics.


  http://arxiv.org/abs/0906.4249

---------------------------------------------------------------

8681. ENERGY MEASURES AND INDICES OF DIRICHLET FORMS, WITH  
APPLICATIONS TO  DERIVATIVES ON SOME FRACTALS

Masanori Hino

We introduce the concept of index for regular Dirichlet forms by means  
of
energy measures, and discuss its properties. In particular, it is  
proved that
the index of strong local regular Dirichlet forms is identical with the
martingale dimension of the associated diffusion processes. As an  
application,
a class of self-similar fractals is taken up as an underlying space.  
We prove
that first-order derivatives can be defined for functions in the  
domain of the
Dirichlet forms and their total energies are represented as the square
integrals of the derivatives.


  http://arxiv.org/abs/0906.4251

---------------------------------------------------------------

8682. ON LARGE DEVIATION REGIMES FOR RANDOM MEDIA MODELS

M. Cranston and  D. Gauthier and  T. S. Mountford

The focus of this article is on the different behavior of large  
deviations of
random subadditive functionals above the mean versus large deviations  
below the
mean in two random media models. We consider the point-to-point first  
passage
percolation time $a_n$ on $\mathbb{Z}^d$ and a last passage  
percolation time
$Z_n$. For these functionals, we have $\lim_{n\to\infty}\frac{a_n}{n}= 
\nu$ and
$\lim_{n\to\infty}\frac{Z_n}{n}=\mu$. Typically, the large deviations  
for such
functionals exhibits a strong asymmetry, large deviations above the  
limiting
value are radically different from large deviations below this  
quantity. We
develop robust techniques to quantify and explain the differences.


  http://arxiv.org/abs/0906.4254

---------------------------------------------------------------

8683. ERGODICITY OF THE 3D STOCHASTIC NAVIER-STOKES EQUATIONS DRIVEN  
BY MILDLY  DEGENERATE NOISE

Lihu Xu and  Marco Romito

We prove that the any Markov solution to the 3D stochastic Navier-Stokes
equations driven by a mildly degenerate noise (i.e.all but finitely many
Fourier modes are forced) is uniquely ergodic. This follows by proving  
strong
Feller regularity and irreducibility.


  http://arxiv.org/abs/0906.4281

---------------------------------------------------------------

8684. PARAMETER ESTIMATION IN DIAGONALIZABLE STOCHASTIC HYPERBOLIC  
EQUATIONS

W. Liu and S. V. Lototsky

A parameter estimation problem is considered for a linear stochastic
hyperbolic equation driven by additive space-time Gaussian white  
noise. The
damping/amplification operator is allowed to be unbounded.
   The estimator is of spectral type and utilizes a finite number of  
the spatial
Fourier coefficients of the solution. The asymptotic properties of the
estimator are studied as the number of the Fourier coefficients  
increases,
while the observation time and the noise intensity are fixed.


  http://arxiv.org/abs/0906.4353

---------------------------------------------------------------

8685. BULK UNIVERSALITY FOR WIGNER HERMITIAN MATRICES WITH  
SUBEXPONENTIAL  DECAY

Laszlo Erdos and  Jose Ramirez and  Benjamin Schlein and  Terence Tao  
and  Van Vu and   Horng-Tzer Yau

We consider the ensemble of $n \times n$ Wigner hermitian matrices
   $H = (h_{\ell k})_{1 \leq \ell,k \leq n}$ that generalize the  
Gaussian
unitary ensemble (GUE). The matrix elements $h_{k\ell} = \bar h_{\ell  
k}$ are
given by $h_{\ell k} = n^{-1/2} (x_{\ell k} + \sqrt{-1} y_{\ell k})$,  
where
$x_{\ell k}, y_{\ell k}$ for $1 \leq \ell < k \leq n$ are i.i.d. random
variables with mean zero and variance 1/2, $y_{\ell\ell}=0$ and  
$x_{\ell \ell}$
have mean zero and variance 1. We assume the distribution of $x_{\ell  
k},
y_{\ell k}$ to have subexponential decay. In a recent paper, four of the
authors recently established that the gap distribution and averaged $k 
$-point
correlation of these matrices were \emph{universal} (and in  
particular, agreed
with those for GUE) assuming additional regularity hypotheses on the  
$x_{\ell
k}, y_{\ell k}$. In another recent paper, the other two authors, using a
different method, established the same conclusion assuming instead  
some moment
and support conditions on the $x_{\ell k}, y_{\ell k}$. In this short  
note we
observe that the arguments of these two papers can be combined to  
establish
universality of the gap distribution and averaged $k$-point  
correlations for
all Wigner matrices (with subexponentially decaying entries), with no  
extra
assumptions.


  http://arxiv.org/abs/0906.4400

---------------------------------------------------------------

8686. STEIN'S METHOD MEETS MALLIAVIN CALCULUS: A SHORT SURVEY WITH  
NEW  ESTIMATES

Ivan Nourdin (PMA) and  Giovanni Peccati (MODAL'X)

We provide an overview of some recent techniques involving the Malliavin
calculus of variations and the so-called ``Stein's method'' for the  
Gaussian
approximations of probability distributions. Special attention is  
devoted to
establishing explicit connections with the classic method of moments: in
particular, we use interpolation techniques in order to deduce some new
estimates for the moments of random variables belonging to a fixed  
Wiener
chaos. As an illustration, a class of central limit theorems  
associated with
the quadratic variation of a fractional Brownian motion is studied in  
detail.


  http://arxiv.org/abs/0906.4419

---------------------------------------------------------------

8687. MOST LIKELY PATHS TO ERROR WHEN ESTIMATING THE MEAN OF A  
REFLECTED  RANDOM WALK

Ken R. Duffy and Sean P. Meyn

It is known that simulation of the mean position of a reflected random  
walk
$\{W_n\}$ exhibits non-standard behavior, even for light-tailed  
increment
distributions with negative drift. The Large Deviation Principle (LDP)  
holds
for deviations below the mean, but for deviations at the usual speed  
above the
mean the rate function is null. This paper takes a deeper look at this
phenomenon. Conditional on a large sample mean, a complete sample path  
LDP
analysis is obtained. Let $I$ denote the rate function for the one  
dimensional
increment process. If $I$ is coercive, then given a large simulated mean
position, under general conditions our results imply that the most  
likely
asymptotic behavior, $\psi$, of the paths $n^{-1} W_{\lfloor tn\rfloor} 
$ is to
be zero apart from on an interval $[T_0,T_1]\subset[0,1]$ and to  
satisfy the
functional equation \nabla I(\ddt\psi(t))=\lambda^*(T_1-t) \quad
\text{whenever} \psi(t)\neq 0. If $I$ is non-coercive, a similar, but  
slightly
more involved, result holds.


  http://arxiv.org/abs/0906.4514

---------------------------------------------------------------

8688. ON THE TIME SCHEDULE OF BROWNIAN FLIGHTS

Athanasios Batakis (MAPMO) and  Michel Zinsmeister (MAPMO)

We are interested on the statistics of the duration of Brownian  
diffusions
started at distance \epsilon from a given boundary and stopped when  
they hit
back the interface.


  http://arxiv.org/abs/0906.4537

---------------------------------------------------------------

8689. COVARIANCE FUNCTION OF VECTOR SELF-SIMILAR PROCESS

Fr\'ed\'eric Lavancier (LMJL) and  Anne Philippe (LMJL) and  Donatas  
Surgailis

The paper obtains the general form of the cross-covariance function of  
vector
fractional Brownian motion with correlated components having different
self-similarity indices.


  http://arxiv.org/abs/0906.4541

---------------------------------------------------------------

8690. SUMSET AND INVERSE SUMSET THEOREMS FOR SHANNON ENTROPY

Terence Tao

Let $G = (G,+)$ be an additive group. The sumset theory of Pl\"unnecke  
and
Ruzsa gives several relations between the size of sumsets $A+B$ of  
finite sets
$A, B$, and related objects such as iterated sumsets $kA$ and  
difference sets
$A-B$, while the inverse sumset theory of Freiman, Ruzsa, and others
characterises those finite sets $A$ for which $A+A$ is small. In this  
paper we
establish analogous results in which the finite set $A \subset G$ is  
replaced
by a discrete random variable $X$ taking values in $G$, and the  
cardinality
$|A|$ is replaced by the Shannon entropy $\Ent(X)$. In particular, we  
classify
the random variable $X$ which have small doubling in the sense that
$\Ent(X_1+X_2) = \Ent(X)+O(1)$ when $X_1,X_2$ are independent copies  
of $X$, by
showing that they factorise as $X = U+Z$ where $U$ is uniformly  
distributed on
a coset progression of bounded rank, and $\Ent(Z) = O(1)$.
   When $G$ is torsion-free, we also establish the sharp lower bound $ 
\Ent(X+X)
\geq \Ent(X) + {1/2} \log 2 - o(1)$, where $o(1)$ goes to zero as $ 
\Ent(X) \to
\infty$.


  http://arxiv.org/abs/0906.4387

---------------------------------------------------------------

8691. EXCURSIONS OF DIFFUSION PROCESSES AND CONTINUED FRACTIONS

Alain Comtet and Yves Tourigny

It is well-known that the excursions of a one-dimensional diffusion  
process
can be studied by considering a certain Riccati equation associated  
with the
process. We show that, in many cases of interest, the Riccati equation  
can be
solved in terms of an infinite continued fraction. We examine the  
probabilistic
significance of the expansion. To illustrate our results, we discuss  
some
examples of diffusions in deterministic and in random environments.


  http://arxiv.org/abs/0906.4651

---------------------------------------------------------------

8692. THE INCLUSION PROCESS: DUALITY AND CORRELATION INEQUALITIES

C. Giardina and  F. Redig and  K. Vafayi

We prove a comparison inequality between a system of independent random
walkers and a system of random walkers which interact by attracting  
eachother
-a process which we call here the symmetric inclusion process (SIP).  
As an
application, correlation inequalities for the SIP, as well as for a  
model of
heat conduction, the so-called Brownian momentum process, are  
obtained. These
inequalities are counterparts of the inequalities (in the opposite  
direction)
for the symmetric exclusion process, confirming that the SIP is a  
natural
bosonic analogue of the symmetric exclusion process (which is  
fermionic). We
discuss stationary measures of the SIP, and an asymmetric version that  
has the
same stationary probability measures, as well as infinite non- 
translation
invariant reversible measures. Finally, we consider a boundary driven  
version
of the SIP for which we prove duality and correlation inequalities.


  http://arxiv.org/abs/0906.4664

---------------------------------------------------------------

8693. THE ORTHOGONAL WEINGARTEN FORMULA IN COMPACT FORM

T. Banica

We present a compact formulation of the orthogonal Weingarten formula,  
with
the traditional quantity
$I(i_1,...,i_{2k}:j_1,...,j_{2k})=\int_{O_n}u_{i_1j_1}...  
u_{i_{2k}j_{2k}} du$
replaced by the more advanced quantity $I(a)=\int_{O_n}\Pi  
u_{ij}^{a_{ij}} du$,
depending on a matrix of exponents $a\in M_n(\mathbb N)$. Among  
consequences,
we establish a number of basic facts regarding the integrals $I(a)$:  
vanishing
condition, sign, possible poles, asymptotic behavior.


  http://arxiv.org/abs/0906.4694

---------------------------------------------------------------

8694. A CLT FOR THE $L^{2}$ MODULI OF CONTINUITY OF LOCAL TIMES OF  
LEVY  PROCESSES

Michael B. Marcus and Jay Rosen

Let $X=\{X_{t},t\in R_{+}\}$ be a symmetric L\'evy process with local  
time
$\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$. When the L\'evy  
exponent
$\psi(\la)$ is regularly varying at infinity with index $1<\beta\leq  
2$ and
satisfies some additional regularity conditions && \sqrt{h\psi^{2}(1/ 
h)} \lc
\int (L^{x+h}_{1}- L^{x}_{1})^{2} dx- E(\int (L^{x+h}_{1}-  
L^{x}_{1})^{2}
dx)\rc\nn && {1 in} \stackrel{\mathcal{L}}{\Longrightarrow}
(8c_{\beta,1})^{1/2} \eta (\int (L_{1}^{x})^{2} dx)^{1/2} \nn, as $h 
\rar 0$,
where $\eta$ is a normal random variable with mean zero and variance  
one that
is independent of $L^{x}_{t}$, and $c_{\beta,1}$ is a known constant.


  http://arxiv.org/abs/0906.4770

---------------------------------------------------------------

8695. LEVY FLIGHTS IN CONFINING POTENTIALS

Piotr Garbaczewski and Vladimir Stephanovich

We analyze confining mechanisms for L\'{e}vy flights. When they evolve  
in
suitable external potentials their variance may exist and show  
signatures of a
superdiffusive transport. Two classes of stochastic jump - type  
processes are
considered: those driven by Langevin equation with L\'{e}vy noise and  
those,
named by us topological L\'{e}vy processes (occurring in systems with
topological complexity like folded polymers or complex networks and  
generically
in inhomogeneous media), whose Langevin representation is unknown and  
possibly
nonexistent. Our major finding is that both above classes of processes  
stay in
affinity and may share common stationary (eventually asymptotic)  
probability
density, even if their detailed dynamical behavior look different. That
generalizes and offers new solutions to a reverse engineering (e.g.  
targeted
stochasticity) problem due to I. Eliazar and J. Klafter [J. Stat.  
Phys. 111,
739, (2003)]: design a L\'{e}vy process whose target pdf equals a priori
preselected one. Our observations extend to a broad class of L\'{e}vy  
noise
driven processes, like e.g. superdiffusion on folded polymers,  
geophysical
flows and even climatic changes.


  http://arxiv.org/abs/0904.4157

---------------------------------------------------------------

8696. SURVIVAL AND COEXISTENCE FOR A MULTITYPE CONTACT PROCESS

J. Theodore Cox and  Rinaldo B. Schinazi

We study the ergodic theory of a multitype contact process with equal  
death
rates and unequal birth rates on the $d$-dimensional integer lattice and
regular trees. We prove that for birth rates in a certain interval  
there is
coexistence on the tree, which by a result of Neuhauser is not  
possible on the
lattice. We also prove a complete convergence result when the larger  
birth rate
falls outside of this interval.


  http://arxiv.org/abs/0906.4845

---------------------------------------------------------------

8697. A BOUNDED DERIVATION METHOD FOR THE MAXIMUM LIKELIHOOD  
ESTIMATION ON THE  PARAMETERS OF WEIBULL DISTRIBUTION

DeTao Mao and Wenyuan Li

For the basic maximum likelihood estimating function of the two  
parameters
Weibull distribution, a simple proof on its global monotonicity is  
given to
ensure the existence and uniqueness of its solution. The boundary of the
function's first-order derivation is defined based on its scale-free  
property.
With a bounded derivation, the possible range of the root of this  
function can
be determined. A novel root-finding algorithm employing these  
established
results is proposed accordingly, its convergence is proved  
analytically as
well. Compared with other typical algorithms for this problem, the  
efficiency
of the proposed algorithm is also demonstrated by numerical experiments.


  http://arxiv.org/abs/0906.4823

---------------------------------------------------------------

8698. A SERIES REPRESENTATION OF MULTISTABLE MULTIFRACTIONAL PROCESSES  
AND  OTHER LOCALISABLE PROCESSES

Ronan Le Gu\'evel (LMJL) and  Jacques L\'evy-V\'ehel (INRIA Saclay -  
Ile  de France)

The study of non-stationary processes whose local form has controlled
properties is a fruitful and important area of research, both in  
theory and
applications. We present here a construction of multifractional  
multistable
processes, based on a series representation of stable stochastic  
integrals. We
consider various particular cases of interest, including multistable L 
\'evy
motion, multistable reverse Ornstein-Uhlenbeck process, log-fractional
multistable motion and linear multistable multifractional motion. We  
also
compute the finite dimensional distributions of those processes.  
Finally, we
display numerical experiments showing graphs of synthesized paths of  
such
processes.


  http://arxiv.org/abs/0906.5042

---------------------------------------------------------------

8699. USING DIFFERENTIAL EQUATIONS TO OBTAIN JOINT MOMENTS OF FIRST- 
PASSAGE  TIMES OF INCREASING LEVY PROCESSES

Mark S. Veillette and  Murad S. Taqqu

Let $\{D(s), s \geq 0 \}$ be a L\'evy subordinator, that is, a non- 
decreasing
process with stationary and independent increments and suppose that  
$D(0) = 0$.
We study the first-hitting time of the process $D$, namely, the  
process $E(t) =
\inf \{s: D(s) > t \}$, $t \geq 0$.
   The process $E$ is, in general, non-Markovian with non-stationary and
non-independent increments. We derive a partial differential equation  
for the
Laplace transform of the $n$-time tail distribution function $P[E(t_1) >
s_1,...,E(t_n) > s_n]$, and show that this PDE has a unique solution  
given
natural boundary conditions. This PDE can be used to derive all $n$-time
moments of the process $E$.


  http://arxiv.org/abs/0906.5083

---------------------------------------------------------------

8700. STUDENTIZED PROCESSES OF U-STATISTICS

Masoud M. Nasari

A uniform in probability approximation is established for Studentized
processes of non degenerate U-statistics of order m greater or equal  
to 2 in
terms of a standard Wiener process. The classical condition that the  
second
moment of kernel of the underlying U-statistic exists is relaxed to  
having 5/3
moments. Furthermore, the conditional expectation of the kernel is  
only assumed
to be in the domain of attraction of the normal law (instead of the  
classical
two moment condition).


  http://arxiv.org/abs/0906.5101

---------------------------------------------------------------

8701. BOUNDS ON THE CONSTANT IN THE MEAN CENTRAL LIMIT THEOREM

Larry Goldstein

Let $X_1,...,X_n$ be independent with mean zero, finite variances
$\sigma_1^2,...,\sigma_n^2$ and finite absolute third moments, $F_n$ the
distribution function of $(X_1+...+X_n)/\sigma$ where $\sigma^2= 
\sum_{i=1}^n
\sigma_i^2$, and $\Phi$ that of the standard normal. Then the $L^1$  
distance
between $F_n$ and $\Phi$ satisfies $$ ||F_n-\Phi||_1 \le
\frac{1}{\sigma^3}\sum_{i=1}^n E|X_i|^3. $$ In particular, when  
$X_1,...,X_n$
are identically distributed with variance $\sigma^2$, $$ ||F_n-\Phi|| 
_1 \le
\frac{E|X_1|^3}{\sigma^3 \sqrt{n}} \quad {for all $n \in \mathbb{N}$,}  
$$
corresponding to an $L^1$ Berry Esseen constant of 1. A lower bound of  
$$
\frac{2 \sqrt{\pi} (2\Phi(1)-1) - (\sqrt{\pi}+\sqrt{2})+ 2
e^{-1/2}\sqrt{2}}{\sqrt{\pi}} =0.535377... $$ on the smallest possible  
constant
is provided.


  http://arxiv.org/abs/0906.5145

---------------------------------------------------------------

8702. STOCHASTIC HOMOGENIZATION OF HOROSPHERIC TREE PRODUCTS

Vadim A. Kaimanovich and  Florian Sobieczky

We construct measures invariant with respect to equivalence relations  
which
are graphed by horospheric products of trees. The construction is  
based on
using conformal systems of boundary measures on treed equivalence  
relations.
The existence of such an invariant measure allows us to establish  
amenability
of horospheric products of random trees.


  http://arxiv.org/abs/0906.5296

---------------------------------------------------------------

8703. GEOMETRIC ERGODICITY AND THE SPECTRAL GAP OF NON-REVERSIBLE  
MARKOV  CHAINS

Ioannis Kontoyiannis and  Sean P. Meyn

We argue that the spectral theory of non-reversible Markov chains may  
often
be more effectively cast within the framework of the naturally  
associated
weighted-$L_\infty$ space $L_\infty^V$, instead of the usual Hilbert  
space
$L_2=L_2(\pi)$, where $\pi$ is the invariant measure of the chain. This
observation is, in part, based on the following results. A discrete- 
time Markov
chain with values in a general state space is geometrically ergodic if  
and only
if its transition kernel admits a spectral gap in $L_\infty^V$. If the  
chain is
reversible, the same equivalence holds with $L_2$ in place of $L_ 
\infty^V$, but
in the absence of reversibility it fails: There are (necessarily
non-reversible, geometrically ergodic) chains that admit a spectral  
gap in
$L_\infty^V$ but not in $L_2$. Moreover, if a chain admits a spectral  
gap in
$L_2$, then for any $h\in L_2$ there exists a Lyapunov function $V_h 
\in L_1$
such that $V_h$ dominates $h$ and the chain admits a spectral gap in
$L_\infty^{V_h}$. The relationship between the size of the spectral  
gap in
$L_\infty^V$ or $L_2$, and the rate at which the chain converges to  
equilibrium
is also briefly discussed.


  http://arxiv.org/abs/0906.5322

---------------------------------------------------------------

8704. TESTING FOR WHITE NOISE UNDER UNKNOWN DEPENDENCE AND ITS  
APPLICATIONS TO  GOODNESS-OF-FIT FOR TIME SERIES MODELS

Xiaofeng Shao

Testing for white noise has been well studied in the literature of
econometrics and statistics. For most of the proposed test statistics,  
such as
the well-known Box-Pierce's test statistic with fixed lag truncation  
number,
the asymptotic null distributions are obtained under independent and
identically distributed assumptions and may not be valid for the  
dependent
white noise. Due to recent popularity of conditional heteroscedastic  
models
(e.g., GARCH models), which imply nonlinear dependence with zero
autocorrelation, there is a need to understand the asymptotic  
properties of the
existing test statistics under unknown dependence. In this paper, we  
showed
that the asymptotic null distribution of Box-Pierce's test statistic  
with
general weights still holds under unknown weak dependence so long as  
the lag
truncation number grows at an appropriate rate with increasing sample  
size.
Further applications to diagnostic checking of the ARMA and FARIMA  
models with
dependent white noise errors are also addressed. Our results go beyond  
earlier
ones by allowing non-Gaussian and conditional heteroscedastic errors  
in the
ARMA and FARIMA models and provide theoretical support for some  
empirical
findings reported in the literature.


  http://arxiv.org/abs/0906.5179

---------------------------------------------------------------

8705. DISTANCE STATISTICS IN QUADRANGULATIONS WITH A BOUNDARY, OR WITH  
A  SELF-AVOIDING LOOP

J. Bouttier and E. Guitter

We consider quadrangulations with a boundary and derive explicit  
expressions
for the generating functions of these maps with either a marked vertex  
at a
prescribed distance from the boundary, or two boundary vertices at a  
prescribed
mutual distance in the map. For large maps, this yields explicit  
formulas for
the bulk-boundary and boundary-boundary correlators in the various  
encountered
scaling regimes: a small boundary, a dense boundary and a critical  
boundary
regime. The critical boundary regime is characterized by a one- 
parameter family
of scaling functions interpolating between the Brownian map and the  
Brownian
Continuum Random Tree. We discuss the cases of both generic and self- 
avoiding
boundaries, which are shown to share the same universal scaling limit.  
We
finally address the question of the bulk-loop distance statistics in the
context of planar quadrangulations equipped with a self-avoiding loop.  
Here
again, a new family of scaling functions describing critical loops is
discovered.


  http://arxiv.org/abs/0906.4892

---------------------------------------------------------------

8706. SPREAD OF MISINFORMATION IN SOCIAL NETWORKS

Daron Acemoglu and  Asuman Ozdaglar and  Ali ParandehGheibi

We provide a model to investigate the tension between information  
aggregation
and spread of misinformation in large societies (conceptualized as  
networks of
agents communicating with each other). Each individual holds a belief
represented by a scalar. Individuals meet pairwise and exchange  
information,
which is modeled as both individuals adopting the average of their pre- 
meeting
beliefs. When all individuals engage in this type of information  
exchange, the
society will be able to effectively aggregate the initial information  
held by
all individuals. There is also the possibility of misinformation,  
however,
because some of the individuals are "forceful," meaning that they  
influence the
beliefs of (some) of the other individuals they meet, but do not  
change their
own opinion. The paper characterizes how the presence of forceful agents
interferes with information aggregation. Under the assumption that even
forceful agents obtain some information (however infrequent) from some  
others
(and additional weak regularity conditions), we first show that  
beliefs in this
class of societies converge to a consensus among all individuals. This
consensus value is a random variable, however, and we characterize its
behavior. Our main results quantify the extent of misinformation in  
the society
by either providing bounds or exact results (in some special cases) on  
how far
the consensus value can be from the benchmark without forceful agents  
(where
there is efficient information aggregation). The worst outcomes obtain  
when
there are several forceful agents and forceful agents themselves  
update their
beliefs only on the basis of information they obtain from individuals  
most
likely to have received their own information previously.


  http://arxiv.org/abs/0906.5007

---------------------------------------------------------------

8707. DERIVATION OF AN EIGENVALUE PROBABILITY DENSITY FUNCTION  
RELATING TO THE  POINCARE DISK

Peter J. Forrester and Manjunath Krishnapur

A result of Zyczkowski and Sommers [J.Phys.A, 33, 2045--2057 (2000)]  
gives
the eigenvalue probability density function for the top N x N sub- 
block of a
Haar distributed matrix from U(N+n). In the case n \ge N, we rederive  
this
result, starting from knowledge of the distribution of the sub-blocks,
introducing the Schur decomposition, and integrating over all  
variables except
the eigenvalues. The integration is done by identifying a recursive  
structure
which reduces the dimension. This approach is inspired by an analogous  
approach
which has been recently applied to determine the eigenvalue  
probability density
function for random matrices A^{-1} B, where A and B are random  
matrices with
entries standard complex normals. We relate the eigenvalue  
distribution of the
sub-blocks to a many body quantum state, and to the one-component  
plasma, on
the pseudosphere.


  http://arxiv.org/abs/0906.5223

---------------------------------------------------------------

8708. STOCHASTIC CALCULUS FOR A TIME-CHANGED SEMIMARTINGALE AND THE  
ASSOCIATED  STOCHASTIC DIFFERENTIAL EQUATIONS

Kei Kobayashi

It is shown that under a certain condition on a semimartingale and a
time-change, any stochastic integral driven by the time-changed  
semimartingale
is a time-changed stochastic integral driven by the original  
semimartingale. As
a direct consequence, a specialized form of the Ito formula is  
derived. When a
standard Brownian motion is the original semimartingale, classical Ito
stochastic differential equations driven by the Brownian motion with  
drift
extend to a larger class of stochastic differential equations  
involving a
time-change with continuous paths. A form of the general solution of  
linear
equations in this new class is established, followed by consideration  
of some
examples analogous to the classical equations. Through these examples,  
each
coefficient of the stochastic differential equations in the new class  
is given
meaning. The new feature is the coexistence of a usual drift term  
along with a
term related to the time-change.


  http://arxiv.org/abs/0906.5385

---------------------------------------------------------------

8709. LONGEST CONVEX CHAINS

Gergely Ambrus and  Imre Barany

Assume $X_n$ is a random sample of $n$ uniform, independent points  
from a
triangle $T$. The longest convex chain, $Y$, of $X_n$ is defined  
naturally. The
length $|Y|$ of $Y$ is a random variable, denoted by $L_n$. In this  
article, we
determine the order of magnitude of the expectation of $L_n$. We show  
further
that $L_n$ is highly concentrated around its mean, and that the  
longest convex
chains have a limit shape.


  http://arxiv.org/abs/0906.5452

---------------------------------------------------------------

8710. ORTHOGONAL SERIES AND LIMIT THEOREMS FOR CANONICAL U- AND V- 
STATISTICS  OF STATIONARY CONNECTED OBSERVATIONS

I.S.Borisov and  N.Volodko

The limit behavior is studied for the distributions of normalized U- and
V-statistics of an arbitrary order with canonical (degenerate)  
kernels, based
on samples of increasing sizes from a stationary sequence of  
observations
satisfying classical mixing conditions. The corresponding limit  
distributions
are represented as infinite multilinear forms of a centered Gaussian  
sequence
with a known covariance matrix.


  http://arxiv.org/abs/0906.5465

---------------------------------------------------------------

8711. POISSON BOUNDARY OF $GL_D(\Q)$

Sara Brofferio (LM-Orsay) and  Bruno Schapira (LM-Orsay)

We construct the Poisson boundary for a random walk supported by the  
general
linear group on the rational numbers as the product of flag manifolds  
over the
$p$-adic fields. To this purpose, we prove a law of large numbers  
using the
Oseledets' multiplicative ergodic theorem.


  http://arxiv.org/abs/0906.5548

---------------------------------------------------------------

8712. ON BUFFON MACHINES AND NUMBERS

Philippe Flajolet and  Maryse Pelletier and  Michele Soria

Buffon's needle experiment is well-known: take a plane on which parallel
lines at unit distance one from the next have been marked, throw a  
needle of
unit length at random, and, finally, declare the experiment a success  
if the
needle intersects one of the lines. Basic calculus implies that the  
probability
of success is 2/pi~0.63661, and the experiment can be regarded as an  
analog
(i.e., continuous) device that stochastically "computes'' 2/pi.  
Generalizing
the experiment and simplifying the computational framework, we ask  
ourselves
which probability distributions can be produced perfectly, from a  
discrete
source of unbiased coin flips. We describe and analyse a few simple  
Buffon
machines that can generate geometric, Poisson, and logarithmic-series
distributions (these are in particular required to transform continuous
Boltzmann samplers of classical combinatorial structures into purely  
discrete
random generators). Say that a number is Buffon if it is the  
probability of
success of a probabilistic experiment based on discrete coin  
flippings. We
provide human-accessible Buffon machines, which require a dozen coin  
flips or
less, on average, and produce experiments whose probabilities are  
expressible
in terms of numbers such as pi, exp(-1), log2, sqrt(3), cos(1/4),  
zeta(5). More
generally, we develop a collection of constructions based on simple
probabilistic mechanisms that enable one to create Buffon experiments  
involving
compositions of exponentials and logarithms, polylogarithms, direct  
and inverse
trigonometric functions, algebraic and hypergeometric functions, as  
well as
functions defined by integrals, such as the Gaussian error function.


  http://arxiv.org/abs/0906.5560

---------------------------------------------------------------

8713. STRONG TAYLOR APPROXIMATION OF STOCHASTIC DIFFERENTIAL EQUATIONS  
AND  APPLICATION TO THE L\'EVY LIBOR MODEL

Antonis Papapantoleon and  Maria Siopacha

In this article we consider the strong approximation of stochastic
differential equations driven by L\'evy processes or general  
semimartingales.
The main ingredients of our method is the perturbation of the SDE and  
the
Taylor expansion of the resulting parameterized curve. We apply this  
method to
develop strong approximation schemes for LIBOR market models. In  
particular, we
derive fast and precise algorithms for the valuation of derivatives in  
LIBOR
models which are more tractable than the simulation of the full SDE. A
numerical example for the L\'evy LIBOR model illustrates our method.


  http://arxiv.org/abs/0906.5581

---------------------------------------------------------------

8714. RELATIVE DENSITY OF THE RANDOM R-FACTOR PROXIMITY CATCH DIGRAPH  
FOR  TESTING SPATIAL PATTERNS OF SEGREGATION AND ASSOCIATION  
(TECHNICAL REPORT)

Elvan Ceyhan and  Carey E. Priebe and  John C. Wierman

Statistical pattern classification methods based on data-random graphs  
were
introduced recently. In this approach, a random directed graph is  
constructed
from the data using the relative positions of the data points from  
various
classes. Different random graphs result from different definitions of  
the
proximity region associated with each data point and different graph  
statistics
can be employed for data reduction. The approach used in this article  
is based
on a parameterized family of proximity maps determining an associated  
family of
data-random digraphs. The relative arc density of the digraph is used  
as the
summary statistic, providing an alternative to the domination number  
employed
previously. An important advantage of the relative arc density is that,
properly re-scaled, it is a U-statistic, facilitating analytic study  
of its
asymptotic distribution using standard U-statistic central limit  
theory. The
approach is illustrated with an application to the testing of spatial  
patterns
of segregation and association. Knowledge of the asymptotic  
distribution allows
evaluation of the Pitman and Hodges-Lehmann asymptotic efficacy, and  
selection
of the proximity map parameter to optimize efficacy. Notice that the  
approach
presented here also has the advantage of validity for data in any  
dimension.


  http://arxiv.org/abs/0906.5436

---------------------------------------------------------------

8715. MARTINGALE DIFFERENCES AND THE METRIC THEORY OF CONTINUED  
FRACTIONS

Alan K. Haynes and Jeffrey D. Vaaler

We investigate a collection of orthonormal functions that encodes  
information
about the continued fraction expansion of real numbers. When suitably  
ordered
these functions form a complete system of martingale differences and  
are a
special case of a class of martingale differences considered by R. F.  
Gundy. By
applying known results for martingales we obtain corresponding metric  
theorems
for the continued fraction expansion of almost all real numbers.


  http://arxiv.org/abs/0906.5428

---------------------------------------------------------------

8716. RANDOM $K$-NONCROSSING RNA STRUCTURES

William Y.C. Chen and  Hillary S.W. Han and Christian M. Reidys

In this paper we derive polynomial time algorithms that generate random
$k$-noncrossing matchings and $k$-noncrossing RNA structures with  
uniform
probability. Our approach employs the bijection between $k$-noncrossing
matchings and oscillating tableaux and the $P$-recursiveness of the
cardinalities of $k$-noncrossing matchings. The main idea is to  
consider the
tableaux sequences as paths of stochastic processes over shapes and to  
derive
their transition probabilities.


  http://arxiv.org/abs/0906.5553

---------------------------------------------------------------

8717. AN EXPERIMENTAL MATHEMATICS PERSPECTIVE ON THE OLD, AND STILL  
OPEN,  QUESTION OF WHEN TO STOP?

Luis A. Medina and Doron Zeilberger

In a recent article in American Scientist, Theodore Hill described a
coin-tossing game whose pay-off is the number of heads over the total  
number of
throws. Suppose that at a given point during the game you have 5 heads  
and 3
tails, should you stop and get 5/8, or should you keep playing, hoping  
to get a
better score? This is still an open problem. In the present article,  
we explore
different strategies to this game from the Experimental Mathematics
perspective.


  http://arxiv.org/abs/0907.0032

---------------------------------------------------------------

8718. EXPONENTIAL INEQUALITIES FOR THE DISTRIBUTIONS OF CANONICAL U-  
AND  V-STATISTICS OF DEPENDENT OBSERVATIONS

I.S.Borisov and  N.Volodko

The exponential inequalities are obtained for the distribution tails of
canonical (degenerate) U- and V-statistics of an arbitrary order based  
on
samples from uniformly strong mixing stationary sequences.


  http://arxiv.org/abs/0907.0058

---------------------------------------------------------------

8719. CONTINUITY ON STOCHASTIC CONTROL PROBLEM WITH STOPPING TIME

Qingshuo Song and  Jie Yang

Stochastic control problem with stopping time in finite time horizon is
considered. In the general framework, the value function is  
characterized as
the unique viscosity solution of Bellman equation on a bounded domain.  
It
requires the continuity of the value function on its bounded domain.  
However,
the necessary and sufficient condition on continuity still remains  
unclear. In
this paper, compared to the existing literature, a more general  
sufficient
condition for the continuity of the value function is achieved by  
studying
pathwise local behavior of underlying stochastic processes on the  
boundary of
the domain. In addition, a simplified proof for the existence of the  
value
function is provided by applying the dynamic programming principle to a
naturally chosen stopping time.


  http://arxiv.org/abs/0907.0062

---------------------------------------------------------------

8720. STABILITY FOR RANDOM MEASURES, POINT PROCESSES AND DISCRETE  
SEMIGROUPS

Youri Davydov and  Ilya Molchanov and  Sergei Zuyev

A scaling operation on non-negative integers can be defined in a  
randomised
way by transforming an integer into the corresponding binomial  
distribution
with success probability being the scaling factor. We explore a similar
(thinning) operation defined on counting measures and characterise the
corresponding discrete stablility property of point processes. It is  
shown that
these processes are exactly Cox (doubly stochastic Poisson) processes  
with
strictly stable random intensity measures. The paper contains spectral  
and
LePage representations for strictly stable measures and characterises  
some
special cases, e.g. independently scattered measures. As consequence,  
spectral
representations are provided for the probability generating functional  
and void
probabilities of discrete stable processes. An alternative cluster
representation for discrete stable processes is also derived using the
so-called Sibuya point processes that constitute a new family of  
purely random
point processes. The obtained results are then applied to explore  
stable random
elements in discrete semigroups, where the scaling is defined by means  
of
thinning of a point process on the basis of the semigroup. Particular  
examples
include discrete stable vectors that generalise the one-dimensional  
case of
discrete random variables studied by Steutel and van Harn (1979) and  
the family
of natural numbers with the multiplication operation, where the primes  
form the
basis.


  http://arxiv.org/abs/0907.0077

---------------------------------------------------------------

8721. A PURE JUMP MARKOV PROCESS WITH A RANDOM SINGULARITY SPECTRUM

Julien Barral and  Nicolas Fournier and  Stephane Jaffard and   
Stephane Seuret

We construct a non-decreasing pure jump Markov process, whose jump  
measure
heavily depends on the values taken by the process. We determine the
singularity spectrum of this process, which turns out to be random and  
to
depend locally on the values taken by the process. The result relies  
on fine
properties of the distribution of Poisson point processes and on  
ubiquity
theorems.


  http://arxiv.org/abs/0907.0104

---------------------------------------------------------------

8722. COHERENT FREQUENTISM

David R. Bickel

The certainty distribution, a fiducial-like distribution of a scalar  
interest
parameter, combines the logical consistency of Bayesian methods with the
reliability of Neyman-Pearson methods. As a probability distribution  
over
parameter space, the certainty distribution is coherent in the sense  
that it
satisfies the axioms of the decision-theoretic and logic-theoretic  
systems
typically cited in support of the Bayesian posterior distribution.  
Since the
probabilities of a certainty distribution by definition are equal to the
coverage rates of the corresponding confidence intervals, the resulting
inferences are uniquely minimax to risk in a betting game designed to  
quantify
inferential reliability.


  http://arxiv.org/abs/0907.0139

---------------------------------------------------------------

8723. SETS OF FINITE PERIMETER AND THE HAUSDORFF-GAUSS MEASURE ON THE  
WIENER  SPACE

Masanori Hino

In Euclidean space, the integration by parts formula for a set of finite
perimeter is expressed by the integration with respect to a type of  
surface
measure. According to geometric measure theory, this surface measure is
realized by the one-codimensional Hausdorff measure restricted on the  
reduced
boundary and/or the measure-theoretic boundary, which may be strictly  
smaller
than the topological boundary. In this paper, we discuss the  
counterpart of
this measure in the abstract Wiener space, which is a typical
infinite-dimensional space. We introduce the concept of the measure- 
theoretic
boundary in the Wiener space and provide the integration by parts  
formula for
sets of finite perimeter. The formula is presented in terms of the  
integration
with respect to the one-codimensional Hausdorff-Gauss measure  
restricted on the
measure-theoretic boundary.


  http://arxiv.org/abs/0907.0056

---------------------------------------------------------------

8724. THE METRIC THEORY OF P-ADIC APPROXIMATION

Alan K. Haynes

Metric Diophantine approximation in its classical form is the study of  
how
well almost all real numbers can be approximated by rationals. There  
is a long
history of results which give partial answers to this problem, but  
there are
still questions which remain unknown. The Duffin-Schaeffer Conjecture  
is an
attempt to answer all of these questions in full, and it has withstood  
more
than fifty years of mathematical investigation. In this paper we  
establish a
strong connection between the Duffin-Schaeffer Conjecture and its p-adic
analogue. Our main theorems are transfer principles which allow us to  
go back
and forth between these two problems. We prove that if the variance  
method from
probability theory can be used to solve the p-adic Duffin-Schaeffer  
Conjecture
for even one prime p, then almost the entire classical Duffin-Schaeffer
Conjecture would follow. Conversely if the variance method can be used  
to prove
the classical conjecture then the p-adic conjecture is true for all  
primes.
Furthermore we are able to unconditionally and completely establish  
the higher
dimensional analogue of this conjecture in which we allow simultaneous
approximation in any finite number and combination of real and p-adic  
fields,
as long as the total number of fields involved is greater than one.  
Finally by
using a mass transference principle for Hausdorff measures we are able  
to
extend all of our results to their corresponding analogues with Haar  
measures
replaced by the Hausdorff measures associated with arbitrary dimension
functions.


  http://arxiv.org/abs/0907.0141



-----------------------------------
Stefano M. Iacus
Department of Economics,
Business and Statistics
University of Milan
Via Conservatorio, 7
I-20123 Milan - Italy
Ph.: +39 02 50321 461
Fax: +39 02 50321 505
http://www.economia.unimi.it/iacus
------------------------------------------------------------------------------------
Please don't send me Word or PowerPoint attachments if not
absolutely necessary. See:
http://www.gnu.org/philosophy/no-word-attachments.html



From pas at lists.imstat.org  Tue Sep  1 02:20:45 2009
From: pas at lists.imstat.org (Probability Abstract Service)
Date: Tue, 1 Sep 2009 09:20:45 +0200
Subject: [PAS] Probability Abstracts 111
Message-ID: <7E0E3207-F5AD-4033-98FF-1A5768698223@unimi.it>

Probability Abstracts 111
This document contains abstracts 8725-9028
from July-1-2009 to August-31-2009.
They have been mailed on Sep 1st, 2009.

This letter can be also found on line at
http://pas.imstat.org/Letters/letter_111.shtml


---------------------------------------------------------------

8725. MULTIVARIATE LOG-CONCAVE DISTRIBUTIONS AS A NEARLY PARAMETRIC  
MODEL

Dominic Schuhmacher and  Andre Huesler and  Lutz Duembgen

In this paper we show that the family P_d of probability distributions  
on R^d
with log-concave densities satisfies a strong continuity condition. In
particular, it turns out that weak convergence within this family  
entails (i)
convergence in total variation distance, (ii) convergence of arbitrary  
moments,
and (iii) pointwise convergence of Laplace transforms. Hence the  
nonparametric
model P_d has similar properties as parametric models such as, for  
instance,
the family of all d-variate Gaussian distributions.


  http://arxiv.org/abs/0907.0250

---------------------------------------------------------------

8726. SDES DRIVEN BY A TIME-CHANGED L\'EVY PROCESS AND THEIR  
ASSOCIATED  TIME-FRACTIONAL ORDER PSEUDO-DIFFERENTIAL EQUATIONS

Marjorie G. Hahn and  Kei Kobayashi and Sabir Umarov

It is known that if a stochastic process is a solution to a classical  
Ito
stochastic differential equation (SDE), then its transition  
probabilities
satisfy in the weak sense the associated Cauchy problem for the forward
Kolmogorov equation. The forward Kolmogorov equation is a parabolic  
partial
differential equation with coefficients determined by the  
corresponding SDE.
Stochastic processes which are scaling limits of continuous time  
random walks
have been connected with time-fractional differential equations.  
However, the
class of SDEs that is associated with time-fractional Kolmogorov type  
equations
is unknown. The present paper shows that in the cases of either time- 
fractional
order or more general time-distributed order differential equations, the
associated class of SDEs can be described within the framework of SDEs  
driven
by semimartingales. These semimartingales are time-changed Levy  
processes where
the independent time-change is given respectively by the inverse of a  
stable
subordinator or the inverse of a mixture of independent stable  
subordinators.


  http://arxiv.org/abs/0907.0253

---------------------------------------------------------------

8727. BROWNIAN AND FRACTIONAL BROWNIAN STOCHASTIC CURRENTS VIA  
MALLIAVIN  CALCULUS

Franco Flandoli and  Ciprian Tudor (CES and  SAMOS)

By using Malliavin calculus and multiple Wiener-It\^o integrals, we  
study the
existence and the regularity of stochastic currents defined as Skorohod
(divergence) integrals with respect to the Brownian motion and to the
fractional Brownian motion. We consider also the multidimensional
multiparameter case and we compare the regularity of the current as a
distribution in negative Sobolev spaces with its regularity in  
Watanabe space.


  http://arxiv.org/abs/0907.0292

---------------------------------------------------------------

8728. A MIN-TYPE STOCHASTIC FIXED-POINT EQUATION RELATED TO THE  
SMOOTHING  TRANSFORMATION

Gerold Alsmeyer and Matthias Meiners

This paper is devoted to the study of the stochastic fixed-point  
equation X
\stackrel{d}{=} \inf_{i \geq 1: T_i > 0} X_i/T_i and the connection  
with its
additive counterpart $X \stackrel{d}{=} \sum_{i\ge 1}T_{i}X_{i}$  
associated
with the smoothing transformation. Here $\stackrel{d}{=}$ means  
equality in
distribution, $T := (T_i)_{i \geq 1}$ is a given sequence of  
nonnegative random
variables and $X, X_1, ...$ is a sequence of nonnegative i.i.d. random
variables independent of $T$. We draw attention to the question of the
existence of nontrivial solutions and, in particular, of special  
solutions
named $\alpha$-regular solutions $(\alpha>0)$. We give a complete  
answer to the
question of when $\alpha$-regular solutions exist and prove that they  
are
always mixtures of Weibull distributions or certain periodic variants.  
We also
give a complete characterization of all fixed points of this kind. A
disintegration method which leads to the study of certain multiplicative
martingales and a pathwise renewal equation after a suitable transform  
are the
key tools for our analysis. Finally, we provide corresponding results  
for the
fixed points of the related additive equation mentioned above. To some  
extent,
these results have been obtained earlier by Iksanov.


  http://arxiv.org/abs/0907.0300

---------------------------------------------------------------

8729. INTERLACEMENT PERCOLATION ON TRANSIENT WEIGHTED GRAPHS

Augusto Teixeira

In this article, we first extend the construction of random  
interlacements,
introduced by A.S. Sznitman in [arXiv:0704.2560], to the more general  
setting
of transient weighted graphs. We prove the Harris-FKG inequality for  
this model
and analyze some of its properties on specific classes of graphs. For  
the case
of non-amenable graphs, we prove that the critical value u_* for the
percolation of the vacant set is finite. We also prove that, once G  
satisfies
the isoperimetric inequality IS_6 (see (1.5)), u_* is positive for the  
product
GxZ (where we endow Z with unit weights). When the graph under  
consideration is
a tree, we are able to characterize the vacant cluster containing some  
fixed
point in terms of a Bernoulli independent percolation process. For the  
specific
case of regular trees, we obtain an explicit formula for the critical  
value
u_*.


  http://arxiv.org/abs/0907.0316

---------------------------------------------------------------

8730. A FUNCTIONAL COMBINATORIAL CENTRAL LIMIT THEOREM

A. D. Barbour and Svante Janson

The paper establishes a functional version of the Hoeffding  
combinatorial
central limit theorem. First, a pre-limiting Gaussian process  
approximation is
defined, and is shown to be at a distance of the order of the  
Lyapounov ratio
from the original random process. Distance is measured by comparison of
expectations of smooth functionals of the processes, and the argument  
is by way
of Stein's method. The pre-limiting process is then shown, under weak
conditions, to converge to a Gaussian limit process. The theorem is  
used to
describe the shape of random permutation tableaux.


  http://arxiv.org/abs/0907.0347

---------------------------------------------------------------

8731. STABILITY PROPERTIES OF LINEAR FILE-SHARING NETWORKS

L. Leskel\"a (MSA) and  Philippe Robert (INRIA) and  Florian Simatos

File-sharing networks are distributed systems used to disseminate  
files among
a subset of the nodes of the Internet. A file is split into several  
pieces
called chunks, the general simple principle is that once a node of the  
system
has retrieved a chunk, it may become a server for this chunk. A  
stochastic
model is considered for arrival times and durations of time to  
download chunks.
One investigates the maximal arrival rate that such a network can  
accommodate,
i.e., the conditions under which the Markov process describing this  
network is
ergodic. Technical estimates related to the survival of interacting  
branching
processes are key ingredients to establish the stability of these  
systems.
Several cases are considered: networks with one and two chunks where a  
complete
classification is obtained and several cases of a network with $n$  
chunks.


  http://arxiv.org/abs/0907.0375

---------------------------------------------------------------

8732. SEMIMARTINGALE DECOMPOSITION OF CONVEX FUNCTIONS OF CONTINUOUS   
SEMIMARTINGALES BY BROWNIAN PERTURBATION

Nastasiya F Grinberg

In this note we prove that the local martingale part of a convex  
function f
of a d-dimensional semimartingale X=M+A can be wrtitten in terms of an  
Ito
stochastic intergral of H(x), some measurable choice of subgradient of  
fat x,
against M, the martingale part of X. This result was first proved by  
Bouleau in
[2]. Here we present a new treatment of the problem.


  http://arxiv.org/abs/0907.0382

---------------------------------------------------------------

8733. MAJORITY DYNAMICS ON TREES AND THE DYNAMIC CAVITY METHOD

Yashodhan Kanoria and Andrea Montanari

An elector sits on each vertex of an infinite tree of degree $k$, and  
has to
decide between two alternatives. At each time step, each elector  
switches to
the opinion of the majority of her neighbors. We analyze this majority  
process
when opinions are initialized to independent and identically  
distributed random
variables.
   In particular, we bound the threshold value of the initial bias  
such that the
process converges to consensus. In order to prove an upper bound, we
characterize the process of a single node in the large $k$-limit. This  
approach
is inspired by the theory of mean field spin-glass and can potentially  
be
generalized to a wider class of models. We also derive a lower bound  
that is
non-trivial for small, odd values of $k$.


  http://arxiv.org/abs/0907.0449

---------------------------------------------------------------

8734. A STRONG LOG-CONCAVITY PROPERTY FOR MEASURES ON BOOLEAN ALGEBRAS

Jeff Kahn and  Michael Neiman

We introduce the antipodal pairs property for probability measures on  
finite
Boolean algebras and prove that conditional versions imply strong  
forms of
log-concavity. We give several applications of this fact, including
improvements of some results of Wagner; a new proof of a theorem of  
Liggett
stating that ultra-log-concavity of sequences is preserved by  
convolutions; and
some progress on a well-known log-concavity conjecture of J. Mason.


  http://arxiv.org/abs/0907.0243

---------------------------------------------------------------

8735. A CUT-OFF PHENOMENON IN LOCATION BASED RANDOM ACCESS GAMES WITH   
IMPERFECT INFORMATION

Hazer Inaltekin and  Mung Chiang and  H. Vincent Poor

This paper analyzes the behavior of selfish transmitters under imperfect
location information. The scenario considered is that of a wireless  
network
consisting of selfish nodes that are randomly distributed over the  
network
domain according to a known probability distribution, and that are  
interested
in communicating with a common sink node using common radio resources.  
In this
scenario, the wireless nodes do not know the exact locations of their
competitors but rather have belief distributions about these locations.
Firstly, properties of the packet success probability curve as a  
function of
the node-sink separation are obtained for such networks. Secondly, a
monotonicity property for the best-response strategies of selfish  
nodes is
identified. That is, for any given strategies of competitors of a  
node, there
exists a critical node-sink separation for this node such that its
best-response is to transmit when its distance to the sink node is  
smaller than
this critical threshold, and to back off otherwise. Finally, necessary  
and
sufficient conditions for a given strategy profile to be a Nash  
equilibrium are
provided.


  http://arxiv.org/abs/0907.0255

---------------------------------------------------------------

8736. SELF-INTERSECTIONS OF RANDOM GEODESICS ON NEGATIVELY CURVED  
SURFACES

Steven P. Lalley

We study the fluctuations of self-intersection counts of random geodesic
segments of length $t$ on a compact, negatively curved surface in the  
limit of
large $t$. If the initial direction vector of the geodesic is chosen  
according
to the \emph{Liouville measure}, then it is not difficult to show that  
the
number $N (t)$ of self-intersections by time $t$ grows like $\kappa  
t^{2}$,
where $\kappa =\kappa_{M}$ is a positive constant depending on the  
surface $M$.
We show that (for a smooth modification of $N (t)$) the fluctuations  
are of
size $t$, and the limit distribution is a weak limit of Gaussian  
quadratic
forms. We also show that the fluctuations of \emph{localized} self- 
intersection
counts (that is, only self-intersections in a fixed subset of $M$ are  
counted)
are typically of size $t^{3/2}$, and the limit distribution is Gaussian.


  http://arxiv.org/abs/0907.0259

---------------------------------------------------------------

8737. REDUCING THE ISING MODEL TO MATCHINGS

Mark Huber (Claremont McKenna College) and Jenny Law (Duke University)

Canonical paths is one of the most powerful tools available to show  
that a
Markov chain is rapidly mixing, thereby enabling approximate sampling  
from
complex high dimensional distributions. Two success stories for the  
canonical
paths method are chains for drawing matchings in a graph, and a chain  
for a
version of the Ising model called the subgraphs world. In this paper,  
it is
shown that a subgraphs world draw can be obtained by taking a draw from
matchings on a graph that is linear in the size of the original graph.  
This
provides a partial answer to why canonical paths works so well for both
problems, as well as providing a new source of algorithms for the  
Ising model.
For instance, this new reduction immediately yields a fully polynomial  
time
approximation scheme for the Ising model on a bounded degree graph  
when the
magnitization is bounded away from 0.


  http://arxiv.org/abs/0907.0477

---------------------------------------------------------------

8738. ZEROS OF A TWO-PARAMETER RANDOM WALK

Davar Khoshnevisan and Pal Revesz

We prove that the number gamma(N) of the zeros of a two-parameter simple
random walk in its first N-by-N time steps is almost surely equal to N  
to the
power 1+o(1) as N goes to infinity. This is in contrast with our  
earlier joint
effort with Z. Shi [4]; that work shows that the number of zero  
crossings in
the first N-by-N time steps is N to the power (3/2)+o(1) as N goes to  
infinity.
We prove also that the number of zeros on the diagonal in the first N  
time
steps is (c+o(1)) log N as N goes to infinity, where c is 2\pi.


  http://arxiv.org/abs/0907.0487

---------------------------------------------------------------

8739. BRANCHING RANDOM WALKS IN SPACE-TIME RANDOM ENVIRONMENT:  
SURVIVAL  PROBABILITY, GLOBAL AND LOCAL GROWTH RATES

Francis Comets and Nobuo Yoshida

We study the survival probability and the growth rate for branching  
random
walks in random environment (BRWRE). The particles perform simple  
symmetric
random walks on the $d$-dimensional integer lattice, while at each  
time unit,
they split into independent copies according to time-space i.i.d.  
offspring
distributions. The BRWRE is naturally associated with the directed  
polymers in
random environment (DPRE), for which the quantity called the free  
energy is
well studied. We discuss the survival probability (both global and  
local) for
BRWRE and give a criterion for its positivity in terms of the free  
energy of
the associated DPRE. We also show that the global growth rate for the  
number of
particles in BRWRE is given by the free energy of the associated DPRE,  
though
the local growth rateis given by the directional free energy.


  http://arxiv.org/abs/0907.0509

---------------------------------------------------------------

8740. UNIFORM ESTIMATES FOR METASTABLE TRANSITION TIMES IN A COUPLED  
BISTABLE  SYSTEM

Florent Barret (CMAP) and  Anton Bovier (IAM) and  Sylvie M\'el\'eard  
(CMAP)

We consider a coupled bistable N-particle system driven by a Brownian  
noise,
with a strong coupling corresponding to the synchronised regime. Our  
aim is to
obtain sharp estimates on the metastable transition times betwen the  
two stable
states, both for fixed N and in the limit when N tends to infinity.  
These
estimates would be the main step for a rigorous understanding of the  
metastable
behavior of infinite dimensional systems, as the stochastically  
perturbed
Ginzburg-Landau equation. The quantities of interest are objects of  
potential
theory, as capacities and equilibrium measure. We prove estimates with  
error
bounds that are uniform in the dimension of the system.


  http://arxiv.org/abs/0907.0537

---------------------------------------------------------------

8741. UPPER LARGE DEVIATIONS FOR MAXIMAL FLOWS THROUGH A TILTED CYLINDER

Marie Theret

We consider the standard first passage percolation model in $\ZZ^d$ for
$d\geq 2$ and we study the maximal flow from the upper half part to  
the lower
half part (respectively from the top to the bottom) of a cylinder  
whose basis
is a hyperrectangle of sidelength proportional to $n$ and whose height  
is
$h(n)$ for a certain height function $h$. We denote this maximal flow by
$\tau_n$ (respectively $\phi_n$). We emphasize the fact that the  
cylinder may
be tilted. We look at the probability that these flows, rescaled by  
the surface
of the basis of the cylinder, are greater than $\nu(\vec{v})+\eps$ for  
some
positive $\eps$, where $\nu(\vec{v})$ is the almost sure limit of the  
rescaled
variable $\tau_n$ when $n$ goes to infinity. On one hand, we prove  
that the
speed of decay of this probability in the case of the variable $\tau_n 
$ depends
on the tail of the distribution of the capacities of the edges: it can  
decays
exponentially fast with $n^{d-1}$, or with $n^{d-1} \min(n,h(n))$, or  
at an
intermediate regime. On the other hand, we prove that this probability  
in the
case of the variable $\phi_n$ decays exponentially fast with the  
volume of the
cylinder as soon as the law of the capacity of the edges admits one  
exponential
moment; the importance of this result is however limited by the fact  
that
$\nu(\vec{v})$ is not in general the almost sure limit of the rescaled  
maximal
flow $\phi_n$, but it is the case at least when the height $h(n)$ of the
cylinder is negligible compared to $n$.


  http://arxiv.org/abs/0907.0614

---------------------------------------------------------------

8742. CENTRAL LIMIT THEOREMS FOR MULTICOLOR URNS WITH DOMINATED COLORS

Patrizia Berti (Dip. di Matematica and  Univ. Modena and  Italy) and   
Irene  Crimaldi (Dip. Matematica, Univ. Bologna, Italy), Luca Pratelli  
(Accademia
   Navale, Livorno, Italy), Pietro Rigo (Dip. Economia politica e Metodi
   quantitativi, Univ. Pavia, Italy)

An urn contains balls of d colors. At each time, a ball is drawn and  
then
replaced together with a random number of balls of the same color.  
Assuming
that some colors are dominated by others, we prove central limit  
theorems. Some
statistical applications are discussed.


  http://arxiv.org/abs/0907.0676

---------------------------------------------------------------

8743. D\'EVIATIONS MOD\'ER\'EES DE LA DISTANCE CHIMIQUE

Olivier Garet (IECN) and  R\'egine Marchand (IECN)

In this paper, we establish moderate deviations for the chemical  
distance in
Bernoulli percolation. The chemical distance between two points is the  
length
of the shortest open path between these two points. Thus, we study the  
size of
random fluctuations around the mean value, and also the asymptotic  
behavior of
this mean value. The estimates we obtain improve our knowledge of the
convergence to the asymptotic shape. Our proofs rely on concentration
inequalities proved by Boucheron, Lugosi and Massart, and also on the
approximation theory of subadditive functions initiated by Alexander.


  http://arxiv.org/abs/0907.0697

---------------------------------------------------------------

8744. MODERATE DEVIATIONS FOR THE CHEMICAL DISTANCE IN BERNOULLI  
PERCOLATION

Olivier Garet (IECN) and  R\'egine Marchand (IECN)

In this paper, we establish moderate deviations for the chemical  
distance in
Bernoulli percolation. The chemical distance between two points is the  
length
of the shortest open path between these two points. Thus, we study the  
size of
random fluctuations around the mean value, and also the asymptotic  
behavior of
this mean value. The estimates we obtain improve our knowledge of the
convergence to the asymptotic shape. Our proofs rely on concentration
inequalities proved by Boucheron, Lugosi and Massart, and also on the
approximation theory of subadditive functions initiated by Alexander.


  http://arxiv.org/abs/0907.0698

---------------------------------------------------------------

8745. ON THE PRESERVATION OF GIBBSIANNESS UNDER SYMBOL AMALGAMATION

Jean-Rene Chazottes and Edgardo Ugalde

Starting from the full-shift on a finite alphabet $A$, suppose we  
confound
some symbols of $A$. This gives a new full shift on a new alphabet $B 
$. The
amalgamation map, call it $\pi$, defines a `factor map', that is, a  
continuous
transformation between $(A^\nn,T_A)$ and $(B^\nn,T_B)$ with the  
property that
$\pi\circ T_A=T_B\circ \pi$, where $T_A$, resp. $T_B$, is the shift  
map on
$A^\nn$, resp. $B^\nn$.
   Given a regular function $\psi:A^\nn\to\rr$ (a `potential'), there  
is a
unique Gibbs measure $\mu_\psi$. In this article, we prove that, for a  
large
class of potentials, the pushforward measure $\mu_\psi\circ\pi^{-1}$  
is still
Gibbsian for a potential $\phi:B^\nn\to\rr$ having a `bit less'  
regularity than
$\psi$. In the special case where $\psi$ is a `2-symbol' potential,  
the Gibbs
measure $\mu_\psi$ is none other than a Markov measure and the  
amalgamation
$\pi$ defines a hidden Markov chain. In that special case, our theorem  
can be
recast by saying that a hidden Markov chain is a Gibbs measure (for a H 
\"older
potential).


  http://arxiv.org/abs/0907.0528

---------------------------------------------------------------

8746. POINCAR\'E INEQUALITY AND EXPONENTIAL INTEGRABILITY OF HITTING  
TIMES FOR  LINEAR DIFFUSIONS

D. Loukianova and  O. Loukianov and  Sh. Song

Let $X$ be a regular linear continuous positively recurrent Markov  
process
with state space $\R$, scale function $S$ and speed measure $m$. For $a 
\in \R$
denote B^+_a&=\sup_{x\geq a} \m(]x,+\infty[)(S(x)-S(a)) B^-_a&=\sup_{x 
\leq a}
\m(]-\infty;x[)(S(a)-S(x)) We study some characteristic relations  
between
$B^+_a$, $B^-_a$, the exponential moments of the hitting times $T_a$  
of $X$,
the Hardy and Poincar\'e inequalities for the Dirichlet form  
associated with
$X$. As a corollary, we establish the equivalence between the  
existence of
exponential moments of the hitting times and the spectral gap of the  
generator
of $X$.


  http://arxiv.org/abs/0907.0762

---------------------------------------------------------------

8747. BOUNDARY HARNACK INEQUALITY FOR ALPHA-HARMONIC FUNCTIONS ON THE   
SIERPI\'NSKI TRIANGLE

Kamil Kaleta and  Mateusz Kwa\'snicki

We prove an uniform boundary Harnack inequality for nonnegative  
functions
harmonic with respect to $\alpha$-stable process on the Sierpi{\'n}ski
triangle, where $\alpha \in (0, 1)$. Our result requires no regularity
assumptions on the domain of harmonicity.


  http://arxiv.org/abs/0907.0793

---------------------------------------------------------------

8748. DUALITY AND INTERTWINING FOR DISCRETE MARKOV KERNELS: A RELATION  
AND  EXAMPLES

Thierry Huillet (LPTM) and  Servet Martinez

We work out some relations between duality and intertwining in the  
context of
discrete Markov chains, fixing up the background of previous relations  
first
established for birth and death chains and their Siegmund duals. In  
view of the
results, the monotone properties resulting from the Siegmund dual of  
birth and
death chains are revisited in some detail, with emphasis on the non  
neutral
Moran model. We also introduce an ultrametric type dual extending the  
Siegmund
kernel. Finally we discuss the sharp dual, following closely the  
Diaconis-Fill
study.


  http://arxiv.org/abs/0907.0840

---------------------------------------------------------------

8749. DIFFUSION APPROXIMATION FOR THE COMPONENTS IN CRITICAL  
INHOMOGENEOUS  RANDOM GRAPHS OF RANK 1

Tatyana S. Turova

Consider the random graph on $n$ vertices $1, ..., n$. Each vertex $i$  
is
assigned a type $X_i$ with $X_1, ..., X_n$ being independent identically
distributed as a nonnegative discrete random variable $X$. We assume  
that ${\bf
E} X^3<\infty$. Given types of all vertices, an edge exists between  
vertices
$i$ and $j$ independent of anything else and with probability $\min \{1,
\frac{X_iX_j}{n}(1+\frac{a}{n^{1/3}}) \}$. We study the critical  
phase, which
is known to take place when ${\bf E} X^2=1$. We prove that normalized by
$n^{-2/3}$ the asymptotic joint distributions of component sizes of  
the graph
equals the joint distribution of the excursions of a reflecting  
Brownian motion
$B^a(s)$ with diffusion coefficient $\sqrt{{\bf E}X{\bf E}X^3}$ and  
drift
$a-\frac{1}{({\bf E}X)^2}s$. This shows that finiteness of ${\bf  
E}X^3$ is the
necessary condition for the diffusion limit. In particular, we  
conclude that
the size of the largest connected component is of order $n^{2/3}$.


  http://arxiv.org/abs/0907.0897

---------------------------------------------------------------

8750. DIFFERENTIABILITY OF QUADRATIC BSDE GENERATED BY CONTINUOUS  
MARTINGALES  AND HEDGING IN INCOMPLETE MARKETS

Peter Imkeller and  Anthony Reveillac and  Anja Richter

In this paper we consider a class of BSDE with drivers of quadratic  
growth,
on a stochastic basis generated by continuous local martingales. We  
first
derive the Markov property of a forward-backward system (FBSDE) if the
generating martingale is a strong Markov process. Then we establish the
differentiability of a FBSDE with respect to the initial value of its  
forward
component. This enables us to obtain the main result of this article  
which from
the perspective of a utility optimization interpretation of the  
underlying
control problem on a financial market takes the following form. The  
control
process of the BSDE steers the system into a random liability  
depending on a
market external uncertainty and this way describes the optimal  
derivative hedge
of the liability by investment in a capital market the dynamics of  
which is
described by the forward component. This delta hedge is described in a  
key
formula in terms of a derivative functional of the solution process  
and the
correlation structure of the internal uncertainty captured by the  
forward
process and the external uncertainty responsible for the market  
incompleteness.
The formula largely extends the scope of validity of the results  
obtained by
several authors in the Brownian setting, designed to give a genuinely
stochastic representation of the optimal delta hedge in the context of  
cross
hedging insurance derivatives generalizing the derivative hedge in the
Black-Scholes model. Of course, Malliavin's calculus needed in the  
Brownian
setting is not available in the general local martingale framework. We  
replace
it by new tools based on stochastic calculus techniques.


  http://arxiv.org/abs/0907.0941

---------------------------------------------------------------

8751. ON THE ORTHOGONAL COMPONENT OF BSDES IN A MARKOVIAN SETTING

Anthony R\'eveillac

In this Note we consider a quadratic backward stochastic differential
equation (BSDE) driven by a continuous martingale $M$ and whose  
generator is a
deterministic function. We prove (in Theorem \ref{theorem:main}) that  
if $M$ is
a strong homogeneous Markov process and if the BSDE has the form  
\eqref{BSDE}
then the unique solution $(Y,Z,N)$ of the BSDE is reduced to $(Y,Z)$,
\textit{i.e.} the orthogonal martingale $N$ is equal to zero showing  
that in a
Markovian setting the "usual" solution $(Y,Z)$ has not to be completed  
by a
strongly orthogonal even if $M$ does not enjoy the martingale  
representation
property.


  http://arxiv.org/abs/0907.1071

---------------------------------------------------------------

8752. A CONSTRUCTIVE APPROACH TO THE MONGE-KANTOROVICH PROBLEM FOR  
CHAINS OF  INFINITE ORDER

Antonio Galves and  Nancy L. Garcia and Clementine Prieur

We propose a constructive approach to solve the Monge-Kantorovich  
problem for
chains of infinite order on a finite alphabet with an additive cost  
function.
   From this constructive description of the Kantorovich coupling we  
obtain, for
any $\epsilon > 0$, a perfect simulation algorithm for sampling from an
$\epsilon$-approximating coupling which assigns to the cost function an
expectation which is $\epsilon$-close to the minimum cost. Our  
approach is
based on a regenerative scheme which enable us to construct the  
Kantorovich
coupling as a mixture of product measures.


  http://arxiv.org/abs/0907.1113

---------------------------------------------------------------

8753. HSU-ROBBINS AND SPITZER'S THEOREMS FOR THE VARIATIONS OF  
FRACTIONAL  BROWNIAN MOTION

Ciprian Tudor (CES and  Samos)

Using recent results on the behavior of multiple Wiener-It\^o  
integrals based
on Stein's method, we prove Hsu-Robbins and Spitzer's theorems for  
sequences of
correlated random variables related to the increments of the fractional
Brownian motion.


  http://arxiv.org/abs/0907.1116

---------------------------------------------------------------

8754. CONVERGENCE TO L\'EVY STABLE PROCESSES UNDER STRONG MIXING  
CONDITIONS

Marta Tyran-Kaminska

For a strictly stationary sequence of random vectors in $\mathbb{R}^d$  
we
study convergence of partial sums processes to L\'evy stable process  
in the
Skorohod space with $J_1$-topology. We identify necessary and sufficient
conditions for such convergence and provide sufficient conditions when  
the
stationary sequence is strongly mixing.


  http://arxiv.org/abs/0907.1185

---------------------------------------------------------------

8755. AN APPLICATION TO CREDIT RISK OF A HYBRID MONTE CARLO-OPTIMAL   
QUANTIZATION METHOD

Giorgia Callegaro and  Abass Sagna (PMA)

In this paper we use a hybrid Monte Carlo-Optimal quantization method to
approximate the conditional survival probabilities of a firm, given a
structural model for its credit defaul, under partial information. We  
consider
the case when the firm's value is a non-observable stochastic process $ 
(V_t)_{t
\geq 0}$ and inverstors in the market have access to a process $ 
(S_t)_{t \geq
0}$, whose value at each time t is related to $(V_s, s \leq t)$. We are
interested in the computation of the conditional survival  
probabilities of the
firm given the "investor information". As a application, we analyse  
the shape
of the credit spread curve for zero coupon bonds in two examples.


  http://arxiv.org/abs/0907.0645

---------------------------------------------------------------

8756. PERIMETER AND AREA OF THE CONVEX HULL OF N PLANAR BROWNIAN MOTIONS

Julien Randon-Furling and  Satya N. Majumdar and Alain Comtet

We compute exactly the mean perimeter and area of the convex hull of N
independent planar Brownian paths each of duration T, both for open  
and closed
paths. We show that the mean perimeter < L_N > = \alpha_N, \sqrt{T}  
and the
mean area <A_N> = \beta_N T for all T. The prefactors \alpha_N and  
\beta_N,
computed exactly for all N, increase very slowly (logarithmically) with
increasing N. This slow growth is a consequence of extreme value  
statistics and
has interesting implication in ecological context in estimating the  
home range
of a herd of animals with population size N.


  http://arxiv.org/abs/0907.0921

---------------------------------------------------------------

8757. DISTRIBUTED RANDOM ACCESS ALGORITHM: SCHEDULING AND CONGESION  
CONTROL

Libin Jiang and  Devavrat Shah and  Jinwoo Shin and  Jean Walrand

This paper provides proofs of the rate stability, Harris recurrence, and
epsilon-optimality of CSMA algorithms where the backoff parameter of  
each node
is based on its backlog. These algorithms require only local  
information and
are easy to implement.
   The setup is a network of wireless nodes with a fixed conflict  
graph that
identifies pairs of nodes whose simultaneous transmissions conflict.  
The paper
studies two algorithms. The first algorithm schedules transmissions to  
keep up
with given arrival rates of packets. The second algorithm controls the  
arrivals
in addition to the scheduling and attempts to maximize the sum of the  
utilities
of the flows of packets at the different nodes. For the first  
algorithm, the
paper proves rate stability for strictly feasible arrival rates and  
also Harris
recurrence of the queues. For the second algorithm, the paper proves the
epsilon-optimality. Both algorithms operate with strictly local  
information in
the case of decreasing step sizes, and operate with the additional  
information
of the number of nodes in the network in the case of constant step size.


  http://arxiv.org/abs/0907.1266

---------------------------------------------------------------

8758. DYNKIN'S ISOMORPHISM THEOREM AND THE STOCHASTIC HEAT EQUATION

Nathalie Eisenbaum and  Mohammud Foondun and Davar Khoshnevisan

Consider the stochastic heat equation $\partial_t u = \sL u + \dot{W} 
$, where
$\sL$ is the generator of a [Borel right] Markov process in duality.  
We show
that the solution is locally mutually absolutely continuous with  
respect to a
smooth perturbation of the Gaussian process that is associated, via  
Dynkin's
isomorphism theorem, to the local times of the replica-symmetric  
process that
corresponds to $\sL$.In the case that $\sL$ is the generator of a L\'evy
process on $\R^d$, our result gives a probabilistic explanation of the  
recent
findings of Foondun et al.


  http://arxiv.org/abs/0907.1316

---------------------------------------------------------------

8759. ON THE DISCRETIZATION OF BACKWARD DOUBLY STOCHASTIC  
DIFFERENTIAL  EQUATIONS

Omar Aboura (CES and  Samos)

In this paper, we are dealing with the approximation of the process  
(Y,Z)
solution to the backward doubly stochastic differential equation with  
the
forward process X . After proving the L2-regularity of Z, we use the  
Euler
scheme to discretize X and the Zhang approach in order to give a  
discretization
scheme of the process (Y,Z).


  http://arxiv.org/abs/0907.1406

---------------------------------------------------------------

8760. EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR FOKKER-PLANCK  
EQUATIONS ON  HILBERT SPACES

Vladimir Bogachev and  Giuseppe Da Prato and  Michael Roeckner

We consider a stochastic differential equation in a Hilbert space with
time-dependent coefficients for which no general existence and  
uniqueness
results are known. We prove, under suitable assumptions, existence and
uniqueness of a measure valued solution, for the corresponding Fokker-- 
Planck
equation. In particular, we verify the Chapman--Kolmogorov equations  
and get an
evolution system of transition probabilities for the stochastic dynamics
informally given by the stochastic differential equation.


  http://arxiv.org/abs/0907.1431

---------------------------------------------------------------

8761. LIMIT DISTRIBUTIONS FOR LARGE P\'OLYA URNS

Brigitte Chauvin and  Nicolas Pouyanne and  R\'eda Sahnoun

We consider a two colors P\'olya urn with balance $S$. Assume it is a
\emph{large} urn \emph{i.e.} the second eigenvalue $m$ of the  
replacement
matrix satisfies $1/2<m/S\leq 1$. After $n$ drawings, the composition  
vector
has asymptotically a first deterministic term of order $n$ and a  
second random
term of order $n^{m/S}$. The object of interest is the limit  
distribution of
this random term.
   The method consists in embedding the discrete time urn in  
continuous time,
getting a two type branching process. The dislocation equations  
associated with
this process lead to a system of two differential equations satisfied  
by the
Fourier transforms of the limit distributions. The resolution is  
carried out
and it turns out that the Fourier transforms are explicitely related  
to Abelian
integrals on the Fermat curve of degree $m$.


  http://arxiv.org/abs/0907.1477

---------------------------------------------------------------

8762. INHOMOGENEITY AND UNIVERSALITY: OFF-CRITICAL BEHAVIOR OF  
INTERFACES

Pierre Nolin

We further study the interfaces arising in a situation of  
inhomogeneity. More
precisely, we identify a characteristic length for the gradient  
percolation
model, that enables us to tighten previous estimates established for  
it. This
allows to construct non-trivial scaling limits: the limiting objects  
share some
properties with critical percolation interfaces, but locally, they  
rather
behave like off-critical percolation interfaces.


  http://arxiv.org/abs/0907.1495

---------------------------------------------------------------

8763. LES PROBABILIT\'ES D\'EFAILLANCE COMME INDICATEURS DE  
PERFORMANCE DES  BARRI\`ERES TECHNIQUES DE S\'ECURIT\'E ? APPROCHE  
ANALYTIQUE

Florent Brissaud (INERIS and  UTT) and  Brice Lanternier (INERIS)

French environmental laws require industrialists to include probability
criteria in risk assessments, especially to define confidence levels  
for risk
management measures. This paper presents the failure probabilities as  
efficient
indicators for technical safety barrier performances. Generic formulas  
are
proposed to evaluate these probabilities, including failure rate,  
barrier
architecture, full and partial proof tests. In many cases, these  
results can be
directly used to assess safety barrier confidence levels.


  http://arxiv.org/abs/0907.1516

---------------------------------------------------------------

8764. A REMARK ON ZEROS OF BROWNIAN MOTION

Weber Michel

Let $ \{W(t), t\ge 0\}$ be a standard Brownian motion. If $I$ is a  
bounded
interval on which $W $ has no zero, an almost sure lower bound to
$\inf\{|W(t)|, t\in I\}$ can be provided, when $I$ is taken from a given
countable family of intervals covering the positive half-line.


  http://arxiv.org/abs/0907.1572

---------------------------------------------------------------

8765. SYMMETRIZATION OF L\'EVY PROCESSES AND APPLICATIONS

Rodrigo Banuelos and  Pedro J. Mendez-Hernandez

It is shown that many of the classical generalized isoperimetric  
inequalities
for the Laplacian when viewed in terms of Brownian motion extend to a  
wide
class of Levy processes. The results are derived from the multiple  
integral
inequalities of Brascamp, Lieb and Luttinger but the probabilistic  
structure of
the processes plays a crucial role in the proofs.


  http://arxiv.org/abs/0907.1598

---------------------------------------------------------------

8766. ARE FRACTIONAL BROWNIAN MOTIONS PREDICTABLE?

Adam Jakubowski

We provide a device, called the local predictor, which extends the  
idea of
the predictable compensator. It is shown that a fBm with the Hurst index
greater than 1/2 coincides with its local predictor while fBm with the  
Hurst
index smaller than 1/2 does not admit any local predictor. The local  
predictor
of a martingale (in particular: Brownian motion) trivially exists and  
equals 0.


  http://arxiv.org/abs/0907.1618

---------------------------------------------------------------

8767. RANDOM WALKS ON DISCRETE CYLINDERS WITH LARGE BASES AND RANDOM   
INTERLACEMENTS

David Windisch

Following the recent work of Sznitman (arXiv:0805.4516), we  
investigate the
microscopic picture induced by a random walk trajectory on a cylinder  
of the
form G_N x Z, where G_N is a large finite connected weighted graph,  
and relate
it to the model of random interlacements on infinite transient  
weighted graphs.
Under suitable assumptions, the set of points not visited by the  
random walk
until a time of order |G_N|^2 in a neighborhood of a point with Z- 
component of
order |G_N| converges in distribution to the law of the vacant set of  
a random
interlacement on a certain limit model describing the structure of the  
graph in
the neighborhood of the point. The level of the random interlacement  
depends on
the local time of a Brownian motion. The result also describes the limit
behavior of the joint distribution of the local pictures in the  
neighborhood of
several distant points with possibly different limit models. As  
examples of
G_N, we treat the d-dimensional box of side length N, the Sierpinski  
graph of
depth N and the d-ary tree of depth N, where d >= 2.


  http://arxiv.org/abs/0907.1627

---------------------------------------------------------------

8768. FLUCTUATIONS OF THE NODAL LENGTH OF RANDOM SPHERICAL HARMONICS

Igor Wigman

Using the multiplicities of the Laplace eigenspace on the sphere (the  
space
of spherical harmonics) we endow the space with Gaussian probability  
measure.
This induces a notion of random Gaussian spherical harmonics of degree  
$n$
having Laplace eigenvalue $E=n(n+1)$. We study the length distribution  
of the
nodal lines of random spherical harmonics.
   It is known that the expected length is of order $n$. It is natural  
to
conjecture that the variance should be of order $n$, due to the natural
scaling. Our principal result is that, due to an unexpected  
cancelation, the
variance of the nodal length of random spherical harmonics is of order
$\log{n}$. This behaviour is consistent to the one predicted by Berry  
for nodal
lines on chaotic billiards (Random Wave Model). In addition we find  
that a
similar result is applicable for "generic" linear statistics of the  
nodal
lines.


  http://arxiv.org/abs/0907.1648

---------------------------------------------------------------

8769. SOME ALMOST SURE RESULTS FOR UNBOUNDED FUNCTIONS OF INTERMITTENT  
MAPS  AND THEIR ASSOCIATED MARKOV CHAINS

Jerome Dedecker (LSTA) and  Sebastien Gouezel (IRMAR) and  Florence  
Merlevede  (LAMA)

We consider a large class of piecewise expanding maps T of [0,1] with a
neutral fixed point, and their associated Markov chain Y_i whose  
transition
kernel is the Perron-Frobenius operator of T with respect to the  
absolutely
continuous invariant probability measure. We give a large class of  
unbounded
functions f for which the partial sums of f\circ T^i satisfy both a  
central
limit theorem and a bounded law of the iterated logarithm. For the  
same class,
we prove that the partial sums of f(Y_i) satisfy a strong invariance  
principle.
When the class is larger, so that the partial sums of f\circ T^i may  
belong to
the domain of normal attraction of a stable law of index p\in (1, 2),  
we show
that the almost sure rates of convergence in the strong law of large  
numbers
are the same as in the corresponding i.i.d. case.


  http://arxiv.org/abs/0907.1403

---------------------------------------------------------------

8770. ALMOST SURE INVARIANCE PRINCIPLE FOR DYNAMICAL SYSTEMS BY  
SPECTRAL  METHODS

Sebastien Gouezel (IRMAR)

We prove the almost sure invariance principle for stationary R^d--valued
processes (with dimension-independent very precise error terms),  
solely under a
strong assumption on the characteristic functions of these processes.  
This
assumption is easy to check for large classes of dynamical systems or  
Markov
chains, using strong or weak spectral perturbation arguments.


  http://arxiv.org/abs/0907.1404

---------------------------------------------------------------

8771. FOREST FIRES ON $\Z_+$ WITH IGNITION ONLY AT 0

Stanislav Volkov

We consider a version of the forest fire model on graph $G$, where each
vertex of a graph becomes occupied with rate one. A fixed vertex $v_0$  
is hit
by lightning with the same rate, and when this occurs, the whole  
cluster of
occupied vertices containing $v_0$ is burnt out. We show that when  
$G=Z_{+}$,
the times between consecutive burnouts at vertex $n$, divided by $\log  
n$,
converge weakly as $n\to\infty$ to a random variable which  
distribution is
$1-\rho(x)$ where $\rho(x)$ is the Dickman function.
   We also show that on transitive graphs with a non-trivial site  
percolation
threshold and one infinite cluster at most, the distributions of the  
time till
the first burnout of {\it any} vertex have exponential tails.
   Finally, we give an elementary proof of an interesting limit:
$\lim_{n\to\infty} \sum_{k=1}^n {n \choose k} (-1)^k \log k -\log\log
n=\gamma$.


  http://arxiv.org/abs/0907.1821

---------------------------------------------------------------

8772. QUEUEING WITH NEIGHBOURS

Vadim Shcherbakov and  Stanislav Volkov

In this paper we study asymptotic behaviour of a growth process  
generated by
a semi-deterministic variant of cooperative sequential adsorption  
model (CSA).
This model can also be viewed as a particular queueing system with local
interactions. We show that quite limited randomness of the model still
generates a rich collection of possible limiting behaviours.


  http://arxiv.org/abs/0907.1826

---------------------------------------------------------------

8773. ESTIMATES ON THE SPEEDUP AND SLOWDOWN FOR A DIFFUSION IN A  
DRIFTED  BROWNIAN POTENTIAL

Gabriel Faraud

We study a model of diffusion in a brownian potential. This model was  
firstly
introduced by T. Brox (1986) as a continuous time analogue of random  
walk in
random environment. We estimate the deviations of this process above  
or under
its typical behavior. Our results rely on different tools such as a
representation introduced by Y. Hu, Z. Shi and M. Yor, Kotani's lemma,
introduced at first by K. Kawazu and H. Tanaka (1997), and a  
decomposition of
hitting times developed in a recent article by A. Fribergh, N. Gantert  
and S.
Popov (2008). Our results are in agreement with their results in the  
discrete
case.


  http://arxiv.org/abs/0907.1864

---------------------------------------------------------------

8774. HIDDEN MARKOV PROCESSES IN THE CONTEXT OF SYMBOLIC DYNAMICS

Mike Boyle (University of Maryland) and Karl Petersen (University of   
North Carolina)

In an effort to aid communication among different fields and perhaps
facilitate progress on problems common to all of them, this article  
discusses
hidden Markov processes from several viewpoints, especially that of  
symbolic
dynamics, where they are known as sofic measures, or continuous shift- 
commuting
images of Markov measures. It provides background, describes known  
tools and
methods, surveys some of the literature, and proposes several open  
problems.


  http://arxiv.org/abs/0907.1858

---------------------------------------------------------------

8775. THE TRIANGLE AND THE OPEN TRIANGLE

Gady Kozma

We show that for percolation on any transitive graph, the triangle  
condition
implies the open triangle condition.


  http://arxiv.org/abs/0907.1959

---------------------------------------------------------------

8776. LP-SOLUTION OF BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS

Auguste Aman (LMAI)

In this paper, our goal is solving backward doubly stochastic  
differential
equation (BDSDE for short) under weak assumptions on the data. The  
first part
of the paper is devoted to the development of some new technical  
aspects of
stochastic calculus related to BDSDEs. Then we derive a priori  
estimates and
prove existence and uniqueness of solutions, extending the results of  
Pardoux
and Peng \cite{PP1} to the case where the solution is taked in $L^{p},  
p>1$ and
the monotonicity conditions are satisfied. This study is limited to
deterministic terminal time.


  http://arxiv.org/abs/0907.1983

---------------------------------------------------------------

8777. ON THE OPTIMAL AMOUNT OF EXPERIMENTATION IN SEQUENTIAL DECISION  
PROBLEMS

Dinah Rosenberg and  Eilon Solan and  Nicolas Vieille

We provide a tight bound on the amount of experimentation under the  
optimal
strategy in sequential decision problems. We show the applicability of  
the
result by providing a bound on the cut-off in a one-arm bandit problem.


  http://arxiv.org/abs/0907.2002

---------------------------------------------------------------

8778. NEW RATES FOR EXPONENTIAL APPROXIMATION AND THE THEOREMS OF R 
\'ENYI AND  YAGLOM

Erol Pek\"oz and Adrian R\"ollin

We introduce two abstract theorems that reduce a variety of complex
exponential distributional approximation problems to the construction of
couplings. These are applied to obtain rates of convergence with  
respect to the
Wasserstein and Kolmogorov metrics for the theorem of R\'enyi on  
random sums
and generalizations of it, hitting times for Markov chains, and to  
obtain a new
rate for the classical theorem of Yaglom on the exponential asymptotic  
behavior
of a critical Galton-Watson process conditioned on non-extinction. The  
primary
tools are an adaptation of Stein's method, Stein couplings, as well as  
the
equilibrium distributional transformation from renewal theory.


  http://arxiv.org/abs/0907.2009

---------------------------------------------------------------

8779. L$^{P}$-SOLUTION OF REFLECTED GENERALIZED BSDES WITH NON- 
LIPSCHITZ  COEFFICIENTS

Auguste Aman (LMAI)

In this paper, we continue in solving reflected generalized backward
stochastic differential equations (RGBSDE for short) and fixed  
terminal time
with use some new technical aspects of the stochastic calculus related  
to the
reflected generalized BSDE. Here, existence and uniqueness of solution  
is
proved under a non-Lipschitz condition on the coefficients.


  http://arxiv.org/abs/0907.2032

---------------------------------------------------------------

8780. NUMERICAL SCHEME FOR BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL  
EQUATIONS

Auguste Aman (LMAI)

We study a discrete-time approximation for solutions of systems of  
decoupled
forward-backward doubly stochastic differential equations (FBDSDEs).  
Assuming
that the coefficients are Lipschitz-continuous, we prove the  
convergence of the
scheme when the step of time discretization, $|\pi|$ goes to zero. The  
rate of
convergence is exactly equal to $|\pi|^{1/2}$. The proof is based on a
generalization of a remarkable result on the $^{2}$-regularity of the  
solution
of the backward equation derived by J. Zhang


  http://arxiv.org/abs/0907.2035

---------------------------------------------------------------

8781. HOMEOMORPHISM OF SOLUTIONS TO BACKWARD DOUBLY SDES AND  
APPLICATIONS

Auguste Aman (LMAI)

In this paper we study the homeomorphic properties of the solutions to  
one
dimensional backward doubly stochastic differential equations under  
suitable
assumptions, where the terminal values depend on a real parameter.  
Then, we
apply them to the solutions for a class of second order quasilinear  
parabolic
stochastic partial differential equations.


  http://arxiv.org/abs/0907.2036

---------------------------------------------------------------

8782. REFLECTED GENERALIZED BACKWARD DOUBLY SDES DRIVEN BY L\'EVY  
PROCESSES  AND APPLICATIONS

Auguste Aman (LMAI)

In this paper, a class of reflected generalized backward doubly  
stochastic
differential equations (reflected GBDSDEs in short) driven by Teugels
martingales associated with L\'{e}vy process and the integral with  
respect to
an adapted continuous increasing process is investigated. We obtain the
existence and uniqueness of solutions to these equations. A  
probabilistic
interpretation for solutions to a class of reflected stochastic partial
differential integral equations (PDIEs in short) with a nonlinear  
Neumann
boundary condition is given.


  http://arxiv.org/abs/0907.2037

---------------------------------------------------------------

8783. STOCHASTIC 2D HYDRODYNAMICAL SYSTEMS: SUPPORT THEOREM

Igor Chueshov and  Annie Millet (SAMOS and  Ces and  Pma)

We deal with a class of abstract nonlinear stochastic models with
multiplicative noise, which covers many 2D hydrodynamical models  
including the
2D Navier-Stokes equation, 2D MHD models and 2D magnetic B\'enard  
problems as
well as some shell models of turbulence. Our main result describes the  
support
of the distribution of solutions. Both inclusions are proved by means  
of a
general result of convergence in probability for non linear stochastic  
PDEs
driven by a Hilbert-valued Brownian motion and some adapted finite  
dimensional
approximation of this process.


  http://arxiv.org/abs/0907.2100

---------------------------------------------------------------

8784. PERFECT SIMULATION FOR STOCHASTIC CHAINS WITH UNBOUNDED VARIABLE  
LENGTH  MEMORY

Alexsandro Gallo

We present a new perfect simulation algorithm for stationary chains  
(indexed
by $\mathbb{Z}$) having unbounded variable length memory. This is the  
class of
infinite memory chains for which the family of transition  
probabilities is
represented through the form of a \emph{probabilistic context tree}. Our
condition is expressed in terms of the structure of the context tree. In
particular, we do not assume the continuity of the family of transition
probabilities. We give an explicit construction of the chain using a  
sequence
of i.i.d. random variables uniformly distributed in $[0,1[$.


  http://arxiv.org/abs/0907.2150

---------------------------------------------------------------

8785. ON THE DOMINATION OF RANDOM WALK ON A DISCRETE CYLINDER BY  
RANDOM  INTERLACEMENTS

Alain-Sol Sznitman

We consider simple random walk on a discrete cylinder with base a large
d-dimensional torus of side-length N, when d is two or more. We  
develop a
stochastic domination control on the local picture left by the random  
walk in
boxes of side-length almost of order N, at certain random times  
comparable to
the square of the number of sites in the base. We show a domination  
control in
terms of the trace left in similar boxes by random interlacements in the
infinite (d+1)-dimensional cubic lattice at a suitably adjusted level.  
As an
application we derive a lower bound on the disconnection time of the  
discrete
cylinder, which as a by-product shows the tightness of the laws of the  
ratio of
the square of the number of sites in the base to the disconnection  
time. This
fact had previously only been established when d is at least 17, in  
arXiv:
math/0701414.


  http://arxiv.org/abs/0907.2184

---------------------------------------------------------------

8786. A PATH GUESSING GAME WITH WAGERING

Marcus Pendergrass

We consider a two-player game in which the first player (the Guesser)  
tries
to guess, edge-by-edge, the path that second player (the Chooser)  
takes through
a directed graph. At each step, the Guesser makes a wager as to the  
correctness
of her guess, and receives a payoff proportional to her wager if she is
correct. We derive optimal strategies for both players for various  
classes of
graphs, and describe the Markov-chain dynamics of the game under  
optimal play.
These results are applied to the infinite-duration Lying Oracle Game,  
in which
the Guesser must use information provided by an unreliable Oracle to  
predict
the outcome of a coin toss.


  http://arxiv.org/abs/0907.2196

---------------------------------------------------------------

8787. ON THE PHILOSOPHY OF CRAM\'ER-RAO-BHATTACHARYA INEQUALITIES IN  
QUANTUM  STATISTICS

K. R. Parthasarathy

To any parametric family of states of a finite level quantum system we
associate a space of Fisher maps and introduce the natural notions of
Cram\'er-Rao-Bhattacharya tensor and Fisher information form. This  
leads us to
an abstract Cram\'er-Rao-Bhattacharya lower bound for the covariance  
matrix of
any finite number of unbiased estimators of parameteric functions. A  
number of
illustrative examples is included. Modulo technical assumptions of  
various
kinds our methods can be applied to infinite level quantum systems as  
well as
parametric families of classical probability distributions on Borel  
spaces.


  http://arxiv.org/abs/0907.2210

---------------------------------------------------------------

8788. OPTIMAL INVESTMENT ON FINITE HORIZON WITH RANDOM DISCRETE ORDER  
FLOW IN  ILLIQUID MARKETS

Paul Gassiat (PMA) and  Huyen Pham (PMA and  CREST) and  Mihai Sirbu

We study the problem of optimal portfolio selection in an illiquid  
market
with discrete order flow. In this market, bids and offers are not  
available at
any time but trading occurs more frequently near a terminal horizon. The
investor can observe and trade the risky asset only at exogenous  
random times
corresponding to the order flow given by an inhomogenous Poisson  
process. By
using a direct dynamic programming approach, we first derive and solve  
the
fixed point dynamic programming equation satisfied by the value  
function, and
then perform a verification argument which provides the existence and
characterization of optimal trading strategies. We prove the  
convergence of the
optimal performance, when the deterministic intensity of the order flow
approaches infinity at any time, to the optimal expected utility for an
investor trading continuously in a perfectly liquid market model with  
no-short
sale constraints.


  http://arxiv.org/abs/0907.2203

---------------------------------------------------------------

8789. A SHAPE THEOREM FOR RIEMANNIAN FIRST-PASSAGE PERCOLATION

Tom LaGatta and Jan Wehr

Riemannian first-passage percolation (FPP) is a continuum analogue of
standard FPP on the lattice, where the discrete passage times of  
standard FPP
are replaced by a random Riemannian metric. We prove a shape theorem  
for this
model--that balls in this metric grow linearly in time--and from this  
conclude
that the metric is complete.


  http://arxiv.org/abs/0907.2228

---------------------------------------------------------------

8790. HEAVY TAIL PHENOMENON AND CONVERGENCE TO STABLE LAWS ITERATED  
LIPSCHITZ  MAPS

Mariusz Mirek

We consider the Markov chain $\{X_n^x\}_{n=0}^\infty$ on $\R^d$  
defined by
the stochastic recursion $X_{n}^{x}=\p_{\theta_{n}}(X_{n-1}^{x})$,  
starting at
$x\in\R^d$, where $\theta_{1}, \theta_{2}, ...$ are i.i.d. random  
variables
taking their values in a matric space $(\Theta, d)$ and
$\p_{\theta_{n}}:\R^d\mapsto\R^d$ are Lipschitz maps. Assume that the  
Markov
chain has a unique stationary measure $\nu$. Under appropriate  
assumptions on
$\p_{\theta_n}$ we will show that the measure $\nu$ has a heavy tail  
with the
exponent $\alpha>0$ i.e. $\nu(\{x\in\R^d: |x|>t\})\asymp t^{-\alpha}$.  
Using
this result we show that properly normalized Birkhoff sums $S_n^x= 
\sum_{k=0}^n
X_k^x$, converge in law to an $\alpha$--stable laws for $\alpha\in(0,  
2]$.


  http://arxiv.org/abs/0907.2261

---------------------------------------------------------------

8791. UNIFORM MODULUS OF CONTINUITY OF RANDOM FIELDS

Yimin Xiao

A sufficient condition for the uniform modulus of continuity of a random
field $X = \{X(t), t \in \R^N\}$ is provided. The result is applicable  
to
random fields with heavy-tailed distribution such as stable random  
fields.


  http://arxiv.org/abs/0907.2291

---------------------------------------------------------------

8792. SPECTRAL ANALYSIS OF MULTI-DIMENSIONAL SELF-SIMILAR MARKOV  
PROCESSES

N. Modarresi and S. Rezakhah

In this paper we consider a wide sense discrete scale invariant  
process with
scale $l>1$. We consider to have $T$ samples at each scale, and choose  
$\alpha$
by the equality $l=\alpha^T$. Our special scheme of sampling is to  
choose our
samples at discrete points $\alpha^k, k\in W$. So we provide a  
discrete time
wide sense scale invariant(DT-SI) process. We find the spectral  
representation
of the covariance function of such DT-SI process. By providing  
harmonic like
representation of multi-dimensional self-similar processes, spectral  
density
function of them are presented. We also consider a discrete time scale
invariance Markov(DT-SIM) process with the above scheme of sampling at  
points
$\alpha^k, k\in {\bf W}$ and show that the spectral density matrix of  
DT-SIM
process and its associated $T$-dimensional self-similar Markov process  
is fully
specified by $\{R_{j}^H(1),R_{j}^H(0),j=0, 1, ..., T-1\}$ where
$R_{j}^H(\tau)=\mathrm{Cov}\big(X(\alpha^{j+\tau}),X(\alpha^j)\big)$


  http://arxiv.org/abs/0907.2295

---------------------------------------------------------------

8793. HEAT KERNEL UPPER BOUNDS ON LONG RANGE PERCOLATION CLUSTERS

Nicholas Crawford and  Allan Sly

In this paper, we derive upper bounds for the heat kernel of the simple
random walk on the infinite cluster of a supercritical long range  
percolation
process. For any $d \geq 1$ and for any exponent $s \in (d, (d+2)  
\wedge 2d)$
giving the rate of decay of the percolation process, we show that the  
return
probability decays like $t^{-\ffrac{d}{s-d}}$ up to logarithmic  
corrections,
where $t$ denotes the time the walk is run. Moreover, our methods also  
yield
generalized bounds on the spectral gap of the dynamics and on the  
diameter of
the largest component in a box.
   Besides its intrinsic interest, the main result is needed for a  
companion
paper studying the scaling limit of simple random walk on the infinite  
cluster.


  http://arxiv.org/abs/0907.2434

---------------------------------------------------------------

8794. A GRAPH-BASED EQUILIBRIUM PROBLEM FOR THE LIMITING DISTRIBUTION  
OF  NON-INTERSECTING BROWNIAN MOTIONS AT LOW TEMPERATURE

Steven Delvaux and  Arno B.J. Kuijlaars

We consider n non-intersecting Brownian motion paths with p prescribed
starting positions at time t=0 and q prescribed ending positions at  
time t=1.
The positions of the paths at any intermediate time are a  
determinantal point
process, which in the case p=1 is equivalent to the eigenvalue  
distribution of
a random matrix from the Gaussian unitary ensemble with external  
source. For
general p and q, we show that if a temperature parameter is  
sufficiently small,
then the distribution of the Brownian paths is characterized in the  
large n
limit by a vector equilibrium problem with an interaction matrix that  
is based
on a bipartite planar graph. Our proof is based on a steepest descent  
analysis
of an associated (p+q) by (p+q) matrix valued Riemann-Hilbert problem  
whose
solution is built out of multiple orthogonal polynomials. A new  
feature of the
steepest descent analysis is a systematic opening of a large number of  
global
lenses.


  http://arxiv.org/abs/0907.2310

---------------------------------------------------------------

8795. 3-CONNECTED CORES IN RANDOM PLANAR GRAPHS

Nikolaos Fountoulakis and Konstantinos Panagiotou

The study of the structural properties of large random planar graphs has
become in recent years a field of intense research in computer science  
and
discrete mathematics. Nowadays, a random planar graph is an important  
and
challenging model for evaluating methods that are developed to study  
properties
of random graphs from classes with structural side constraints. In  
this paper
we focus on the structure of random biconnected planar graphs  
regarding the
sizes of their 3-connected building blocks, which we call cores. In  
fact, we
prove a general theorem regarding random biconnected graphs. If B_n is  
a graph
drawn uniformly at random from a class B of labeled biconnected  
graphs, then we
show that with probability 1-o(1) B_n belongs to exactly one of the  
following
categories:
   (i) Either there is a unique giant core in B_n, that is, there is a  
0 < c < 1
such that the largest core contains ~ cn vertices, and every other core
contains at most n^a vertices, where 0 < a < 1; (ii) or all cores of B_n
contain O(log n) vertices.
   Moreover, we find the critical condition that determines the  
category to
which B_n belongs, and also provide sharp concentration results for  
the counts
of cores of all sizes between 1 and n. As a corollary, we obtain that  
a random
biconnected planar graph belongs to category (i), where in particular  
c =
0.765... and a = 2/3.


  http://arxiv.org/abs/0907.2326

---------------------------------------------------------------

8796. ON DIVERGENCE FORM SPDES WITH GROWING COEFFICIENTS IN  
$W^{1}_{2}$ SPACES  WITHOUT WEIGHTS

N.V. Krylov

We consider divergence form uniformly parabolic SPDEs with bounded and
measurable leading coefficients and possibly growing lower-order  
coefficients
in the deterministic part of the equations. We look for solutions  
which are
summable to the second power with respect to the usual Lebesgue  
measure along
with their first derivatives with respect to the spatial variable.


  http://arxiv.org/abs/0907.2467

---------------------------------------------------------------

8797. ON THE STRUCTURE OF GAUSSIAN RANDOM VARIABLES

Ciprian Tudor (CES and  Samos)

We study when a given Gaussian random variable on a given probability  
space
$(\Omega, {\cal{F}}, P) $ is equal almost surely to $\beta_{1}$ where $ 
\beta $
is a Brownian motion defined on the same (or possibly extended)  
probability
space. As a consequences of this result, we prove that the  
distribution of a
random variable (satisfying in addition a certain property) in a  
finite sum of
Wiener chaoses cannot be normal. This result also allows to understand  
better
some characterization of the Gaussian variables obtained via Malliavin
calculus.


  http://arxiv.org/abs/0907.2501

---------------------------------------------------------------

8798. WEAK CONVERGENCE FOR THE STOCHASTIC HEAT EQUATION DRIVEN BY  
GAUSSIAN  WHITE NOISE

Xavier Bardina and  Maria Jolis and Lluis Quer-Sardanyons

In this paper, we consider a quasi-linear stochastic heat equation on
$[0,1]$, with Dirichlet boundary conditions and controlled by the  
space-time
white noise. We formally replace the random perturbation by a family  
of noisy
inputs depending on a parameter $n\in \mathbb{N}$ such that  
approximate the
white noise in some sense. Then, we provide sufficient conditions  
ensuring that
the real-valued {\it mild} solution of the SPDE perturbed by this  
family of
noises converges in law, in the space $\mathcal{C}([0,T]\times [0,1])$  
of
continuous functions, to the solution of the white noise driven SPDE.  
Making
use of a suitable continuous functional of the stochastic convolution  
term, we
show that it suffices to tackle the linear problem. For this, we prove  
that the
corresponding family of laws is tight and we identify the limit law by  
showing
the convergence of the finite dimensional distributions. We have also
considered two particular families of noises to that our result  
applies. The
first one involves a Poisson process in the plane and has been  
motivated by a
one-dimensional result of Stroock, which states that the family of  
processes $n
\int_0^t (-1)^{N(n^2 s)} ds$, where $N$ is a standard Poisson process,
converges in law to a Brownian motion. The second one is constructed  
in terms
of the kernels associated to the extension of Donsker's theorem to the  
plane.


  http://arxiv.org/abs/0907.2508

---------------------------------------------------------------

8799. THE DIVERSITY OF A DISTRIBUTED GENOME IN BACTERIAL POPULATIONS

F. Baumdicker and  W. R. Hess and  P. Pfaffelhuber

The distributed genome hypothesis states that the set of genes in a
population of bacteria is distributed over all individuals that belong  
to the
specific taxon. It implies that certain genes can be gained and lost  
from
generation to generation. We use the random genealogy given by a Kingman
coalescent in order to superimpose events of gene gain and loss along  
ancestral
lines. Gene gains occur at constant rate along ancestral lines. We  
assume that
gained genes have never been present in the population before. Gene  
losses
occur at a rate proportional to the number of genes present along the  
ancestral
line. In this "infinitely many genes model" we derive moments for  
several
statistics within a sample: the average number of genes per  
individual, the
average number of genes differing between individuals, the number of
incongruent pairs of genes, the total number of different genes in the  
sample
and the gene frequency spectrum. We demonstrate that the model gives a
reasonable fit with gene frequency data from marine cyanobacteria.


  http://arxiv.org/abs/0907.2572

---------------------------------------------------------------

8800. EXTREMAL SOLUTIONS FOR STOCHASTIC EQUATIONS INDEXED BY NEGATIVE  
INTEGERS  AND TAKING VALUES IN COMPACT GROUPS

Takao Hirayama and Kouji Yano

Stochastic equations indexed by negative integers and taking values in
compact groups are studied. Extremal solutions of the equations are
characterized in terms of infinite products of independent random  
variables.
This result is applied to characterize several properties of the set  
of all
solutions in terms of the law of the driving noise.


  http://arxiv.org/abs/0907.2587

---------------------------------------------------------------

8801. ON A ZERO-ONE LAW FOR THE NORM PROCESS OF TRANSIENT RANDOM WALK

Ayako Matsumoto and Kouji Yano

A zero-one law of Engelbert--Schmidt type is proven for the norm  
process of a
transient random walk. An invariance principle for random walk local  
times and
a limit version of Jeulin's lemma play key roles.


  http://arxiv.org/abs/0907.2588

---------------------------------------------------------------

8802. LOCAL LIMIT OF PACKABLE GRAPHS

Itai Benjamini and  Nicolas Curien

We adapt some of the planar results into higher dimensions. In  
particular, it
is shown that every unbiased local limit of graphs sphere packed in  
R^d is
d-parabolic (under some additional boundedness assumptions). We then  
extend
parts of the circle packing theory into higher dimensions and derive few
geometric corollaries. E.g. every infinite graph ``well'' packed in  
R^d has
either strictly positive isoperimetric (Cheeger) constant or admits  
arbitrarily
large finite sets W with boundary size which satisfies |\partial W| <
|W|^{(d-1)/d + o(1)}, were "well" is a local bounded geometry  
assumption. Some
open problems and conjectures are gathered at the end.


  http://arxiv.org/abs/0907.2609

---------------------------------------------------------------

8803. A CLT FOR THE THIRD INTEGRATED MOMENT OF BROWNIAN LOCAL TIME  
INCREMENTS

Jay Rosen

Let $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote the local  
time of
Brownian motion. Our main result is to show that for each fixed $t$ $$ 
{\int
(L^{x+h}_t- L^x_t)^3 dx-12h\int (L^{x+h}_t - L^x_t)L^x_t dx\over h^2}
\stackrel{\mathcal{L}}{\Longrightarrow}\sqrt{192}(\int  
(L^x_t)^3dx)^{1/2}\eta$$
as $h\to 0$, where $\eta$ is a normal random variable with mean zero and
variance one that is independent of $L^{x}_{t}$. This generalizes our  
previous
result for the second moment. We also explain why our approach will  
not work
for higher moments


  http://arxiv.org/abs/0907.2693

---------------------------------------------------------------

8804. STOCHASTIC TAYLOR EXPANSIONS AND HEAT KERNEL ASYMPTOTICS

Fabrice Baudoin

These notes focus on the applications of the stochastic Taylor  
expansion of
solutions of stochastic differential equations to the study of heat  
kernels in
small times. As an illustration of these methods we provide a new heat  
kernel
proof of the Chern-Gauss-Bonnet theorem.


  http://arxiv.org/abs/0907.2711

---------------------------------------------------------------

8805. EXPLICIT SOLUTIONS OF G-HEAT EQUATION WITH A CLASS OF INITIAL  
CONDITIONS  BY G-BROWNIAN MOTION

Mingshang Hu

We obtain the viscosity solution of G-heat equation with the initial
condition $\phi(x)=x^{n}$ for each integer $n\geq1$ using the method of
G-Brownian motion.


  http://arxiv.org/abs/0907.2748

---------------------------------------------------------------

8806. GENERALIZED BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS  
DRIVEN BY  L\'EVY PROCESSES WITH NON-LIPSCHITZ COEFFICIENTS

Auguste Aman (LMAI) and  Jean Marc Owo (LMAI)

We prove an existence and uniqueness result for generalized backward  
doubly
stochastic differential equations driven by L\'evy processes with non- 
Lipschitz
assumptions.


  http://arxiv.org/abs/0907.2785

---------------------------------------------------------------

8807. SHARPNESS OF THE PERCOLATION TRANSITION IN THE TWO-DIMENSIONAL  
CONTACT  PROCESS

Jacob van den Berg

For ordinary (independent) percolation on a large class of lattices it  
is
well-known that below the critical percolation parameter the cluster  
size
distribution has exponential decay, and that power-law behaviour of this
distribution can only occur at the critical value. This behaviour is  
often
called `sharpness of the percolation transition'.
   For theoretical reasons as well as motivated by applied research,  
there is an
increasing interest in percolation models with (weak) dependencies. For
instance, biologists and agricultural researchers have used (stationary
distributions of) certain two-dimensional contact-like processes to  
model
vegetation patterns in an arid landscape. In that context, occupied  
clusters
are interpreted as patches of vegetation. For some of these models it  
has been
reported in the literature that computer simulations indicate power-law
behaviour in some interval of positive length of a model parameter.  
This would
mean that in these models the percolation transition is not sharp.
   This motivated us to investigate similar questions for the ordinary  
('basic')
two-dimensional contact process with parameter the infection rate. We  
show,
using techniques from papers on Voronoi and Johnson-Mehl tessellations  
by
Bollob\'as and Riordan, that for the upper invariant measure of the  
contact
process the percolation transition is sharp.


  http://arxiv.org/abs/0907.2843

---------------------------------------------------------------

8808. CONDITIONAL LIMIT THEOREMS FOR ORDERED RANDOM WALKS

D. Denisov and  V. Wachtel

In a recent paper of Eichelsbacher and Koenig (2008) the model of  
ordered
random walks has been considered. There it has been shown that, under  
certain
moment conditions, one can construct a k-dimensional random walk  
conditioned to
stay in a strict order at all times. Moreover, they have shown that the
rescaled random walk converges to the Dyson Brownian motion. In the  
present
paper we find the optimal moment assumptions for the construction of the
conditional random walk and generalise the limit theorem for this  
conditional
process.


  http://arxiv.org/abs/0907.2854

---------------------------------------------------------------

8809. ON SOJOURN TIMES IN THE FINITE CAPACITY $M/M/1$ QUEUE WITH  
PROCESSOR  SHARING

Qiang Zhen and  Charles Knessl

We consider a processor shared $M/M/1$ queue that can accommodate at  
most a
finite number $K$ of customers. We give an exact expression for the  
sojourn
time distribution in the finite capacity model, in terms of a Laplace
transform. We then give the tail behavior, for the limit $K\to\infty$,  
by
locating the dominant singularity of the Laplace transform.


  http://arxiv.org/abs/0907.2908

---------------------------------------------------------------

8810. CORRELATION AND BRASCAMP-LIEB INEQUALITIES FOR MARKOV SEMIGROUPS

F. Barthe and  D. Cordero-Erausquin and  M. Ledoux and  B. Maurey

This paper builds upon several recent works, where semigroup proofs of
Brascamp-Lieb inequalities are provided in various settings (Euclidean  
space,
spheres and symmetric groups). Our aim is twofold. Firstly, we provide a
general, unifying, framework based on Markov generators, in order to  
cover a
variety of examples of interest going beyond previous investigations.  
Secondly,
we put forward the combinatorial reasons for which unexpected  
exponents occur
in these inequalities.


  http://arxiv.org/abs/0907.2858

---------------------------------------------------------------

8811. THE GEOMETRY OF EUCLIDEAN CONVOLUTION INEQUALITIES AND ENTROPY

Dario Cordero-Erausquin and  Michel Ledoux

The goal of this note is to show that some convolution type  
inequalities from
Harmonic Analysis and Information Theory, such as Young's convolution
inequality (with sharp constant), Nelson's hypercontractivity of the  
Hermite
semi-group or Shannon's inequality, can be reduced to a simple  
geometric study
of frames of $\R^2$. We shall derive directly entropic inequalities,  
which were
recently proved to be dual to the Brascamp-Lieb convolution type  
inequalities.


  http://arxiv.org/abs/0907.2861

---------------------------------------------------------------

8812. ASYMPTOTIC EXPANSIONS FOR THE CONDITIONAL SOJOURN TIME  
DISTRIBUTION IN  THE $M/M/1$-PS QUEUE

Qiang Zhen and  Charles Knessl

We consider the $M/M/1$ queue with processor sharing. We study the
conditional sojourn time distribution, conditioned on the customer's  
service
requirement, in various asymptotic limits. These include large time  
and/or
large service request, and heavy traffic, where the arrival rate is only
slightly less than the service rate. The asymptotic formulas relate  
to, and
extend, some results of Morrison \cite{MO} and Flatto \cite{FL}.


  http://arxiv.org/abs/0907.2910

---------------------------------------------------------------

8813. WEAK APPROXIMATION OF FRACTIONAL SDES: THE DONSKER SETTING

Xavier Bardina and  Samy Tindel and  Carles Rovira

In this note, we take up the study of weak convergence for stochastic
differential equations driven by a (Liouville) fractional Brownian  
motion $B$
with Hurst parameter $H\in(1/3,1/2)$. In the current paper, we  
approximate the
$d$-dimensional fBm by the convolution of a rescaled random walk with
Liouville's kernel. We then show that the corresponding differential  
equation
converges in law to a fractional SDE driven by $B$.


  http://arxiv.org/abs/0907.3030

---------------------------------------------------------------

8814. BOOTSTRAP PERCOLATION IN HIGH DIMENSIONS

Jozsef Balogh and  Bela Bollobas and  Robert Morris

In r-neighbour bootstrap percolation on a graph G, a set of initially
infected vertices A \subset V(G) is chosen independently at random, with
density p, and new vertices are subsequently infected if they have at  
least r
infected neighbours. The set A is said to percolate if eventually all  
vertices
are infected. Our aim is to understand this process on the grid,  
[n]^d, for
arbitrary functions n = n(t), d = d(t) and r = r(t), as t -> infinity.  
The main
question is to determine the critical probability p_c([n]^d,r) at which
percolation becomes likely, and to give bounds on the size of the  
critical
window. In this paper we study this problem when r = 2, for all  
functions n and
d satisfying d \gg log n.
   The bootstrap process has been extensively studied on [n]^d when d  
is a fixed
constant and 2 \le r \le d, and in these cases p_c([n]^d,r) has  
recently been
determined up to a factor of 1 + o(1) as n -> infinity. At the other  
end of the
scale, Balogh and Bollobas determined p_c([2]^d,2) up to a constant  
factor, and
Balogh, Bollobas and Morris determined p_c([n]^d,d) asymptotically if  
d > (log
log n)^{2+\eps}, and gave much sharper bounds for the hypercube.
   Here we prove the following result: let x be the smallest positive  
root of
the equation \sum_{k=0}^\infty (-1)^k x^k / (2^{k^2-k} k!) = 0, so x  
\approx
1.166. Then
   (16x/d^2 + (log d)/d^{5/2) 2^{-2\sqrt{d}} < p_c([2]^d,2) < (16x/d^2  
+ 5(log
d)^2/d^{5/2}) 2^{-2\sqrt{d}} if d is sufficiently large, and moreover we
determine a sharp threshold for the critical probability p_c([n]^d,2)  
for every
function n = n(d) with d \gg log n.


  http://arxiv.org/abs/0907.3097

---------------------------------------------------------------

8815. THE MAXWELL-BOLTZMANN DISTRIBUTION IS NOT THE EQUILIBRIUM ON A   
HYPERBOLOID

S. G. Rajeev

We give a geometric formulation of the Fokker-Planck-Kramer equations  
for a
particle moving on a Lie algebra under the influence of a dissipative  
and a
random force. Special cases of interest are fluid mechanics, the  
Stochastic
Loewner Equation and the rigid body. We find that the Boltzmann  
distribution,
although a static solution, is not normalizable when the algebra is not
unimodular. This is because the invariant measure of integration in  
momentum
space is not the standard one. We solve the special case of the upper
half-plane (hyperboloid) explicitly: there is another equilibrium  
solution to
the Fokker-Planck equation, which is integrable. It breaks rotation  
invariance;
moreover, the most likely value for velocity is not zero.


  http://arxiv.org/abs/0907.2401

---------------------------------------------------------------

8816. FROM A DICHOTOMY FOR IMAGES TO HAAGERUP'S INEQUALITY

Iosif Pinelis

Let X be a compact topological space, and let D be a subset of X. Let  
Y be a
Hausdorff topological space. Let f be a continuous map of the closure  
of D to Y
such that f(D) is open. Let E be any connected subset of the  
complement (to Y)
of the boundary of D. Then f(D) either contains E or is contained in the
complement of E.
   Applications of this dichotomy principle are given, in particular for
holomorphic maps, including maximum and minimum modulus principles, an  
inverse
boundary correspondence, and a proof of Haagerup's inequality for the  
absolute
power moments of linear combinations of independent Rademacher random
variables.


  http://arxiv.org/abs/0907.2960

---------------------------------------------------------------

8817. LARGE DEVIATIONS FOR FLOWS OF INTERACTING BROWNIAN MOTIONS

A.A.Dorogovtsev and  O.V.Ostapenko

We establish the large deviation principle (LDP) for stochastic flows of
interacting Brownian motions. In particular, we consider smoothly  
correlated
flows, coalescing flows and Brownian motion stopped at a hitting moment.


  http://arxiv.org/abs/0907.3207

---------------------------------------------------------------

8818. SCALING LIMITS OF RANDOM PLANAR MAPS WITH LARGE FACES

Jean-Fran\c{c}ois Le Gall (LM-Orsay) and  Gr\'egory Miermont (DMA)

We discuss asymptotics for large random planar maps under the  
assumption that
the distribution of the degree of a typical face is in the domain of  
attraction
of a stable distribution with index $\alpha\in(1,2)$. When the number  
$n$ of
vertices of the map tends to infinity, the asymptotic behavior of  
distances
from a distinguished vertex is described by a random process called the
continuous distance process, which can be constructed from a centered  
stable
process with no negative jumps and index $\alpha$. In particular, the  
profile
of distances in the map, rescaled by the factor $n^{?1/2\alpha}$,  
converges to
a random measure defined in terms of the distance process. With the same
rescaling of distances, the vertex set viewed as a metric space  
converges in
distribution as $n\to\infty$, at least along suitable subsequences,  
towards a
limiting random compact metric space whose Hausdorff dimension is  
equal to
$2\alpha$.


  http://arxiv.org/abs/0907.3262

---------------------------------------------------------------

8819. Q-EXCHANGEABILITY VIA QUASI-INVARIANCE

Alexander Gnedin and Grigori Olshanski

For positive q, the q-exchangeability is introduced as quasi- 
invariance under
permutations, with a special cocycle. This allows us to extend the q- 
analogue
of de Finetti's theorem for binary sequences (arXiv:0905.0367) to the  
general
real-valued sequences. In contrast to the classical case with q=1, the  
order on
the reals plays for the q-analogues a significant role. An explicit
construction of ergodic q-exchangeable measures involves a random  
shuffling of
the set N={1,2,..} by iteration of the geometric choice. For q  
distinct from 1,
the shuffling yields a probability measure Q that is supported by the  
group of
bijections of N, and has the property of quasi-invariance under both  
left and
right multiplications by finite permutations. We establish connections  
of the
q-exchangeability to certain transient Markov chains on the q-Pascal  
pyramids
and to invariant random flags over the Galois fields.


  http://arxiv.org/abs/0907.3275

---------------------------------------------------------------

8820. HIGH LEVEL EXCURSION SET GEOMETRY FOR NON-GAUSSIAN INFINITELY  
DIVISIBLE  RANDOM FIELDS

Robert J Adler and  Gennady Samorodnitsky and  Jonathan E Taylor

We consider smooth, infinitely divisible random fields $X(t), t\in M)$,
$M\subset \real^d$, with regularly varying L\'evy measure, and are  
interested
in the geometric characteristics of the excursion sets  
\begin{eqnarray*} A_u =
\{t\in M: X(t) >u\} \end{eqnarray*} over high levels $u$.
   For a large class of such random fields we compute the $u\to\infty$
asymptotic joint distribution of the numbers of critical points, of  
various
types, of $X$ in $A_u$, conditional on $A_u$ being non-empty. This  
allows us,
for example, to obtain the asymptotic conditional distribution of the  
Euler
characteristic of the excursion set.
   In a significant departure from the Gaussian situation, the high  
level
excursion sets for these random fields can have quite a complicated  
geometry.
Whereas in the Gaussian case non-empty excursion sets are, with high
probability, roughly ellipsoidal, in the more general infinitely  
divisible
setting almost any shape is possible.


  http://arxiv.org/abs/0907.3359

---------------------------------------------------------------

8821. DISORDER CHAOS AND MULTIPLE VALLEYS IN SPIN GLASSES

Sourav Chatterjee

We prove that the Sherrington-Kirkpatrick model of spin glasses is  
chaotic
under small perturbations of the couplings at any temperature in the  
absence of
an external field. The result is proved for two kinds of  
perturbations: (a)
distorting the couplings via Ornstein-Uhlenbeck flows, and (b)  
replacing a
small fraction of the couplings by independent copies. We further  
prove that
the S-K model exhibits multiple valleys in its energy landscape, i.e.  
there are
many states with near-minimal energy that are mutually nearly  
orthogonal. We
show that the variance of the free energy of the S-K model is  
unusually small
at any temperature. (By `unusually small' we mean that it is much  
smaller than
the number of sites; in other words, it beats the classical Gaussian
concentration inequality, a phenomenon that we call  
`superconcentration'.) We
prove that the bond overlap in the Edwards-Anderson model of spin  
glasses is
not chaotic under perturbations of the couplings, even large  
perturbations.
Lastly, we obtain sharp lower bounds on the variance of the free  
energy in the
E-A model on any bounded degree graph, generalizing a result of Wehr and
Aizenman and establishing the absence of superconcentration in this  
class of
models. Our techniques apply for the p-spin models and the Random  
Field Ising
Model as well, although we do not work out the details in these cases.


  http://arxiv.org/abs/0907.3381

---------------------------------------------------------------

8822. SPIN NEEDLETS SPECTRAL ESTIMATION

Daryl Geller and  Xiaohong Lan and Domenico Marinucci

We consider the statistical analysis of random sections of a spin fibre
bundle over the sphere. These may be thought of as random fields that  
at each
point p in $S^2$ take as a value a curve (e.g. an ellipse) living in the
tangent plane at that point $T_{p}S^2$, rather than a number as in  
ordinary
situations. The analysis of such fields is strongly motivated by  
applications,
for instance polarization experiments in Cosmology. To investigate  
such fields,
spin needlets were recently introduced by Geller and Marinucci (2008)  
and
Geller et al. (2008). We consider the use of spin needlets for spin  
angular
power spectrum estimation, in the presence of noise and missing  
observations,
and we provide Central Limit Theorem results, in the high frequency  
sense; we
discuss also tests for bias and asymmetries with an asymptotic  
justification.


  http://arxiv.org/abs/0907.3369

---------------------------------------------------------------

8823. A BIJECTION THEOREM FOR DOMINO TILING WITH DIAGONAL IMPURITIES

Fumihiko Nakano and Taizo Sadahiro

We consider the dimer problem on a non-bipartite graph $G$, where  
there are
two types of dimers one of which we regard impurities. Results of  
simulations
using Markov chain seem to indicate that impurities are tend to  
distribute on
the boundary, which we set as a conjecture. We first show that there  
is a
bijection between the set of dimer coverings on $G$ and the set of  
spanning
forests on two graphs which are made from $G$, with configuration of  
impurities
satisfying a pairing condition. This bijection can be regarded as a  
extension
of the Temperley bijection. We consider local move consisting of two
operations, and by using the bijection mentioned above, we prove local  
move
connectedness. We further obtained some bound of the number of dimer  
coverings
and the probability finding an impurity at given edge, by extending the
argument in our previous result.


  http://arxiv.org/abs/0907.3252

---------------------------------------------------------------

8824. OPTIMAL EXECUTION PROBLEM WITH MARKET IMPACT

Takashi Kato

We study the optimal execution problem in the market model in  
consideration
of market impact. First we study the discrete-time model and describe  
the value
function with respect to the trader's optimization problem. Then, by  
shortening
the intervals of execution times, we derive the value function of the
continuous-time model and study some properties of them (continuity,  
semi-group
property and the characterization as the viscosity solution of HJB.)  
We show
that the properties of the continuous-time value function vary by the  
strength
of market impact. Moreover we introduce some examples of this model,  
which tell
us that the forms of the optimal execution strategies entirely change  
according
to the amount of the security holding.


  http://arxiv.org/abs/0907.3282

---------------------------------------------------------------

8825. DE FINETTI THEOREMS FOR EASY QUANTUM GROUPS

Teodor Banica and  Stephen Curran and Roland Speicher

We study sequences of noncommutative random variables which are  
invariant
under ``quantum transformations'' coming from an orthogonal quantum  
group
satisfying the ``easiness'' condition axiomatized in our previous  
paper. For 10
easy quantum groups, we obtain de Finetti type theorems characterizing  
the
joint distribution of any infinite, quantum invariant sequence. In  
particular,
we give a new and unified proof of the classical results of de Finetti  
and
Freedman for the easy groups S_n, O_n, which is based on the  
combinatorial
theory of cumulants. We also recover the free de Finetti theorem of K 
\"ostler
and Speicher, and the characterization of operator-valued free  
semicircular
families due to Curran. We consider also finite sequences, and prove an
approximation result in the spirit of Diaconis and Freedman.


  http://arxiv.org/abs/0907.3314

---------------------------------------------------------------

8826. SRB MEASURES FOR CERTAIN MARKOV PROCESSES

Wael Bahsoun and Pawel Gora

We study Markov processes generated by iterated function systems  
(IFS). The
constituent maps of the IFS are monotonic transformations of the  
interval with
common fixed points at 0 and 1. We first obtain an upper bound on the  
number of
SRB (Sinai-Ruelle-Bowen) measures for the IFS. Then theorems are given  
to
analyze properties of the ergodic invariant measures $\delta_0$ and $ 
\delta_1$.
In particular, sufficient conditions for $\delta_0$ and/or $\delta_1$  
to be SRB
measures are given. We apply our results to asset market games.


  http://arxiv.org/abs/0907.3372

---------------------------------------------------------------

8827. OPTIMAL EXECUTION PROBLEM WITH MARKET IMPACT

Takashi Kato

We study the optimal execution problem in the market model in  
consideration
of market impact. First we study the discrete-time model and describe  
the value
function with respect to the trader's optimization problem. Then, by  
shortening
the intervals of execution times, we derive the value function of the
continuous-time model and study some properties of them (continuity,  
semi-group
property and the characterization as the viscosity solution of HJB.)  
We show
that the properties of the continuous-time value function vary by the  
strength
of market impact. Moreover we introduce some examples of this model,  
which tell
us that the forms of the optimal execution strategies entirely change  
according
to the amount of the security holding.


  http://arxiv.org/abs/0907.3282

---------------------------------------------------------------

8828. FRACTIONAL NORMAL INVERSE GAUSSIAN PROCESS

Arun Kumar and P. Vellaisamy

Normal inverse Gaussian (NIG) process was introduced by Barndorff- 
Nielsen
(1997) by subordinating Brownian motion with drift to an inverse  
Gaussian
process. Increments of NIG process are independent and stationary. In  
this
paper, we introduce dependence between the increments of NIG process, by
subordinating fractional Brownian motion to an inverse Gaussian  
process and
call it fractional normal inverse Gaussian (FNIG) process. The basic  
properties
of this process are discussed. Its marginal distributions are scale  
mixtures of
normal laws, infinitely divisible for the Hurst parameter 1/2<=H< 1  
and are
heavy tailed. First order increments of the process are stationary and  
possess
long-range dependence (LRD) property. It is shown that they have  
persistence of
signs LRD property also. A generalization of the FNIG process called n- 
FNIG
process is also discussed which allows Hurst parameter H in the  
interval (n-1,
n). Possible applications to mathematical finance and hydraulics are  
also
pointed out


  http://arxiv.org/abs/0907.3637

---------------------------------------------------------------

8829. FLOW OF DIFFEOMORPHISMS FOR SDES WITH UNBOUNDED H\"OLDER  
CONTINUOUS  DRIFT

F. Flandoli and  M. Gubinelli and  E. Priola

We consider a SDE with a smooth multiplicative non-degenerate noise  
and a
possibly unbounded Holder continuous drift term. We prove existence of  
a global
flow of diffeomorphisms by means of a special transformation of the  
drift of
Ito-Tanaka type. The proof requires non-standard elliptic estimates in  
Holder
spaces. As an application of the stochastic flow, we obtain a
Bismut-Elworthy-Li type formula for the first derivatives of the  
associated
diffusion semigroup.


  http://arxiv.org/abs/0907.3668

---------------------------------------------------------------

8830. SYSTEMS OF ONE-DIMENSIONAL RANDOM WALKS IN A COMMON RANDOM  
ENVIRONMENT

Jonathon Peterson

We consider a system of independent one-dimensional random walks in a  
common
random environment under the condition that the random walks are  
transient with
positive speed $v_P$. We give upper bounds on the quenched probability  
that at
least one of the random walks started in the interval $[An, Bn]$ has  
traveled a
distance of less than $(v_P - \epsilon)n$. This leads to both a  
uniform law of
large numbers and a hydrodynamic limit. We also identify a family of
distributions on the configuration of particles (parameterized by  
particle
density) which are stationary under the (quenched) dynamics of the  
random walks
and show that these are the limiting distributions for the system when  
started
from a certain natural collection of distributions.


  http://arxiv.org/abs/0907.3680

---------------------------------------------------------------

8831. A SPECTRAL ANALYSIS OF THE SEQUENCE OF FIRING PHASES IN  
STOCHASTIC  INTEGRATE-AND-FIRE OSCILLATORS

Peter Baxendale and John Mayberry

Integrate and fire oscillators are widely used to model the generation  
of
action potentials in neurons. In this paper, we discuss small noise  
asymptotic
results for a class of stochastic integrate and fire oscillators  
(SIFs) in
which the buildup of membrane potential in the neuron is governed by a  
Gaussian
diffusion process. To analyze this model, we study the asymptotic  
behavior of
the spectrum of the firing phase transition operator. We begin by  
proving
strong versions of a law of large numbers and central limit theorem  
for the
first passage-time of the underlying diffusion process across a  
general time
dependent boundary. Using these results, we obtain asymptotic  
approximations of
the transition operator's eigenvalues. We also discuss connections  
between our
results and earlier numerical investigations of SIFs.


  http://arxiv.org/abs/0907.3700

---------------------------------------------------------------

8832. EVOLUTION IN PREDATOR-PREY SYSTEMS

Rick Durrett and John Mayberry

We study the adaptive dynamics of predator prey systems modeled by a
dynamical system in which the characteristics are allowed to evolve by  
small
mutations. When only the prey are allowed to evolve, and the size of the
mutational change tends to 0, the system does not exhibit long term prey
coexistence and the parameters of the resident prey type converges to  
the
solution of an ODE. When only the predators are allowed to evolve,  
coexistence
of predators occurs. In this case, depending on the parameters being  
varied we
see (i) the number of coexisting predators remains tight and the  
differences of
the parameters from a reference species converge in distribution to a  
limit, or
(ii) the number of coexisting predators tends to infinity, and we  
conjecture
that the differences converge to a deterministic limit.


  http://arxiv.org/abs/0907.3702

---------------------------------------------------------------

8833. HIGH MOMENTS OF LARGE WIGNER RANDOM MATRICES AND ASYMPTOTIC  
PROPERTIES  OF THE SPECTRAL NORM

O. Khorunzhiy

We further modify the method proposed by Ya. Sinai and A. Soshnikov and
developed by A. Ruzmaikina to study the high moments of large Wigner  
random
matrices. Our result concern the asymptotic estimates of the high  
moments of
n-dimensional real symmetric random matrices whose elements have  
symmetric
distribution such that the 12+delta-th moment exists.


  http://arxiv.org/abs/0907.3743

---------------------------------------------------------------

8834. ON THE ONE DIMENSIONAL CRITICAL "LEARNING FROM NEIGHBOURS" MODEL

Antar Bandyopadhyay and  Rahul Roy and Anish Sarkar

We consider a model of a discrete time "interacting particle system"  
on the
integer line where infinitely many changes are allowed at each  
instance of
time. We describe the model using chameleons of two different colours,  
{\it
viz}., red ($R$) and blue ($B$). At each instance of time each chameleon
performs an independent but identical coin toss experiment with  
probability
$\alpha$ to decide whether to change its colour or not. If the coin  
lands head
then the creature retains its colour (this is to be interpreted as a
"success"), otherwise it observes the colours and coin tosses of its two
nearest neighbours and changes its colour only if, among its neighbors  
and
including itself, the proportion of successes of the other colour is  
larger
than the proportion of successes of its own colour. This produces a  
Markov
chain with infinite state space ${R, B}^{\Zbold}$. This model was  
first studied
by Chatterjee and Xu (2004) where different colours had different  
success
probabilities. In this work we consider the "critical" case where the  
success
probability, $\alpha$, is the same irrespective of the colour of the  
chameleon.
We show that starting from any initial translation invariant  
distribution of
colours the Markov chain converges to a limit of a single colour,  
i.e., even at
the critical case there is no "coexistence" of the two colours at the  
limit.
Moreover we show that starting with an i.i.d. colour distribution the  
limiting
distribution gives some advantage to the "underdog".


  http://arxiv.org/abs/0907.3828

---------------------------------------------------------------

8835. ON HELE-SHAW PROBLEMS ARISING AS SCALING LIMITS

Pavel Etingof

We discuss conjectural scaling limits of discrete 2-dimensional  
aggregation
models conditioned on a semi-axis considered by Levine and Peres in
arXiv:0712.3378. These are certain problems about Hele-Show flows. We  
study
moment properties of their solutions, and solve some of them using  
conformal
mappings. In particular, we predict the exact formula for the
computer-generated shape on the left side of Fig. 4 in arXiv:0712.3378.


  http://arxiv.org/abs/0907.3856

---------------------------------------------------------------

8836. WRIGHT-FISHER DIFFUSION IN ONE DIMENSION

Charles L. Epstein and Rafe Mazzeo

We analyze the diffusion processes associated to equations of Wright- 
Fisher
type in one spatial dimension. These are defined by a degenerate  
second order
operator on the interval [0, 1], where the coefficient of the second  
order term
vanishes simply at the endpoints, and the first order term is an
inward-pointing vector field. We consider various aspects of this  
problem,
motivated by applications in population genetics, including a sharp  
regularity
theory for the zero flux boundary conditions, as well as a derivation  
of the
precise asymptotics for solutions of this equation, both as t goes to  
0 and
infinity, and as x goes to 0, 1.


  http://arxiv.org/abs/0907.3881

---------------------------------------------------------------

8837. HARD CORE ENTROPY: LOWER BOUNDS

Kari Eloranta

We establish lower bounds for the entropy of the Hard Core Model on a  
few 2d
lattices $\scriptstyle {\rm {\bf L}}.$ In this model the allowed  
configurations
inside $\scriptstyle \{0,1\}^{{\rm {\bf L}}}$ are the one's in which the
nearest neighbor $\scriptstyle 1$'s are forbidden. Our method which is  
based on
a sequential fill-in scheme is unbiassed and thereby yields in principle
arbitrarily good estimates for the topological entropy. The procedure  
also
gives some detailed information on the support of the measure of maximal
entropy.


  http://arxiv.org/abs/0907.4035

---------------------------------------------------------------

8838. BINOMIAL APPROXIMATIONS FOR BARRIER OPTIONS OF ISRAELI STYLE

Yan Dolinsky and Yuri Kifer

We show that prices and shortfall risks of game (Israeli) barrier  
options in
a sequence of binomial approximations of the Black--Scholes (BS) market
converge to the corresponding quantities for similar game barrier  
options in
the BS market with path dependent payoffs and the speed of convergence  
is
estimated, as well. The results are new also for usual American style  
options
and they are interesting from the computational point of view, as  
well, since
in binomial markets these quantities can be obtained via dynamical  
programming
algorithms. The paper continues the study of [11]and [7] but requires
substantial additional arguments in view of pecularities of barrier  
options
which, in particular, destroy the regularity of payoffs needed in the  
above
papers.


  http://arxiv.org/abs/0907.4136

---------------------------------------------------------------

8839. AN INTRODUCTION TO STOCHASTIC PDES

Martin Hairer

These notes are based on a series of lectures given first at the  
University
of Warwick in spring 2008 and then at the Courant Institute in spring  
2009. It
is an attempt to give a reasonably self-contained presentation of the  
basic
theory of stochastic partial differential equations, taking for  
granted basic
measure theory, functional analysis and probability theory, but  
nothing else.
   The approach taken in these notes is to focus on semilinear parabolic
problems driven by additive noise. These can be treated as stochastic  
evolution
equations in some infinite-dimensional Banach or Hilbert space that  
usually
have nice regularising properties and they already form a very rich  
class of
problems with many interesting properties. Furthermore, this class of  
problems
has the advantage of allowing to completely pass under silence many  
subtle
problems arising from stochastic integration in infinite-dimensional  
spaces.


  http://arxiv.org/abs/0907.4178

---------------------------------------------------------------

8840. LOCALIZATION FOR A CLASS OF LINEAR SYSTEMS

Yukio Nagahata and Nobuo Yoshida

We consider a class of continuous-time stochastic growth models on
$d$-dimensional lattice with non-negative real numbers as possible  
values per
site. The class contains examples such as binary contact path process  
and
potlatch process. We show the equivalence between the slow population  
growth
and localization property that the time integral of the replica overlap
diverges. We also prove, under reasonable assumptions, a localization  
property
in a stronger form that the spatial distribution of the population  
does not
decay uniformly in space.


  http://arxiv.org/abs/0907.4200

---------------------------------------------------------------

8841. THE RANK OF DILUTED RANDOM GRAPHS

Charles Bordenave and Marc Lelarge

We investigate the rank of the adjacency matrix of large diluted random
graphs: for a sequence of graphs converging locally to a tree, we give  
new
formulas for the asymptotic of the multiplicity of the eigenvalue 0. In
particular, the result depends only on the limiting tree structure,  
showing
that the normalized rank is 'continuous at infinity'. Our work also  
gives a new
formula for the mass at zero of the spectral measure of a Galton- 
Watson tree.
Our techniques of proofs borrow ideas from analysis of algorithms,  
random
matrix theory, statistical physics and analysis of Schrodinger  
operators on
trees.


  http://arxiv.org/abs/0907.4244

---------------------------------------------------------------

8842. HAUSDORFF MEASURE OF ARCS AND BROWNIAN MOTION ON BROWNIAN  
SPATIAL TREES

David A. Croydon

A Brownian spatial tree is defined to be a pair $(\mathcal{T},\phi)$,  
where
$\mathcal{T}$ is the rooted real tree naturally associated with a  
Brownian
excursion and $\phi$ is a random continuous function from $\mathcal{T} 
$ into
$\mathbb{R}^d$ such that, conditional on $\mathcal{T}$, $\phi$ maps  
each arc of
$\mathcal{T}$ to the image of a Brownian motion path in $\mathbb{R}^d$  
run for
a time equal to the arc length. It is shown that, in high dimensions,  
the
Hausdorff measure of arcs can be used to define an intrinsic metric
$d_{\mathcal{S}}$ on the set $\mathcal{S}:=\phi(\mathcal{T})$.  
Applications of
this result include the recovery of the spatial tree $(\mathcal{T}, 
\phi)$ from
the set $\mathcal{S}$ alone, which implies in turn that a Dawson-- 
Watanabe
super-process can be recovered from its range. Furthermore,  
$d_{\mathcal{S}}$
can be used to construct a Brownian motion on $\mathcal{S}$, which is  
proved to
be the scaling limit of simple random walks on related discrete  
structures. In
particular, a limiting result for the simple random walk on the  
branching
random walk is obtained.


  http://arxiv.org/abs/0907.4260

---------------------------------------------------------------

8843. SCALING LIMITS FOR CRITICAL INHOMOGENEOUS RANDOM GRAPHS WITH  
FINITE  THIRD MOMENTS

Shankar Bhamidi and  Remco van der Hofstad and  Johan van Leeuwaarden

We identify the scaling limits for the sizes of the largest components  
at
criticality for inhomogeneous random graphs when the degree exponent $ 
\tau$
satisfies $\tau>4$. We see that the sizes of the (rescaled) components  
converge
to the excursion lengths of an inhomogeneous Brownian motion,  
extending results
of \cite{Aldo97}. We rely heavily on martingale convergence  
techniques, and
concentration properties of (super)martingales. This paper is part of a
programme to study the critical behavior in inhomogeneous random  
graphs of
so-called rank-1 initiated in \cite{Hofs09a}.


  http://arxiv.org/abs/0907.4279

---------------------------------------------------------------

8844. TIME-REVERSAL AND ELLIPTIC BOUNDARY VALUE PROBLEMS

Zhen-Qing Chen and  Tusheng Zhang

In this paper, we prove that there exists a unique, bounded continuous  
weak
solution to the Dirichlet boundary value problem for a general class of
second-order elliptic operators with singular coefficients, which does  
not
necessarily have the maximum principle. Our method is probabilistic.  
The time
reversal of symmetric Markov processes and the theory of Dirichlet  
forms play a
crucial role in our approach.


  http://arxiv.org/abs/0907.4301

---------------------------------------------------------------

8845. NOTES ON USING CONTROL VARIATES FOR ESTIMATION WITH REVERSIBLE  
MCMC  SAMPLERS

Ioannis Kontoyiannis and  Petros Dellaportas

A general methodology is presented for the construction and effective  
use of
control variates for reversible MCMC samplers. The values of the  
coefficients
of the optimal linear combination of the control variates are  
computed, and
adaptive, consistent MCMC estimators are derived for these optimal
coefficients. All methodological and asymptotic arguments are rigorously
justified. Numerous MCMC simulation examples from Bayesian inference
applications demonstrate that the resulting variance reduction can be  
quite
dramatic.


  http://arxiv.org/abs/0907.4160

---------------------------------------------------------------

8846. THE SCALING WINDOW FOR A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE

Hamed Hatami and Michael Molloy

We consider a random graph on a given degree sequence ${\cal D}$,  
satisfying
certain conditions. We focus on two parameters $Q=Q({\cal D}),  
R=R({\cal D})$.
Molloy and Reed proved that Q=0 is the threshold for the random graph  
to have a
giant component. We prove that if $|Q|=O(n^{-1/3} R^{2/3})$ then, with  
high
probability, the size of the largest component of the random graph  
will be of
order $\Theta(n^{2/3}R^{-1/3})$. If $|Q|$ is asymptotically larger than
$n^{-1/3}R^{2/3}$ then the size of the largest component is  
asymptotically
smaller or larger than $n^{2/3}R^{-1/3}$. Thus, we establish that the  
scaling
window is $|Q|=O(n^{-1/3} R^{2/3})$.


  http://arxiv.org/abs/0907.4211

---------------------------------------------------------------

8847. DENSE PACKING ON UNIFORM LATTICES

Kari Eloranta

We study the Hard Core Model on the graphs ${\rm {\bf \scriptstyle G}}$
obtained from Archimedean tilings i.e. configurations in $\scriptstyle
\{0,1\}^{{\rm {\bf G}}}$ with the nearest neighbor 1's forbidden. Our
particular aim in choosing these graphs is to obtain insight to the  
geometry of
the densest packings in a uniform discrete set-up. We establish  
density bounds,
optimal configurations reaching them in all cases, and introduce a
probabilistic cellular automaton that generates the legal  
configurations. Its
rule involves a parameter which can be naturally characterized as  
packing
pressure. It can have a critical value but from packing point of view  
just as
interesting are the noncritical cases. These phenomena are related to  
the
exponential size of the set of densest packings and more specifically  
whether
these packings are maximally symmetric, simple laminated or  
essentially random
packings.


  http://arxiv.org/abs/0907.4247

---------------------------------------------------------------

8848. ON THE DISTRIBUTION OF A SECOND CLASS PARTICLE IN THE ASYMMETRIC  
SIMPLE  EXCLUSION PROCESS

Craig A. Tracy and Harold Widom

We give an exact expression for the distribution of the position X(t)  
of a
single second class particle in the asymmetric simple exclusion  
process (ASEP)
where initially the second class particle is located at the origin and  
the
first class particles occupy the sites {1,2,...}.


  http://arxiv.org/abs/0907.4395

---------------------------------------------------------------

8849. STEIN'S METHOD OF EXCHANGEABLE PAIRS WITH APPLICATION TO THE  
CURIE-WEISS  MODEL

Sourav Chatterjee and  Qi-Man Shao

Let $(W, W')$ be an exchangeable pair. Assume that $$E(W-W' | W) =
g(W)+r(W),$$ where $g(W)$ is a dominated term and $r(W)$ is  
negligible. Let
$G(t) = \int_0^t g(s) ds$ and define $p(t) = c_1 e^{-c_0 G(t)}$, where  
$c_0$ is
a properly chosen constant and $c_1 = 1/\int_{-\infty}^\infty p(t) dt 
$. Let $Y$
be a random variable with the probability density function $p$. It is  
proved
that $W$ converges to $Y$ in distribution when the conditional second  
moment of
$(W-W')$ given $W$ satisfies a law of large numbers. A Berry-Esseen  
type bound
is also given. We use this technique to obtain a Berry-Esseen error  
bound of
order $1/\sqrt{n}$ in the non-central limit theorem for the  
magnetization in
the Curie-Weiss ferromagnet at the critical temperature.


  http://arxiv.org/abs/0907.4450

---------------------------------------------------------------

8850. LARGE DEVIATION IN HARNACK TYPE DIRICHLET SPACES

Ann-Kathrin Jarecki

In the framework of Harnack type Dirichlet forms, we prove a large  
deviation
principle for the asymptotics of reversible Markov processes with rate  
function
given by the energy of the paths.


  http://arxiv.org/abs/0907.4479

---------------------------------------------------------------

8851. UPPER BOUND FOR LARGE DEVIATIONS OF REVERSIBLE DIFFUSION PROCESSES

Ann-Kathrin Jarecki

For a Markov process associated with a diffusion type Dirichlet form  
an upper
bound is shown for the law of the finite dimensional distributions of  
the
process. Under some more assumptions on the underlaying space this is  
also
shown for the law of the Markov process itself. In the last section we  
want to
give an application to the Wasserstein diffusion.


  http://arxiv.org/abs/0907.4483

---------------------------------------------------------------

8852. BOUNDING RELATIVE ENTROPY BY THE RELATIVE ENTROPY OF LOCAL   
SPECIFICATIONS IN PRODUCT SPACES

Katalin Marton

For a class of density functions $q^n(x^n)$ on $\Bbb R^n$ we prove an
inequality between relative entropy and the sum of average conditional  
relative
entropies of the following form: For any density function $p^n(x^n)$  
on $\Bbb
R^n$, $D(p^n||q^n)\leq Const. \sum_{i=1}^n \Bbb E D(p_i(\cdot|Y_1,...,
Y_{i-1},Y_{i+1},..., Y_n) || Q_i(\cdot|Y_1,..., Y_{i-1},Y_{i+1},...,  
Y_n)),$
where $p_i(\cdot|y_1,..., y_{i-1},y_{i+1},..., y_n)$ and $Q_i(\cdot| 
x_1,...,
x_{i-1},x_{i+1},..., x_n)$ denote the local specifications for $p^n$  
resp.
$q^n$, i.e., the conditional density functions of the $i$'th  
coordinate, given
the other coordinates. The constant depends on the properties of the  
local
specifications of $q^n$. The above inequality implies a logarithmic  
Sobolev
inequality for $q^n$. We get an explicit lower bound for the logarithmic
Sobolev constant of $q^n$ under the assumptions that: (i) the local
specifications of $q^n$ satisfy logarithmic Sobolev inequalities with  
constants
$\rho_i$, and (ii) they also satisfy some condition expressing that  
the mixed
partial derivatives of the Hamiltonian of $q^n$ are not too large  
relative to
the logarithmic Sobolev constants $\rho_i$. Condition (ii) may be  
weaker than
that used in Otto and Reznikoff's recent paper on the estimation of  
logarithmic
Sobolev constants of spin systems.


  http://arxiv.org/abs/0907.4491

---------------------------------------------------------------

8853. ON MARKOV CHAINS INDUCED BY PARTITIONED TRANSITION PROBABILITY  
MATRICES

Thomas Kaijser

Let S be a denumerable state space and let P be a transition probability
matrix on S. If a denumerable set M of nonnegative matrices is such  
that the
sum of the matrices is equal to P, then we call M a partition of P.  
Let K
denote the set of probability vectors on S. To every partition M of P  
we can
associate a transition probability function on K defined in such a way  
that if
p in K and m in M are such that ||pm|| > 0, then, with probability || 
pm|| the
vector p is transferred to the vector pm/||pm||. Here ||.|| denotes the
l_1-norm. In this paper we investigate convergence in distribution for  
Markov
chains generated by transition probability functions induced by  
partitions of
transition probability matrices. An important application of the  
convergence
results obtained is to filtering processes of partially observed  
Markov chains.


  http://arxiv.org/abs/0907.4502

---------------------------------------------------------------

8854. RETURN PROBABILITIES OF RANDOM WALKS AMONG POLYNOMIAL LOWER TAIL  
RANDOM  CONDUCTANCES

Omar Boukhadra

We study models of continuous-time, symmetric, $\Z^{d}$-valued random  
walks
in random environments, driven by a field of i.i.d. random nearest- 
neighbor
conductances $\omega_{xy}\in[0,1]$ with a power law with an exponent $ 
\gamma$
near 0. We are interested in estimating the quenched decay of the return
probability $P_\omega^{t}(0,0)$, as $t$ tends to $+\infty$. We show  
that for
$\gamma> \frac{d}{2}$, the standard bound turns out to be of the correct
logarithmic order. As an expected concequence, the same result holds  
for the
discrete-time case.


  http://arxiv.org/abs/0907.4525

---------------------------------------------------------------

8855. RECURRENCE AND TRANSIENCE OF BRANCHING RANDOM WALKS ARE  
DYNAMICALLY  STABLE

Sebastian M\"uller

Consider a sequence of i.i.d. random variables $X_n$ where each random
variable is refreshed independently according to a Poisson clock. At  
any fixed
time $t$ the law of the sequence is the same as for the sequence at  
time 0 but
at random times almost sure properties of the sequence may be  
violated. If
there are such \emph{exceptional times} we say that the property is
\emph{dynamically sensitive}, otherwise we call it \emph{dynamically  
stable}.
In this note we consider branching random walks on Cayley graphs and  
prove that
recurrence and transience are dynamically stable. Our proof combines  
techniques
from the theory of branching random walks with those of dynamical  
percolation.


  http://arxiv.org/abs/0907.4557

---------------------------------------------------------------

8856. THE T-MARTIN BOUNDARY OF REFLECTED RANDOM WALKS ON A HALF-SPACE

Irina Ignatiouk-Robert

The t-Martin boundary of a random walk on a half-space with reflected
boundary conditions is identified. It is shown in particular that the  
t-Martin
boundary of such a random walk is not stable in the following sense :  
for
different values of t, the t-Martin compactifications are not  
homeomorphic to
each other.


  http://arxiv.org/abs/0907.4592

---------------------------------------------------------------

8857. INVARIANT RANDOM FIELDS IN VECTOR BUNDLES AND APPLICATION TO  
COSMOLOGY

Anatoliy Malyarenko

We develop the theory of invariant random fields in vector bundles. The
spectral decomposition of an invariant random field in a homogeneous  
vector
bundle generated by an induced representation of a compact connected  
Lie group
$G$ is obtained. We discuss an application to the theory of cosmic  
microwave
background, where $G=SO(3)$. A theorem about equivalence of two  
different
groups of assumptions in cosmological theories is proved.


  http://arxiv.org/abs/0907.4620

---------------------------------------------------------------

8858. DISJOINT HAMILTON CYCLES IN THE RANDOM GEOMETRIC GRAPH

Xavier P\'erez-Gim\'enez and  Nicholas C. Wormald

We prove a conjecture of Penrose about the standard random geometric  
graph
process, in which n vertices are placed at random on the unit square  
and edges
are sequentially added in increasing order of lengths taken in the l_p  
norm. We
show that the first edge that makes the random geometric graph  
Hamiltonian is
a.a.s. exactly the same one that gives 2-connectivity. We also extend  
this
result to arbitrary connectivity, by proving that the first edge in  
the process
that creates a k-connected graph coincides a.a.s. with the first edge  
that
causes the graph to contain k/2 pairwise edge-disjoint Hamilton cycles  
(for
even k), or (k-1)/2 Hamilton cycles plus one perfect matching, all of  
them
pairwise edge-disjoint (for odd k).


  http://arxiv.org/abs/0907.4459

---------------------------------------------------------------

8859. LIMIT THEOREMS FOR VERTEX-REINFORCED JUMP PROCESSES ON REGULAR  
TREES

Andrea Collevecchio

Consider a vertex-reinforced jump process defined on a regular tree,  
where
each vertex has exactly $b$ children, with $b \ge 3$. We prove the  
strong law
of large numbers and the central limit theorem for the distance of the  
process
from the root. Notice that it is still unknown if vertex-reinforced jump
process is transient on the binary tree.


  http://arxiv.org/abs/0907.4854

---------------------------------------------------------------

8860. STOCHASTIC FLOWS OF SDES WITH IRREGULAR DRIFTS AND STOCHASTIC  
TRANSPORT  EQUATIONS

Xicheng Zhang

In this article we study (possibly degenerate) stochastic differential
equations (SDE) with irregular (or discontiuous) drifts, and prove  
that under
certain conditions on the coefficients, there exists a unique almost  
everywhere
stochastic invertible flow associated with the SDE in the sense of  
Lebesgue
measure. In the case of constant diffusions and BV drifts, we obtain  
such a
result by studying the related stochastic transport equation. In the  
case of
non-constant diffusions and Sobolev drifts, we use a direct method. In
particular, we extend the recent results on ODEs with non-smooth  
vector fields
to SDEs.


  http://arxiv.org/abs/0907.4866

---------------------------------------------------------------

8861. THE MONOTONE CUMULANTS

Takahiro Hasebe and Hayato Saigo

In the present paper we define the notion of generalized cumulants which
gives a universal framework for commutative, free, Boolean, and  
especially,
monotone probability theories. The uniqueness of generalized cumulants  
holds
for each independence, and hence, generalized cumulants are equal to  
the usual
cumulants in commutative, free and Boolean cases. The way we define
(generalized) cumulants is so elementary that we need neither partition
lattices nor generating functions. This new approach open the way to  
introduce
monotone cumulants and we obtain quite simple proof of central limit  
theorem
and Poisson's law of small numbers in monotone probability theory.


  http://arxiv.org/abs/0907.4896

---------------------------------------------------------------

8862. INVARIANT MEASURES AND DECAY OF CORRELATIONS OF A CLASS OF  
ERGODIC  PROBABILISTIC CELLULAR AUTOMATA

Cristian Coletti (CMCC) and  Pierre Tisseur (CMCC)

Using an extended version of the duality concept between two stochastic
processes, we give new ergodicity conditions for two states  
probabilistic
cellular automata (PCA) of any dimensions and any radius. Under these
assumptions, in the one dimensional case, we study some properties of  
the
unique invariant measure and show that it is shift mixing. Also, the  
decay of
correlation is studied in detail. In this sense, the extended concept of
duality gives exponential decay of correlation. When the extended  
concept of
duality can not be applied we are able to get, once again, exponential  
decay of
correlation using well known results from the theory of branching  
processes.


  http://arxiv.org/abs/0907.4841

---------------------------------------------------------------

8863. BAYESIAN ESTIMATE OF THE ZERO-DENSITY FREQUENCY OF A CS FOUNTAIN

D Calonico and  F Levi and  L Lorini and G Mana

Caesium fountain frequency-standards realize the second in the  
International
System of Units with a relative uncertainty approaching 10^-16. Among  
the main
contributions to the accuracy budget, cold collisions play an  
important role
because of the atomic density shift of the reference atomic  
transition. This
paper describes an application of the Bayesian analysis of the clock  
frequency
to estimate the density shift and describes how the Bayes theorem  
allows the a
priori knowledge of the sign of the collisional coefficient to be  
rigourously
embedded into the analysis. As an application, data from the INRIM  
caesium
fountain are used and the Bayesian and orthodox analyses are compared.  
The
Bayes theorem allows the orthodox uncertainty to be reduced by 28% and
demonstrates to be an important tool in primary frequency-metrology.


  http://arxiv.org/abs/0907.4849

---------------------------------------------------------------

8864. DIRICHLET POLYNOMIALS: SOME OLD AND RECENT RESULTS, AND THEIR  
INTERPLAY  IN NUMBER THEORY

Michel Weber

In the first part of the paper, we present and discuss the interplay of
Dirichlet polynomials in some classical problems of number theory,  
notably the
Lindel\"of Hypothesis. We review some typical properties of their  
means and
continue with some investigations concerning their supremum  
properties. Their
random counterpart is next considered in the second part of the paper.  
An
analysis of their supremum properties, which is entirely based on  
methods of
stochastic processes, is presented. Some complementary results and  
related
questions are included in the last section of the paper.


  http://arxiv.org/abs/0907.4931

---------------------------------------------------------------

8865. AN ANALOGUE OF THE L\'EVY-CRAM\'ER THEOREM FOR MULTI- 
DIMENSIONAL  RAYLEIGH DISTRIBUTIONS

Thu Nguyen

In the present paper we prove that every k-dimensional Cartesian  
product of
Kingman convolutions can be embedded into a k-dimensional symmetric  
convolution
(k=1, 2, ...) and obtain an analogue of the Cram\'er-L\'evy theorem for
multi-dimensional Rayleigh distributions. A new and more general class  
of
multi-dimensional Rayleygh distributions and associated higher  
dimensional
Bessel processes are introduced and studied. This class of processes  
inherits
the well-known characteristics of Brownian motions: They are independent
stationary "increments" processes with continuous sample paths.


  http://arxiv.org/abs/0907.5035

---------------------------------------------------------------

8866. THE WEAK COUPLING LIMIT OF DISORDERED COPOLYMER MODELS

Francesco Caravenna and  Giambattista Giacomin

A copolymer is a chain of repetitive units (monomers) that are almost
identical, but they differ in their degree of affinity for certain  
solvents.
This difference leads to striking phenomena when the polymer  
fluctuates in a
non-homogeneous medium, for example made up by two solvents separated  
by an
interface. One may observe, for instance, the localization of the  
polymer at
the interface between the two solvents. A discrete model of such  
system, based
on the simple symmetric random walk on Z, has been investigated in  
[Bolthausen
and den Hollander, Ann. Probab. 25 (1997), 1334-1366], notably in the  
weak
polymer-solvent coupling limit, where the convergence of the discrete  
model
toward a continuum model, based on Brownian motion, has been  
established. This
result is remarkable because it strongly suggests a universal feature of
copolymer models. In this work we prove that this is indeed the case.  
More
precisely, we determine the weak coupling limit for a general class of  
discrete
copolymer models, obtaining as limits a one-parameter (\alpha \in  
(0,1)) family
of continuum models, based on \alpha-stable regenerative sets.


  http://arxiv.org/abs/0907.5076

---------------------------------------------------------------

8867. LAW OF LARGE NUMBERS FOR THE MAXIMAL FLOW THROUGH TILTED  
CYLINDERS IN  TWO-DIMENSIONAL FIRST PASSAGE PERCOLATION

Rapha\"el Rossignol and  Marie Th\'eret

Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random  
capacities.
We prove a law of large numbers for the maximal flow crossing a  
rectangle in
$\mathbb{R}^2$ when the side lengths of the rectangle go to infinity.  
The value
of the limit depends on the asymptotic behaviour of the ratio of the  
height of
the cylinder over the length of its basis. This law of large numbers  
extends
the law of large numbers obtained by Grimmett and Kesten (1984) for  
rectangles
of particular orientation.


  http://arxiv.org/abs/0907.5112

---------------------------------------------------------------

8868. STANDARD DEVIATION OF THE LONGEST COMMON SUBSEQUENCE

J\"uri Lember and  Heinrich Matzinger

Let $L_n$ be the length of the longest common subsequence of two  
independent
i.i.d. sequences of Bernoulli variables of length $n$. We prove that  
the order
of the standard deviation of $L_n$ is $\sqrt{n}$, provided the  
parameter of the
Bernoulli variables is small enough. This validates Waterman's  
conjecture in
this situation [Philos. Trans. R. Soc. Lond. Ser. B 344 (1994)  
383--390]. The
order conjectured by Chvatal and Sankoff [J. Appl. Probab. 12 (1975)  
306--315],
however, is different.


  http://arxiv.org/abs/0907.5137

---------------------------------------------------------------

8869. BRUNET-DERRIDA PARTICLE SYSTEMS, FREE BOUNDARY PROBLEMS AND  
WIENER-HOPF  EQUATIONS

Rick Durrett and Daniel Remenik

We consider a branching-selection system in $\rr$ with $N$ particles  
which
give birth independently at rate 1 and where after each birth the  
leftmost
particle is erased, keeping the number of particles constant. We show  
that, as
$N\to\infty$, the empirical measure process associated to the system  
converges
in distribution to a deterministic measure-valued process whose  
densities solve
a free boundary integro-differential equation. We also show that this  
equation
has a unique traveling wave solution traveling at speed $c$ or no such  
solution
depending on whether $c>a$ or $c\leq a$, where $a$ is the asymptotic  
speed of
the branching random walk obtained by ignoring the removal of the  
leftmost
particles in our process. The traveling wave solutions correspond to  
solutions
of Wiener-Hopf equations.


  http://arxiv.org/abs/0907.5180

---------------------------------------------------------------

8870. ON ASEP WITH STEP BERNOULLI INITIAL CONDITION

Craig A. Tracy and  Harold Widom

This paper extends results of earlier work on ASEP to the case of step
Bernoulli initial condition. The main results are a representation in  
terms of
a Fredholm determinant for the probability distribution of a fixed  
particle,
and asymptotic results which in particular establish KPZ universality  
for this
probability in one regime. (And, as a corollary, for the current  
fluctuations.)


  http://arxiv.org/abs/0907.5192

---------------------------------------------------------------

8871. ON INFINITELY COHOMOLOGOUS TO ZERO OBSERVABLES

Amanda de Lima and Daniel Smania

We show that for a large class of piecewise expanding maps T, the  
bounded
p-variation observables u_0 that admits an infinite sequence of bounded
p-variation observables u_i satisfying u_i(x)= u_{i+1}(Tx) -u_{i+1}(x)  
are
constant. The method of the proof consists in to find a suitable  
Hilbert basis
for L^2(hm), where hm is the unique absolutely continuous invariant  
probability
of T. In terms of this basis, the action of the Perron-Frobenious and  
the
Koopan operator on L^2(hm) can be easily understood. This result  
generalizes
earlier results by Bamon, Kiwi, Rivera-Letelier and Urzua in the case  
T(x)= n x
mod 1, n in N-{0,1} and Lipchitizian observables u_0.


  http://arxiv.org/abs/0907.5013

---------------------------------------------------------------

8872. A DISCUSSION ON MEAN EXCESS PLOTS

Souvik Ghosh and  Sidney I Resnick

A widely used tool in the study of risk, insurance and extreme values  
is the
mean excess plot. One use is for validating a Generalized Pareto model  
for the
excess distribution. This paper investigates some theoretical and  
practical
aspects of the use of the mean excess plot.


  http://arxiv.org/abs/0907.5236

---------------------------------------------------------------

8873. A HISTORICAL LAW OF LARGE NUMBERS FOR THE MARCUS LUSHNIKOV PROCESS

St\'ephanie Jacquot

The Marcus-Lushnikov process is a finite stochastic particle system,  
in which
each particle is entirely characterized by its mass. Each pair of  
particles
with masses $x$ and $y$ merges into a single particle at a given rate  
$K(x,y)$.
Under certain assumptions, this process converges to the solution to
Smoluchowski equation, as the number of particles increases to  
infinity. The
Marcus-Lushnikov process gives at each time the distribution of masses  
of the
particles present in the system, but does not retain the history of  
formation
of the particles. In this paper, we set up a historical analogue of the
Marcus-Lushnikov process (built according the rules of construction of  
the
usual Markov-Lushnikov process) each time giving what we call the  
historical
tree of a particle. The historical tree of a particle present in the
Marcus-Lushnikov process at a given time $t$ encodes information about  
the
times and masses of the coagulation events that have formed that  
particle. We
prove a law of large numbers for the empirical distribution of such  
historical
trees. The limit is a natural measure on trees which is constructed  
from a
solution to Smoluchowski coagulation equation.


  http://arxiv.org/abs/0907.5305

---------------------------------------------------------------

8874. A METRIC ANALYSIS OF CRITICAL HAMILTON--JACOBI EQUATIONS IN THE   
STATIONARY ERGODIC SETTING

Andrea Davini and  Antonio Siconolfi

We adapt the metric approach to the study of stationary ergodic
Hamilton-Jacobi equations, for which a notion of admissible random
(sub)solution is defined. For any level of the Hamiltonian greater  
than or
equal to a distinguished critical value, we define an intrinsic random
semidistance and prove that an asymptotic norm does exist. Taking as  
source
region a suitable class of closed random sets, we show that the Lax  
formula
provides admissible subsolutions. This enables us to relate the  
degeneracies of
the critical stable norm to the existence/nonexistence of exact or  
approximate
critical admissible solutions.


  http://arxiv.org/abs/0907.5332

---------------------------------------------------------------

8875. WEAK KAM THEORY TOPICS IN THE STATIONARY ERGODIC SETTING

Andrea Davini and  Antonio Siconolfi

We perform a qualitative analysis of the critical equation associated  
with a
stationary ergodic Hamiltonian through a stochastic version of the  
metric
method, where the notion of closed random stationary set, issued from
stochastic geometry, plays a major role. Our purpose is to give an  
appropriate
notion of random Aubry set, to single out characterizing conditions  
for the
existence of exact or approximate correctors, and write down  
representation
formulae for them. For the last task, we make use of a Lax--type  
formula,
adapted to the stochastic environment. This material can be regarded  
as a first
step of a long--term project to develop a random analog of Weak KAM  
Theory,
generalizing what done in the periodic case or, more generally, when the
underlying space is a compact manifold.


  http://arxiv.org/abs/0907.5334

---------------------------------------------------------------

8876. PROFILES OF PERMUTATIONS

Michael Lugo

This paper develops an analogy between the cycle structure of, on the  
one
hand, random permutations with cycle lengths restricted to lie in an  
infinite
set $S$ with asymptotic density $\sigma$ and, on the other hand,  
permutations
selected according to the Ewens distribution with parameter $\sigma$. In
particular we show that the asymptotic expected number of cycles of  
random
permutations of $[n]$ with all cycles even, with all cycles odd, and  
chosen
from the Ewens distribution with parameter 1/2 are all ${1 \over 2}  
\log n +
O(1)$, and the variance is of the same order. Furthermore, we show  
that in
permutations of $[n]$ chosen from the Ewens distribution with parameter
$\sigma$, the probability of a random element being in a cycle longer  
than
$\gamma n$ approaches $(1-\gamma)^\sigma$ for large $n$. The same  
limit law
holds for permutations with cycles carrying multiplicative weights  
with average
$\sigma$. We draw parallels between the Ewens distribution and the
asymptotic-density case and explain why these parallels should exist  
using
permutations drawn from weighted Boltzmann distributions.


  http://arxiv.org/abs/0907.5351

---------------------------------------------------------------

8877. SELF-INTERACTING DIFFUSIONS IV: RATE OF CONVERGENCE

Michel Benaim (UNINE) and  Olivier Raimond (MODAL'X)

Self-interacting diffusions are processes living on a compact Riemannian
manifold defined by a stochastic differential equation with a drift term
depending on the past empirical measure of the process. The  
asymptotics of this
measure is governed by a deterministic dynamical system and under  
certain
conditions it converges almost surely towards a deterministic measure  
(see
Bena\"im, Ledoux, Raimond (2002) and Bena\"im, Raimond (2005)). We are
interested here in the rate of this convergence. A central limit  
theorem is
proved. In particular, this shows that greater is the interaction  
repelling
faster is the convergence.


  http://arxiv.org/abs/0907.5468

---------------------------------------------------------------

8878. UPPER LARGE DEVIATIONS FOR THE MAXIMAL FLOW THROUGH A DOMAIN OF   
$\MATHBB{R}^D$ IN FIRST PASSAGE PERCOLATION

Rapha\"el Cerf and Marie Th\'eret

We consider the standard first passage percolation model in the rescaled
graph $\mathbb{Z}^d/n$ for $d\geq 2$, and a domain $\Omega$ of boundary
$\Gamma$ in $\mathbb{R}^d$. Let $\Gamma^1$ and $\Gamma^2$ be two  
disjoint open
subsets of $\Gamma$, representing the parts of $\Gamma$ through which  
some
water can enter and escape from $\Omega$. We investigate the asymptotic
behaviour of the flow $\phi_n$ through a discrete version $\Omega_n$ of
$\Omega$ between the corresponding discrete sets $\Gamma^1_n$ and $ 
\Gamma^2_n$.
We prove that under some conditions on the regularity of the domain  
and on the
law of the capacity of the edges, the upper large deviations of $\phi_n/
n^{d-1}$ above a certain constant are of volume order.


  http://arxiv.org/abs/0907.5499

---------------------------------------------------------------

8879. LOWER LARGE DEVIATIONS FOR THE MAXIMAL FLOW THROUGH A DOMAIN OF   
$\MATHBB{R}^D$ IN FIRST PASSAGE PERCOLATION

Rapha\"el Cerf and Marie Th\'eret

We consider the standard first passage percolation model in the rescaled
graph $\mathbb{Z}^d/n$ for $d\geq 2$, and a domain $\Omega$ of boundary
$\Gamma$ in $\mathbb{R}^d$. Let $\Gamma^1$ and $\Gamma^2$ be two  
disjoint open
subsets of $\Gamma$, representing the parts of $\Gamma$ through which  
some
water can enter and escape from $\Omega$. We investigate the asymptotic
behaviour of the flow $\phi_n$ through a discrete version $\Omega_n$ of
$\Omega$ between the corresponding discrete sets $\Gamma^1_n$ and $ 
\Gamma^2_n$.
We prove that under some conditions on the regularity of the domain  
and on the
law of the capacity of the edges, the lower large deviations of $\phi_n/
n^{d-1}$ below a certain constant are of surface order.


  http://arxiv.org/abs/0907.5501

---------------------------------------------------------------

8880. LAW OF LARGE NUMBERS FOR THE MAXIMAL FLOW THROUGH A DOMAIN OF  $ 
\MATHBB{R}^D$ IN FIRST PASSAGE PERCOLATION

Rapha\"el Cerf and Marie Th\'eret

We consider the standard first passage percolation model in the rescaled
graph $\mathbb{Z}^d/n$ for $d\geq 2$, and a domain $\Omega$ of boundary
$\Gamma$ in $\mathbb{R}^d$. Let $\Gamma^1$ and $\Gamma^2$ be two  
disjoint open
subsets of $\Gamma$, representing the parts of $\Gamma$ through which  
some
water can enter and escape from $\Omega$. We investigate the asymptotic
behaviour of the flow $\phi_n$ through a discrete version $\Omega_n$ of
$\Omega$ between the corresponding discrete sets $\Gamma^1_n$ and $ 
\Gamma^2_n$.
We prove that under some conditions on the regularity of the domain  
and on the
law of the capacity of the edges, $\phi_n$ converges almost surely  
towards a
constant $\phi_{\Omega}$, which is the solution of a continuous non- 
random
min-cut problem. Moreover, we give a necessary and sufficient  
condition on the
law of the capacity of the edges to ensure that $\phi_{\Omega} >0$.


  http://arxiv.org/abs/0907.5504

---------------------------------------------------------------

8881. MONOTONICITY PROPERTIES OF THE ASYMPTOTIC RELATIVE EFFICIENCY  
BETWEEN  COMMON CORRELATION STATISTICS IN THE BIVARIATE NORMAL MODEL

Raymond Molzon and Iosif Pinelis

Pearson's is the most common correlation statistic, used mainly in  
parametric
settings. Most common among nonparametric correlation statistics are  
Spearman's
and Kendall's. We show that for bivariate normal i.i.d. samples the  
pairwise
asymptotic relative efficiency between these three statistics depends
monotonically on the population correlation coefficient. This  
monotonicity is a
corollary to a stronger result. The proofs rely on the use of  
l'Hospital-type
rules for monotonicity patterns.


  http://arxiv.org/abs/0907.5448

---------------------------------------------------------------

8882. CONDITIONALLY MONOTONE INDEPENDENCE

Takahiro Hasebe

We define the notion of conditionally monotone product as a part of
conditionally free product, which naturally includes monotone and  
Boolean
products. Then we define conditionally monotone cumulants which are  
useful to
calculate the limit distributions in central limit theorem and  
Poisson's law of
small numbers. Moreover, we introduce deformed convolutions arising  
from the
conditionally monotone convolution of probability measures and compute  
the
limit distributions. In order to understand the validity of cumulants,  
we
discuss what are cumulants of a given convolution product in general.


  http://arxiv.org/abs/0907.5473

---------------------------------------------------------------

8883. LOSS OF MEMORY OF RANDOM FUNCTIONS OF MARKOV CHAINS AND  
LYAPUNOV  EXPONENTS

Pierre Collet and Florencia Leonardi

In this paper we prove that the asymptotic rate of exponential loss of  
memory
of a random function of a Markov chain $(Z_{t})_{t\in\Z}$ is bounded  
above by
the difference of the first two Lyapunov exponents of a certain  
product of
matrices. We also show that this bound is in fact realized, namely for  
almost
all realization of the process $(Z_{t})_{t\in\Z}$, we can find symbols  
where
the asymptotic exponential rate of loss of memory attains the  
difference of the
first two Lyapunov exponents. This shows that the process has infinite  
memory
and leads to a lower bound on the asymptotic exponential loss of  
memory which
is saturated (and equal to the upper bound for an adequate choice of the
symbols) on a set of full measure.


  http://arxiv.org/abs/0908.0077

---------------------------------------------------------------

8884. SCALING LIMITS OF ANISOTROPIC HASTINGS-LEVITOV CLUSTERS

Fredrik Johansson and  Alan Sola and  Amanda Turner

We consider a variation of the standard Hastings-Levitov model HL(0), in
which growth is anisotropic. Two natural scaling limits are  
established and we
give precise descriptions of the effects of the anisotropy. We show  
that the
limit shapes can be realised as Loewner hulls and that the evolution of
harmonic measure on the cluster boundary can be described by the  
solution to a
deterministic ordinary differential equation related to the Loewner  
equation.
We also characterise the stochastic fluctuations around the  
deterministic limit
flow.


  http://arxiv.org/abs/0908.0086

---------------------------------------------------------------

8885. A STOCHASTIC MIN-DRIVEN COALESCENCE PROCESS AND ITS  
HYDRODYNAMICAL LIMIT

Anne-Laure Basdevant (IMT) and  Philippe Laurencot (IMT) and  James R.  
Norris  (DPMMS), Clement Rau (IMT)

A stochastic system of particles is considered in which the sizes of the
particles increase by successive binary mergers with the constraint  
that each
coagulation event involves a particle with minimal size. Convergence  
of a
suitably renormalised version of this process to a deterministic  
hydrodynamical
limit is shown and the time evolution of the minimal size is studied  
for both
deterministic and stochastic models.


  http://arxiv.org/abs/0908.0129

---------------------------------------------------------------

8886. SAMPLING CONDITIONED HYPOELLIPTIC DIFFUSIONS

Martin Hairer and  Andrew M. Stuart and  Jochen Voss

A series of recent articles introduced a method to construct stochastic
partial differential equations (SPDEs) which are invariant with  
respect to the
distribution of a given conditioned diffusion. These works are  
restricted to
the case of elliptic diffusions where the drift has a gradient  
structure, and
the resulting SPDE is of second order parabolic type.
   The present article extends this methodology to allow the  
construction of
SPDEs which are invariant with respect to the distribution of a class of
hypoelliptic diffusion processes, subject to a bridge conditioning.  
This allows
the treatment of more realistic physical models, for example one can  
use the
resulting SPDE to study transitions between meta-stable states in  
mechanical
systems with friction and noise. In this situation the restriction of  
the drift
being a gradient can also be lifted.


  http://arxiv.org/abs/0908.0162

---------------------------------------------------------------

8887. ON THE SPEED OF SPREAD FOR FRACTIONAL REACTION-DIFFUSION EQUATIONS

Hans Engler

The fractional reaction diffusion equation u_t + Au = g(u) is discussed,
where A is a fractional differential operator on the real line with  
order
\alpha between 0 and 2, the C^1 function g vanishes at 0 and 1, and  
either g is
non-negative on (0,1) or g < 0 near 0. In the case of non-negative g,  
it is
shown that solutions with initial support on the positive half axis  
spread into
the left half axis with unbounded speed if g satisfies some weak growth
condition near 0 in the case \alpha > 1, or if g is merely positive on a
sufficiently large interval near 1 in the case \alpha < 1. On the  
other hand,
it shown that solutions spread with finite speed if g'(0) < 0. The  
proofs use
comparison arguments and a new family of traveling wave solutions for  
this
class of problems.


  http://arxiv.org/abs/0908.0024

---------------------------------------------------------------

8888. A STRONG PAIR CORRELATION BOUND IMPLIES THE CLT FOR SINAI  
BILLIARDS

Mikko Stenlund

For Dynamical Systems, a strong bound on multiple correlations implies  
the
Central Limit Theorem (CLT) [ChMa]. In Chernov's paper [Ch2], such a  
bound is
derived for dynamically Holder continuous observables of dispersing  
Billiards.
Here we weaken the regularity assumption and subsequently show that  
the bound
on multiple correlations follows directly from the bound on pair  
correlations.
Thus, a strong bound on pair correlations alone implies the CLT, for a  
wider
class of observables. The result is extended to Anosov diffeomorphisms  
in any
dimension.


  http://arxiv.org/abs/0908.0027

---------------------------------------------------------------

8889. APPROXIMATING EIGENVECTORS BY SUBSAMPLING

Noureddine El Karoui and  Alexandre d'Aspremont

We show that averaging eigenvectors of randomly sampled submatrices
efficiently approximates the true eigenvectors of the original matrix  
under
certain conditions on the incoherence of the spectral decomposition.  
This
incoherence assumption is typically milder than those made in matrix  
completion
and allows eigenvectors to be sparse. We discuss applications to  
spectral
methods in dimensionality reduction and information retrieval.


  http://arxiv.org/abs/0908.0137

---------------------------------------------------------------

8890. ON THE ROLE OF SPARSITY IN COMPRESSED SENSING AND RANDOM MATRIX  
THEORY

Roman Vershynin

We discuss applications of some concepts of Compressed Sensing in the  
recent
work on invertibility of random matrices due to Rudelson and the  
author. We
sketch an argument leading to the optimal bound N^{-1/2} on the median  
of the
smallest singular value of an N by N matrix with random independent  
entries. We
highlight the parts of the argument where sparsity ideas played a key  
role.


  http://arxiv.org/abs/0908.0257

---------------------------------------------------------------

8891. LAYERING AND WETTING TRANSITIONS FOR AN SOS INTERFACE

Kenneth S. Alexander and  Fran\c{c}ois Dunlop and  Salvador Miracle-Sol 
\'e

We study the solid-on-solid interface model above a horizontal wall in  
three
dimensional space, with an attractive interaction when the interface  
is in
contact with the wall, at low temperatures. There is no bulk external  
field.
The system presents a sequence of layering transitions, whose levels  
increase
with the temperature, before reaching the wetting transition.


  http://arxiv.org/abs/0908.0321

---------------------------------------------------------------

8892. UNIVERSAL GAUSSIAN FLUCTUATIONS OF NON-HERMITIAN MATRIX ENSEMBLES

Ivan Nourdin (PMA) and  Giovanni Peccati (MODAL'X)

We prove multi-dimensional central limit theorems for the spectral  
moments
(of arbitrary degrees) associated with random matrices with real- 
valued i.i.d.
entries, satisfying some appropriate moment conditions. Our techniques  
rely on
a universality principle for the Gaussian Wiener chaos, recently  
proved by the
authors together with Gesine Reinert, as well as on some combinatorial
estimates. Unlike other related results in the probabilistic  
literature, we do
not require that the law of the entries has a density with respect to  
the
Lebesgue measure. In particular, our results apply to the ensemble of  
Bernoulli
random matrices.


  http://arxiv.org/abs/0908.0391

---------------------------------------------------------------

8893. OPTIMAL TRANSPORT AND TESSELLATION

Martin Huesmann

Optimal transport from the volume measure to a convex combination of  
Dirac
measures yields a tessellation of a Riemannian manifold into pieces of
arbitrary relative size. This tessellation is studied for the cost  
functions
$c_p(z,y)=\frac{1}{p}d^p(z,y)$ and $1\leq p<\infty$. Geometric  
descriptions of
the tessellations for all $p$ is obtained for compact subsets of the  
Euclidean
space and the sphere. For $p=1$ this approach yields Laguerre  
tessellations for
all compact Riemannian manifolds.


  http://arxiv.org/abs/0908.0442

---------------------------------------------------------------

8894. THE STATISTICAL MECHANICS OF STRETCHED POLYMERS

Dmitry Ioffe and Yvan Velenik

We describe some recent results concerning the statistical properties  
of a
self-interacting polymer stretched by an external force. We  
concentrate mainly
on the cases of purely attractive or purely repulsive self- 
interactions, but
our results are stable under suitable small perturbations of these  
pure cases.
We provide in particular a precise description of the stretched phase  
(local
limit theorems for the end-point and local observables, invariance  
principle,
microscopic structure). Our results also characterize precisely the
(non-trivial, direction-dependent) critical force needed to trigger the
collapsed/stretched phase transition in the attractive case. We also  
describe
some recent progress: first, the determination of the order of the phase
transition in the attractive case; second, a proof that a semi- 
directed polymer
in quenched random environment is diffusive in dimensions 4 and higher  
when the
temperature is high enough. In addition, we correct an incomplete  
argument from
one of our earlier works.


  http://arxiv.org/abs/0908.0452

---------------------------------------------------------------

8895. ON LINEAR EVOLUTION EQUATIONS WITH CYLINDRICAL L\'EVY NOISE

Enrico Priola and Jerzy Zabczyk

We study an infinite-dimensional
   Ornstein-Uhlenbeck process $(X_t)$ in a given
   Hilbert space $H$. This is driven by a cylindrical symmetric L\'evy  
process
without a Gaussian component and taking values in a Hilbert space $U$  
which
usually contains $H$. We give if and only if conditions under which  
$X_t$ takes
values in $H$ for some $t>0$ or for all $t>0$. Moreover, we prove
irreducibility for $(X_t)$.


  http://arxiv.org/abs/0908.0356

---------------------------------------------------------------

8896. ON THE SHORT TIME ASYMPTOTIC OF THE STOCHASTIC ALLEN-CAHN EQUATION

Hendrik Weber

A description of the short time behavior of solutions of the Allen-Cahn
equation with a smoothened additive noise is presented. The key result  
is that
in the sharp interface limit solutions move according to motion by mean
curvature with an additional stochastic forcing. This extends a  
similar result
of Funaki in spatial dimension $n=2$ to arbitrary dimensions.


  http://arxiv.org/abs/0908.0580

---------------------------------------------------------------

8897. UPPER AND LOWER BOUNDS IN EXPONENTIAL TAUBERIAN THEOREMS

Jochen Voss

In this text we study, for positive random variables, the relation  
between
the behaviour of the Laplace transform near infinity and the  
distribution near
zero. A result of de Bruijn shows that $E(e^{-\lambda X}) \sim
e^{r\sqrt{\lambda}}$ for $\lambda\to\infty$ and $P(X\leq\eps) \sim  
\e^{s/\eps}$
for $\eps\downarrow0$ are in some sense equivalent and gives a  
relation between
the constants $r$ and $s$. We illustrate how this result can be used  
to obtain
simple large deviation results. For use in more complex situations we  
also give
a generalisation of de Bruijn's result to the case when the upper and  
lower
limits are different from each other.


  http://arxiv.org/abs/0908.0642

---------------------------------------------------------------

8898. EXACT SOLUTION OF A TWO-TYPE BRANCHING PROCESS: CLONE SIZE  
DISTRIBUTION  IN CELL DIVISION KINETICS

Tibor Antal and P. L. Krapivsky

We study a two-type branching process which provides excellent  
description of
experimental data on cell dynamics in skin tissue (Clayton et al.,  
2007). The
model involves only a single type of progenitor cell, and does not  
require
support from a self-renewed population of stem cells. The progenitor  
cells
divide and may differentiate into post-mitotic cells. We derive an exact
solution of this model in terms of generating functions for the total  
number of
cells, and for the number of cells of different types. We also deduce  
large
time asymptotic behaviors drawing on our exact results, and on an  
independent
diffusion approximation.


  http://arxiv.org/abs/0908.0484

---------------------------------------------------------------

8899. RECURRENCE AND ERGODICITY OF RANDOM WALKS ON LINEAR GROUPS AND  
ON  HOMOGENEOUS SPACES

Y. Guivarc'h and C. R. E. Raja

We discuss recurrence and ergodicity properties of random walks and
associated skew products for large classes of locally compact groups and
homogeneous spaces. In particular we show that a closed subgroup of a  
product
of finitely many linear groups over local fields supports a recurrent  
random
walk if and only if it has at most quadratic growth. We give also a  
detailed
analysis of ergodicity properties for special classes of random walks on
homogeneous spaces. The structure of closed subgroups of linear groups  
over
local fields and the properties of group actions with respect to  
stationary
measures play an important role in the proofs.


  http://arxiv.org/abs/0908.0637

---------------------------------------------------------------

8900. A GENERAL STRONG LAW OF LARGE NUMBERS FOR ADDITIVE ARITHMETIC  
FUNCTIONS

Istvan Berkes and  Michel Weber

Let $f(n)$ be a strongly additive complex valued arithmetic function.  
Under
mild conditions on $f$, we prove the following weighted strong law of  
large
numbers: if $ X,X_1,X_2,... $ is any sequence of integrable i.i.d.  
random
variables, then $$ \lim_{N\to \infty} {\sum_{n=1}^N f(n) X_n \over 
\sum_{n=1}^N
f(n)} \buildrel{a.s.}\over{=} \E X . $$


  http://arxiv.org/abs/0908.0680

---------------------------------------------------------------

8901. OPTIMAL SCALINGS FOR LOCAL METROPOLIS--HASTINGS CHAINS ON  
NONPRODUCT  TARGETS IN HIGH DIMENSIONS

Alexandros Beskos and  Gareth Roberts and  Andrew Stuart

We investigate local MCMC algorithms, namely the random-walk  
Metropolis and
the Langevin algorithms, and identify the optimal choice of the local  
step-size
as a function of the dimension $n$ of the state space, asymptotically as
$n\to\infty$. We consider target distributions defined as a change of  
measure
from a product law. Such structures arise, for instance, in inverse  
problems or
Bayesian contexts when a product prior is combined with the  
likelihood. We
state analytical results on the asymptotic behavior of the algorithms  
under
general conditions on the change of measure. Our theory is motivated by
applications on conditioned diffusion processes and inverse problems  
related to
the 2D Navier--Stokes equation.


  http://arxiv.org/abs/0908.0865

---------------------------------------------------------------

8902. ASYMPTOTIC OPTIMALITY OF ISOPERIMETRIC CONSTANTS WITH RESPECT  
TO  $L^{2}(\PI)$-SPECTRAL GAPS

Achim Wuebker

In this paper we investigate the existence of $L^{2}(\pi)$-spectral  
gaps for
$\pi$-irreducible, positive recurrent Markov chains on general state  
space. We
obtain necessary and sufficient conditions for the existence of
$L^{2}(\pi)$-spectral gaps in terms of a sequence of isoperimetric  
constants
and establish their asymptotic behavior. It turns out that in some  
cases the
spectral gap can be understood in terms of convergence of an induced
probability flow to the uniform flow. The obtained theorems can be  
interpreted
as mixing results and yield sharp estimates for the spectral gap of  
some Markov
chains.


  http://arxiv.org/abs/0908.0867

---------------------------------------------------------------

8903. $L^{2}$-SPECTRAL GAPS, WEAK-REVERSIBLE AND VERY WEAK-REVERSIBLE  
MARKOV  CHAINS

Achim Wuebker and  Zakhar Kabluchko

The theory of $L^2$-spectral gaps for reversible Markov chains has been
studied by many authors. In this paper we consider positive recurrent  
general
state space Markov chains with stationary transition probabilities.  
Replacing
the assumption of reversibility by a less strong one, we still obtain  
a simple
necessary and sufficient condition for the spectral gap property of the
associated Markov operator in terms of isoperimetric constant.  
Moreover, we
define a new sequence of isoperimetric constants which provides a  
necessary and
sufficient condition for the existence of a spectral gap in a very  
general
setting. Finally, these results are used to obtain simple sufficient  
conditions
for the existence of a spectral gap in terms of the first and second  
order
transition probabilities.


  http://arxiv.org/abs/0908.0888

---------------------------------------------------------------

8904. $L^{2}$-SPECTRAL GAPS FOR TIME DISCRETE REVERSIBLE MARKOV CHAINS

Achim Wuebker

In this paper we study the spectral properties of Markov-operator on
$L^{2}$-spaces. Lawler and Sokal (Trans. Amer. Math. Soc., 1988, 309,  
pp.
557-580) used isoperimetric constants for discrete and continuous time  
Markov
chains to obtain a spectral gap at 1. For time discrete Markov chains  
this does
not exclude periodic behavior. We define a new constant measuring the  
distance
from periodicity and give necessary and sufficient conditions for the  
existence
of a global spectral gap in terms of this constant.


  http://arxiv.org/abs/0908.0897

---------------------------------------------------------------

8905. ROBUST MEAN-VARIANCE HEDGING IN THE SINGLE PERIOD MODEL

R. Tevzadze and T. Uzunashvili

We give an explicit solution of robust mean-variance hedging problem  
in the
single period model for some type of contingent claims. The alternative
approach is also considered.


  http://arxiv.org/abs/0908.0840

---------------------------------------------------------------

8906. EFFICIENT IMPORTANCE SAMPLING FOR BINARY CONTINGENCY TABLES

Jose H. Blanchet

Importance sampling has been reported to produce algorithms with  
excellent
empirical performance in counting problems. However, the theoretical  
support
for its efficiency in these applications has been very limited. In  
this paper,
we propose a methodology that can be used to design efficient importance
sampling algorithms for counting and test their efficiency rigorously.  
We apply
our techniques after transforming the problem into a rare-event  
simulation
problem--thereby connecting complexity analysis of counting problems  
with
efficiency in the context of rare-event simulation. As an illustration  
of our
approach, we consider the problem of counting the number of binary  
tables with
fixed column and row sums, $c_j$'s and $r_i$'s, respectively, and total
marginal sums $d=\sum_jc_j$. Assuming that $\max_jc_j=o(d^{1/2})$, $\sum
c_j^2=O(d)$ and the $r_j$'s are bounded, we show that a suitable  
importance
sampling algorithm, proposed by Chen et al. [J. Amer. Statist. Assoc.  
100
(2005) 109--120], requires $O(d^3\varepsilon^{-2}\delta^{-1})$  
operations to
produce an estimate that has $\varepsilon$-relative error with  
probability
$1-\delta$. In addition, if $\max_jc_j=o(d^{1/4-\delta_0})$ for some
$\delta_0>0$, the same coverage can be guaranteed with
$O(d^3\varepsilon^{-2}\log(\delta^{-1}))$ operations.


  http://arxiv.org/abs/0908.0999

---------------------------------------------------------------

8907. A PROBABILISTIC STUDY OF NEURAL COMPLEXITY

Jerome Buzzi (LM-Orsay) and  Lorenzo Zambotti (PMA)

G. Edelman, O. Sporns, and G. Tononi have introduced in theoretical  
biology
the neural complexity of a family of random variables. They have  
defined it as
a specific average of mutual information over subsystems. We show that  
their
choice of weights satisfies two natural properties, namely  
exchangeability and
additivity. This paper classifies all functionals satisfying these two
properties (which we call intricacies) in terms of probability laws on  
the unit
interval and studies the growth rate of maximal intricacies when the  
size of
the system goes to infinity. For systems of a fixed size, we show that  
the
maximizers are non-unique and that the maximal value is not approached  
by
exchangeable laws.


  http://arxiv.org/abs/0908.1006

---------------------------------------------------------------

8908. SELLING A STOCK AT THE ULTIMATE MAXIMUM

Jacques du Toit and  Goran Peskir

Assuming that the stock price $Z=(Z_t)_{0\leq t\leq T}$ follows a  
geometric
Brownian motion with drift $\mu\in\mathbb{R}$ and volatility $ 
\sigma>0$, and
letting $M_t=\max_{0\leq s\leq t}Z_s$ for $t\in[0,T]$, we consider the  
optimal
prediction problems \[V_1=\inf_{0\leq\tau\leq
T}\mathsf{E}\biggl(\frac{M_T}{Z_{\tau}}\biggr)\quadand\quad
V_2=\sup_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{Z_{\tau}}{M_T}\biggr), 
\] where
the infimum and supremum are taken over all stopping times $\tau$ of $Z 
$. We
show that the following strategy is optimal in the first problem: if $ 
\mu\leq0$
stop immediately; if $\mu\in (0,\sigma^2)$ stop as soon as $M_t/Z_t$  
hits a
specified function of time; and if $\mu\geq\sigma^2$ wait until the  
final time
$T$. By contrast we show that the following strategy is optimal in the  
second
problem: if $\mu\leq\sigma^2/2$ stop immediately, and if $\mu> 
\sigma^2/2$ wait
until the final time $T$. Both solutions support and reinforce the  
widely held
financial view that ``one should sell bad stocks and keep good ones.''  
The
method of proof makes use of parabolic free-boundary problems and local
time--space calculus techniques. The resulting inequalities are  
unusual and
interesting in their own right as they involve the future and as such  
have a
predictive element.


  http://arxiv.org/abs/0908.1014

---------------------------------------------------------------

8909. AN OPERATOR APPROACH FOR MARKOV CHAIN WEAK APPROXIMATIONS WITH  
AN  APPLICATION TO INFINITE ACTIVITY L\'{E}VY DRIVEN SDES

Hideyuki Tanaka and  Arturo Kohatsu-Higa

Weak approximations have been developed to calculate the expectation  
value of
functionals of stochastic differential equations, and various numerical
discretization schemes (Euler, Milshtein) have been studied by many  
authors. We
present a general framework based on semigroup expansions for the  
construction
of higher-order discretization schemes and analyze its rate of  
convergence. We
also apply it to approximate general L\'{e}vy driven stochastic  
differential
equations.


  http://arxiv.org/abs/0908.1021

---------------------------------------------------------------

8910. ASYMPTOTIC NORMALITY OF PLUG-IN LEVEL SET ESTIMATES

David M. Mason and  Wolfgang Polonik

We establish the asymptotic normality of the $G$-measure of the  
symmetric
difference between the level set and a plug-in-type estimator of it  
formed by
replacing the density in the definition of the level set by a kernel  
density
estimator. Our proof will highlight the efficacy of Poissonization  
methods in
the treatment of large sample theory problems of this kind.


  http://arxiv.org/abs/0908.1045

---------------------------------------------------------------

8911. GAUSSIAN PERTURBATIONS OF CIRCLE MAPS: A SPECTRAL APPROACH

John Mayberry

In this work, we examine spectral properties of Markov transition  
operators
corresponding to Gaussian perturbations of discrete time dynamical  
systems on
the circle. We develop a method for calculating asymptotic expressions  
for
eigenvalues (in the zero noise limit) and show that changes to the  
number or
period of stable orbits for the deterministic system correspond to  
changes in
the number of limiting modulus 1 eigenvalues of the transition  
operator for the
perturbed process. We call this phenomenon a $\lambda$-bifurcation.  
Asymptotic
expressions for the corresponding eigenfunctions and eigenmeasures are  
also
derived and are related to Hermite functions.


  http://arxiv.org/abs/0908.1058

---------------------------------------------------------------

8912. A CONTINUOUS ANALOGUE OF THE INVARIANCE PRINCIPLE AND ITS ALMOST  
SURE  VERSION

E.E. Permyakova

We deal with random processes obtained from a homogeneous random  
process with
independent increments by replacement of the time scale and by  
multiplication
by a norming constant. We prove the convergence in distribution of these
processes to Wiener process in Skorokhod space endowed by the topology  
of
uniform convergence. An integral type almost sure version of this  
theorem is
obtained.


  http://arxiv.org/abs/0908.1072

---------------------------------------------------------------

8913. FUNCTIONAL LIMIT THEOREMS FOR LEVY PROCESSES AND THEIR ALMOST- 
SURE  VERSIONS

E.E. Permyakova

In this paper we prove a criterion of convergence in distribution in
Skorokhod space. We apply this criterion to some special Levy  
processes and
obtain almost-sure versions of limit theorems for these processes.


  http://arxiv.org/abs/0908.1074

---------------------------------------------------------------

8914. TOTAL PROGENY IN KILLED BRANCHING RANDOM WALK

Louigi Addario-Berry and  Nicolas Broutin

We consider a branching random walk for which the maximum position of a
particle in the n'th generation, M_n, has zero speed on the linear  
scale: M_n/n
--> 0 as n --> infinity. We further remove (``kill'') any particle whose
displacement is negative, together with its entire descendence. The  
size $Z$ of
the set of un-killed particles is almost surely finite. In this paper,  
we
confirm a conjecture of Aldous that Exp[Z] < infinity while Exp[Z log
Z]=infinity. The proofs rely on precise large deviations estimates and  
ballot
theorem-style results for the sample paths of random walks.


  http://arxiv.org/abs/0908.1083

---------------------------------------------------------------

8915. ASYMPTOTIC BEHAVIOR OF THE FINITE-SIZE MAGNETIZATION AS A  
FUNCTION OF  THE SPEED OF APPROACH TO CRITICALITY

Richard S. Ellis and  Jonathan Machta and Peter Tak-Hun Otto

The main focus of this paper is to determine whether the thermodynamic
magnetization is a physically relevant estimator of the finite-size
magnetization. This is done by comparing the asymptotic behaviors of  
these two
quantities along parameter sequences converging to either a second- 
order point
or the tricritical point in the mean-field Blume-Capel model. We show  
that the
thermodynamic magnetization and the finite-size magnetization are  
asymptotic
when the parameter alpha governing the speed at which the sequence  
approaches
criticality is below a certain threshold alpha_0. However, when alpha  
exceeds
alpha_0, the thermodynamic magnetization converges to 0 much faster  
than the
finite-size magnetization. The asymptotic behavior of the finite-size
magnetization is proved via a moderate deviation principle when 0 <  
alpha <
alpha_0 and via a weak-convergence limit when alpha > alpha_0. To the  
best of
our knowledge, our results are the first rigorous confirmation of the
statistical mechanical theory of finite-size scaling for a mean-field  
model.


  http://arxiv.org/abs/0908.1103

---------------------------------------------------------------

8916. ON THE UNIQUENESS OF CLASSICAL SOLUTIONS OF CAUCHY PROBLEMS

Erhan Bayraktar and  Hao Xing

Given that the terminal condition is of at most linear growth, it is  
well
known that a Cauchy problem admits a unique classical solution when the
coefficient multiplying the second derivative (i.e., the volatility)  
is also a
function of at most linear growth. In this note, we give a condition  
on the
volatility that is necessary and sufficient for a Cauchy problem to  
admit a
unique solution.


  http://arxiv.org/abs/0908.1086

---------------------------------------------------------------

8917. CRAM\'{E}R TYPE MODERATE DEVIATION FOR THE MAXIMUM OF THE  
PERIODOGRAM  WITH APPLICATION TO SIMULTANEOUS TESTS IN GENE EXPRESSION  
TIME SERIES

Weidong Liu and  Qi Man Shao

In this paper, Cram\'{e}r type moderate deviations for the maximum of  
the
periodogram and its studentized version are derived. The results are  
then
applied to a simultaneous testing problem in gene expression time  
series. It is
shown that the level of the simultaneous tests is accurate provided  
that the
number of genes $G$ and the sample size $n$ satisfy $G=\exp(o(n^{1/3})) 
$.


  http://arxiv.org/abs/0908.1145

---------------------------------------------------------------

8918. ABSORBING-STATE PHASE TRANSITION FOR STOCHASTIC SANDPILES AND  
ACTIVATED  RANDOM WALKS

Leonardo T. Rolla and  Vladas Sidoravicius

We study the long-time behavior of conservative interacting particle  
systems
in $\mathbb Z$: The Activated Random Walk Model for reaction-diffusion  
systems
and the Stochastic Sandpile. Our main result states that both systems  
locally
fixate when the initial density of particles is small enough,  
establishing the
existence of a non-trivial phase transition in the density parameter.  
This fact
is predicted by theoretical physics arguments and supported by numerical
analysis.


  http://arxiv.org/abs/0908.1152

---------------------------------------------------------------

8919. A CIESIELSKI-TAYLOR TYPE IDENTITY FOR POSITIVE SELF-SIMILAR  
MARKOV  PROCESSES

A.E. Kyprianou and P. Patie

The aim of this note is to give a straightforward proof of a general  
version
of the Ciesielski-Taylor identity for positive self-similar Markov  
processes of
the spectrally negative type which umbrellas all previously known
Ciesielski-Taylor identities within the latter class. The approach  
makes use of
three fundamental features. Firstly a new transformation which maps a  
subset of
the family of Laplace exponents of spectrally negative L\'evy  
processes into
itself. Secondly some classical features of fluctuation theory for  
spectrally
negative L\'evy processes as well as more recent fluctuation  
identities for
positive self-similar Markov processes.


  http://arxiv.org/abs/0908.1157

---------------------------------------------------------------

8920. A SHARP ANALYSIS OF THE MIXING TIME FOR RANDOM WALK ON ROOTED  
TREES

Jason Fulman

We define an analog of Plancherel measure for the set of rooted  
unlabeled
trees on n vertices, and a Markov chain which has this measure as its
stationary distribution. Using the combinatorics of commutation  
relations, we
show that order n^2 steps are necessary and suffice for convergence to  
the
stationary distribution.


  http://arxiv.org/abs/0908.1141

---------------------------------------------------------------

8921. SHARP HEAT KERNEL ESTIMATES FOR RELATIVISTIC STABLE PROCESSES IN  
OPEN  SETS

Zhen-Qing Chen and  Panki Kim and Renming Song

In this paper, we establish sharp two-sided estimates for the transition
densities of relativistic stable processes (or equivalently, for the  
heat
kernels of the operators $m-(m^{2/\alpha}-\Delta)^{\alpha/2}$) in  
$C^{1, 1}$
open sets. The estimates are uniform in $m\in (0, M]$ for each fixed  
$M>0$.
Letting $m\downarrow 0$, the estimates given in this paper recover the
Dirichlet heat kernel estimates for $-(-\Delta)^{\alpha/2}$ in  
$C^{1,1}$-open
sets obtained in \cite{CKS}. Sharp two-sided estimates are also  
obtained for
Green functions of relativistic stable processes in half-space-like  
$C^{1,1}$
open sets and bounded $C^{1,1}$ open sets.


  http://arxiv.org/abs/0908.1509

---------------------------------------------------------------

8922. THE TWO-TYPE CONTINUUM RICHARDSON MODEL: NON-DEPENDENCE OF THE  
SURVIVAL  OF BOTH TYPES ON THE INITIAL CONFIGURATION

Sebastian Carstens and Thomas Richthammer

We consider the model of Deijfen et al. for the competing growth of two
infection types in R^d, based on the Richardson model on Z^d. Stochastic
ball-shaped infection outbursts transmit the infection type of the  
center of
the ball to all points of the ball that are not yet infected. Relevant
parameters of the model are the initial infection configuration, the
(type-dependent) growth rates and the radius distribution of the  
infection
outbursts. The main question is that of coexistence: For what values  
of the
parameters is there a positive probability that both types grow  
unboundedly? It
is known that for this question the initial configuration basically is
irrelevant, provided certain technical assumptions on the radius  
distribution
are satisfied. Here we show how to get rid of these assumptions,  
introducing a
slight generalization of the model, where immune regions and delayed  
initial
infection configurations are allowed.


  http://arxiv.org/abs/0908.1551

---------------------------------------------------------------

8923. BOUNDARY HARNACK PRINCIPLE FOR $\DELTA + \DELTA^{\ALPHA/2}$

Zhen-Qing Chen and  Panki Kim and  Renming Song and Zoran Vondra\v{c}ek

For $d\geq 1$ and $\alpha \in (0, 2)$, consider the family of pseudo
differential operators $\{\Delta+ b \Delta^{\alpha/2}; b\in [0, 1]\}$  
on $\R^d$
that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/ 
2}$. In
this paper, we establish a uniform boundary Harnack principle (BHP) with
explicit boundary decay rate for nonnegative functions which are  
harmonic with
respect to $\Delta +b \Delta^{\alpha/2}$ (or equivalently, the sum of a
Brownian motion and an independent symmetric $\alpha$-stable process  
with
constant multiple $b^{1/\alpha}$) in $C^{1, 1}$ open sets. Here a  
"uniform" BHP
means that the comparing constant in the BHP is independent of $b\in  
[0, 1]$.
Along the way, a uniform Carleson type estimate is established for  
nonnegative
functions which are harmonic with respect to $\Delta + b  
\Delta^{\alpha/2}$ in
Lipschitz open sets. Our method employs a combination of probabilistic  
and
analytic techniques.


  http://arxiv.org/abs/0908.1559

---------------------------------------------------------------

8924. CONFORMAL LOOP ENSEMBLES AND THE STRESS-ENERGY TENSOR. II.  
CONSTRUCTION  OF THE STRESS-ENERGY TENSOR

Benjamin Doyon

This is the second part of a work aimed at constructing the stress- 
energy
tensor of conformal field theory (CFT) as a local "object" in  
conformal loop
ensembles (CLE). This work lies in the wider context of re- 
constructing quantum
field theory from mathematically well-defined ensembles of random  
objects. In
the present paper, based on results of the first part, we identify the
stress-energy tensor in the dilute regime of CLE. This is done by  
deriving both
its conformal Ward identities for single insertion in CLE probability
functions, and its properties under conformal transformations  
involving the
Schwarzian derivative. We also give the one-point function of the  
stress-energy
tensor in terms of a notion of partition function, and we show that  
this agrees
with standard CFT arguments. The construction is in the same spirit as  
that
found in the context of SLE(8/3) by the author, Riva and Cardy (2006),  
which
had to do with the case of zero central charge. The present construction
generalises this to all central charges between 0 and 1, including all  
minimal
models. This generalisation is non-trivial: the application of these  
ideas to
the CLE context requires the introduction of a renormalised  
probability, and
the derivation of the transformation properties and of the one-point  
function
do not have counterparts in the SLE context.


  http://arxiv.org/abs/0908.1511

---------------------------------------------------------------

8925. THE UNIQUENESS OF SYMMETRIZING MEASURE AND LINEAR DIFFUSIONS

Xing Fang and  Jiangang Ying and  Minzhi Zhao

In this short article, we shall study one-dimensional local Dirichlet  
spaces.
One result, which has its independent interest, is to prove that  
irreducibility
implies the uniqueness of symmetrizing measure for right Markov  
processes. The
other result is to give a representation for any 1-dim local,  
irreducible and
regular Dirichlet space and a necessary and sufficient condition for a
Dirichlet space to be regular subspace of another Dirichlet space.


  http://arxiv.org/abs/0908.1607

---------------------------------------------------------------

8926. PERFECT SIMULATION OF VERVAAT PERPETUITIES

James Allen Fill and Mark Huber

We use coupling into and from the past to sample perfectly in a simple  
and
provably fast fashion from the Vervaat family of perpetuities. The  
family
includes the Dickman distribution, which arises both in number theory  
and in
the analysis of the Quickselect algorithm, which was the motivation  
for our
work.


  http://arxiv.org/abs/0908.1733

---------------------------------------------------------------

8927. STATIC LARGE DEVIATIONS OF BOUNDARY DRIVEN EXCLUSION PROCESSES

Jonathan Farfan

We prove that the stationary measure associated to a boundary driven
exclusion process in any dimension satisfies a large deviation  
principle with
rate function given by the quasi potential of the Freidlin and  
Wentzell theory.


  http://arxiv.org/abs/0908.1798

---------------------------------------------------------------

8928. LACK OF STRONG COMPLETENESS FOR STOCHASTIC FLOWS

Xue-Mei Li and Michael Scheutzow

It is well-known that a stochastic differential equation (SDE) on a  
Euclidean
space driven by a Brownian motion with Lipschitz coefficients  
generates a
stochastic flow of homeomorphisms. When the coefficients are only  
locally
Lipschitz, then a maximal continuous flow still exists but explosion  
in finite
time may occur. If -- in addition -- the coefficients grow at most  
linearly,
then this flow has the property that for each fixed initial condition  
$x$, the
solution exists for all times almost surely. If the exceptional set of  
measure
zero can be chosen independently $x$, then the maximal flow is called  
{\em
strongly complete}. The question, whether an SDE with locally Lipschitz
continuous coefficients satisfying a linear growth condition is strongly
complete was open for many years. In this paper, we construct a 2- 
dimensional
SDE with coefficients which are even bounded (and smooth) and which is  
{\em
not} strongly complete thus answering the question in the negative.


  http://arxiv.org/abs/0908.1839

---------------------------------------------------------------

8929. STEIN'S METHOD FOR DEPENDENT RANDOM VARIABLES OCCURRING IN  
STATISTICAL  MECHANICS

Peter Eichelsbacher and Matthias L\"owe

We obtain rates of convergence in limit theorems of partial sums $S_n$  
for
certain sequences of dependent, identically distributed random  
variables, which
arise naturally in statistical mechanics, in particular, in the  
context of the
Curie-Weiss models. Under appropriate assumptions there exists a real  
number
$\alpha$, a positive real number $\mu$, and a positive integer $k$  
such that
$(S_n- n \alpha)/n^{1 - 1/2k}$ converges weakly to a random variable  
with
density proportional to $\exp(-\mu |x|^{2k} /(2k)!)$. We develop  
Stein's method
for exchangeable pairs for a rich class of distributional approximations
including the Gaussian distributions as well as the non-Gaussian limit
distributions with density proportional to $\exp(-\mu |x|^{2k} /(2k)!) 
$. Our
results include the optimal Berry-Esseen rate in the Central Limit  
Theorem for
the total magnetization in the classical Curie-Weiss model, for high
temperatures as well as at the critical temperature $\beta_c=1$, where  
the
Central Limit Theorem fails. Moreover, we analyze Berry-Esseen bounds  
as the
temperature $1/ \beta_n$ converges to one and obtain a threshold for  
the speed
of this convergence. Single spin distributions satisfying the
Griffiths-Hurst-Sherman (GHS) inequality like models of liquid helium or
continuous Curie-Weiss models are considered.


  http://arxiv.org/abs/0908.1909

---------------------------------------------------------------

8930. REPLICA SYMMETRY AND COMBINATORIAL OPTIMIZATION

Johan W\"astlund

We establish the soundness of the replica symmetric ansatz (introduced  
by M.
Mezard and G. Parisi) for minimum matching and the traveling salesman  
problem
in the pseudo-dimension d mean field model for d\geq 1. The case d=1  
of minimum
matching corresponds to the pi^2/6 limit for the assignment problem  
established
by D. Aldous in 2001, and the analogous limit for the d=1 case of TSP  
was
recently established by the author with a different method. We  
introduce a
game-theoretical framework by which we prove the correctness of the
replica-cavity prediction of the corresponding limits also for d>1.


  http://arxiv.org/abs/0908.1920

---------------------------------------------------------------

8931. HIGH ORDER DISCRETIZATION SCHEMES FOR STOCHASTIC VOLATILITY MODELS

Benjamin Jourdain (CERMICS) and  Mohamed Sbai (CERMICS)

In usual stochastic volatility models, the process driving the  
volatility of
the asset price evolves according to an autonomous one-dimensional  
stochastic
differential equation. We assume that the coefficients of this  
equation are
smooth. Using It\^o's formula, we get rid, in the asset price  
dynamics, of the
stochastic integral with respect to the Brownian motion driving this  
SDE.
Taking advantage of this structure, we propose - a scheme, based on the
Milstein discretization of this SDE, with order one of weak trajectorial
convergence for the asset price, - a scheme, based on the Ninomiya- 
Victoir
discretization of this SDE, with order two of weak convergence for the  
asset
price. We also propose a specific scheme with improved convergence  
properties
when the volatility of the asset price is driven by an Orstein-Uhlenbeck
process. We confirm the theoretical rates of convergence by numerical
experiments and show that our schemes are well adapted to the  
multilevel Monte
Carlo method introduced by Giles [2008a,b].


  http://arxiv.org/abs/0908.1926

---------------------------------------------------------------

8932. FILTERING EQUATIONS FOR PARTIALLY OBSERVABLE DIFFUSION PROCESSES  
WITH  LIPSCHITZ CONTINUOUS COEFFICIENTS

N.V. Krylov

We present several results on smoothness in $L_{p}$ sense of filtering
densities under the Lipschitz continuity assumption on the  
coefficients of a
partially observable diffusion processes. We obtain them by rewriting in
divergence form filtering equation which are usually considered in  
terms of
formally adjoint to operators in nondivergence form.


  http://arxiv.org/abs/0908.1935

---------------------------------------------------------------

8933. A CHARACTERIZATION THEOREM FOR THE DISTRIBUTION OF A CONTINUOUS  
LOCAL  MARTINGALE AND RELATED LIMIT THEOREMS

Andriy Yurachkivsky

The main result of the article reads: the distribution of a continuous
starting from zero local martingale whose quadratic characteristic is  
almost
surely absolutely continuous with respect to some non-random increasing
continuous function is determined by the distribution of the quadratic
characteristic. Functional limit theorem based on this  
characterization are
proved.


  http://arxiv.org/abs/0908.1939

---------------------------------------------------------------

8934. AN APPLICATION OF DISC PACKING TO STATISTICAL MECHANICS

Matthew Kahle

We construct stable configurations of n overlapping discs of radius r  
in a
unit square, with r = O(1/n). By a result of Diaconis, Lebeau, and  
Michel, this
result is best possible, up to a constant factor. A consequence is  
that the
Metropolis algorithm, a well-studied Markov chain on the hardcore  
model, is not
irreducible in this range of parameters.


  http://arxiv.org/abs/0908.1830

---------------------------------------------------------------

8935. A COMPREHENSIVE CONNECTION BETWEEN THE BASIC RESULTS AND  
PROPERTIES  DERIVED FROM TWO KINDS OF TOPOLOGIES OF A RANDOM LOCALLY  
CONVEX MODULE

Tiexin Guo

The purpose of this paper is to make a comprehensive connection  
between the
basic results and properties derived from the two kinds of topologies  
(namely
the $(\epsilon,\lambda)-$topology introduced by the author and locally
$L^{0}-$convex topology recently introduced by Filipovi$\acute{c}$ et.  
al) of a
random locally convex module. First, we give an extremely simple proof  
of the
known Hahn-Banach extension theorem of $L^{0}-$linear functions as  
well as its
continuous variants. Then we give the essential relations between the
hyperplane separation theorems in [Filipovi$\acute{c}$ et. al, J. Funct.
Anal.256(2009)3996--4029] and a basic strict separation theorem in  
[Guo et. al,
Nonlinear Anal. 71(2009)3794--3804], and in the process obtain a  
useful and
surprising fact that a random locally convex module with the countable
concatenation property must have the same completeness under the two
topologies! Based on the relation between the two kinds of  
completeness, we
further present the central part of this paper: we prove that most of  
the
previously established deep results of random conjugate spaces of  
random normed
modules under the $(\epsilon,\lambda)-$topology are still valid under  
the
locally $L^{0}-$convex topology, which considerably enriches financial
applications of random normed modules.


  http://arxiv.org/abs/0908.1843

---------------------------------------------------------------

8936. STATIC LARGE DEVIATIONS OF BOUNDARY DRIVEN EXCLUSION PROCESSES

Jonathan Farfan

We prove that the stationary measure associated to a boundary driven
exclusion process in any dimension satisfies a large deviation  
principle with
rate function given by the quasi potential of the Freidlin and  
Wentzell theory.


  http://arxiv.org/abs/0908.1798

---------------------------------------------------------------

8937. LACK OF STRONG COMPLETENESS FOR STOCHASTIC FLOWS

Xue-Mei Li and Michael Scheutzow

It is well-known that a stochastic differential equation (SDE) on a  
Euclidean
space driven by a Brownian motion with Lipschitz coefficients  
generates a
stochastic flow of homeomorphisms. When the coefficients are only  
locally
Lipschitz, then a maximal continuous flow still exists but explosion  
in finite
time may occur. If -- in addition -- the coefficients grow at most  
linearly,
then this flow has the property that for each fixed initial condition  
$x$, the
solution exists for all times almost surely. If the exceptional set of  
measure
zero can be chosen independently $x$, then the maximal flow is called  
{\em
strongly complete}. The question, whether an SDE with locally Lipschitz
continuous coefficients satisfying a linear growth condition is strongly
complete was open for many years. In this paper, we construct a 2- 
dimensional
SDE with coefficients which are even bounded (and smooth) and which is  
{\em
not} strongly complete thus answering the question in the negative.


  http://arxiv.org/abs/0908.1839

---------------------------------------------------------------

8938. STEIN'S METHOD FOR DEPENDENT RANDOM VARIABLES OCCURRING IN  
STATISTICAL  MECHANICS

Peter Eichelsbacher and Matthias L\"owe

We obtain rates of convergence in limit theorems of partial sums $S_n$  
for
certain sequences of dependent, identically distributed random  
variables, which
arise naturally in statistical mechanics, in particular, in the  
context of the
Curie-Weiss models. Under appropriate assumptions there exists a real  
number
$\alpha$, a positive real number $\mu$, and a positive integer $k$  
such that
$(S_n- n \alpha)/n^{1 - 1/2k}$ converges weakly to a random variable  
with
density proportional to $\exp(-\mu |x|^{2k} /(2k)!)$. We develop  
Stein's method
for exchangeable pairs for a rich class of distributional approximations
including the Gaussian distributions as well as the non-Gaussian limit
distributions with density proportional to $\exp(-\mu |x|^{2k} /(2k)!) 
$. Our
results include the optimal Berry-Esseen rate in the Central Limit  
Theorem for
the total magnetization in the classical Curie-Weiss model, for high
temperatures as well as at the critical temperature $\beta_c=1$, where  
the
Central Limit Theorem fails. Moreover, we analyze Berry-Esseen bounds  
as the
temperature $1/ \beta_n$ converges to one and obtain a threshold for  
the speed
of this convergence. Single spin distributions satisfying the
Griffiths-Hurst-Sherman (GHS) inequality like models of liquid helium or
continuous Curie-Weiss models are considered.


  http://arxiv.org/abs/0908.1909

---------------------------------------------------------------

8939. REPLICA SYMMETRY AND COMBINATORIAL OPTIMIZATION

Johan W\"astlund

We establish the soundness of the replica symmetric ansatz (introduced  
by M.
Mezard and G. Parisi) for minimum matching and the traveling salesman  
problem
in the pseudo-dimension d mean field model for d\geq 1. The case d=1  
of minimum
matching corresponds to the pi^2/6 limit for the assignment problem  
established
by D. Aldous in 2001, and the analogous limit for the d=1 case of TSP  
was
recently established by the author with a different method. We  
introduce a
game-theoretical framework by which we prove the correctness of the
replica-cavity prediction of the corresponding limits also for d>1.


  http://arxiv.org/abs/0908.1920

---------------------------------------------------------------

8940. HIGH ORDER DISCRETIZATION SCHEMES FOR STOCHASTIC VOLATILITY MODELS

Benjamin Jourdain (CERMICS) and  Mohamed Sbai (CERMICS)

In usual stochastic volatility models, the process driving the  
volatility of
the asset price evolves according to an autonomous one-dimensional  
stochastic
differential equation. We assume that the coefficients of this  
equation are
smooth. Using It\^o's formula, we get rid, in the asset price  
dynamics, of the
stochastic integral with respect to the Brownian motion driving this  
SDE.
Taking advantage of this structure, we propose - a scheme, based on the
Milstein discretization of this SDE, with order one of weak trajectorial
convergence for the asset price, - a scheme, based on the Ninomiya- 
Victoir
discretization of this SDE, with order two of weak convergence for the  
asset
price. We also propose a specific scheme with improved convergence  
properties
when the volatility of the asset price is driven by an Orstein-Uhlenbeck
process. We confirm the theoretical rates of convergence by numerical
experiments and show that our schemes are well adapted to the  
multilevel Monte
Carlo method introduced by Giles [2008a,b].


  http://arxiv.org/abs/0908.1926

---------------------------------------------------------------

8941. FILTERING EQUATIONS FOR PARTIALLY OBSERVABLE DIFFUSION PROCESSES  
WITH  LIPSCHITZ CONTINUOUS COEFFICIENTS

N.V. Krylov

We present several results on smoothness in $L_{p}$ sense of filtering
densities under the Lipschitz continuity assumption on the  
coefficients of a
partially observable diffusion processes. We obtain them by rewriting in
divergence form filtering equation which are usually considered in  
terms of
formally adjoint to operators in nondivergence form.


  http://arxiv.org/abs/0908.1935

---------------------------------------------------------------

8942. A CHARACTERIZATION THEOREM FOR THE DISTRIBUTION OF A CONTINUOUS  
LOCAL  MARTINGALE AND RELATED LIMIT THEOREMS

Andriy Yurachkivsky

The main result of the article reads: the distribution of a continuous
starting from zero local martingale whose quadratic characteristic is  
almost
surely absolutely continuous with respect to some non-random increasing
continuous function is determined by the distribution of the quadratic
characteristic. Functional limit theorem based on this  
characterization are
proved.


  http://arxiv.org/abs/0908.1939

---------------------------------------------------------------

8943. AN APPLICATION OF DISC PACKING TO STATISTICAL MECHANICS

Matthew Kahle

We construct stable configurations of n overlapping discs of radius r  
in a
unit square, with r = O(1/n). By a result of Diaconis, Lebeau, and  
Michel, this
result is best possible, up to a constant factor. A consequence is  
that the
Metropolis algorithm, a well-studied Markov chain on the hardcore  
model, is not
irreducible in this range of parameters.


  http://arxiv.org/abs/0908.1830

---------------------------------------------------------------

8944. A COMPREHENSIVE CONNECTION BETWEEN THE BASIC RESULTS AND  
PROPERTIES  DERIVED FROM TWO KINDS OF TOPOLOGIES OF A RANDOM LOCALLY  
CONVEX MODULE

Tiexin Guo

The purpose of this paper is to make a comprehensive connection  
between the
basic results and properties derived from the two kinds of topologies  
(namely
the $(\epsilon,\lambda)-$topology introduced by the author and locally
$L^{0}-$convex topology recently introduced by Filipovi$\acute{c}$ et.  
al) of a
random locally convex module. First, we give an extremely simple proof  
of the
known Hahn-Banach extension theorem of $L^{0}-$linear functions as  
well as its
continuous variants. Then we give the essential relations between the
hyperplane separation theorems in [Filipovi$\acute{c}$ et. al, J. Funct.
Anal.256(2009)3996--4029] and a basic strict separation theorem in  
[Guo et. al,
Nonlinear Anal. 71(2009)3794--3804], and in the process obtain a  
useful and
surprising fact that a random locally convex module with the countable
concatenation property must have the same completeness under the two
topologies! Based on the relation between the two kinds of  
completeness, we
further present the central part of this paper: we prove that most of  
the
previously established deep results of random conjugate spaces of  
random normed
modules under the $(\epsilon,\lambda)-$topology are still valid under  
the
locally $L^{0}-$convex topology, which considerably enriches financial
applications of random normed modules.


  http://arxiv.org/abs/0908.1843

---------------------------------------------------------------

8945. RANDOM MATRICES: UNIVERSALITY OF LOCAL EIGENVALUE STATISTICS UP  
TO THE  EDGE

Terence Tao and  Van Vu

This is a continuation of our earlier paper on the universality of the
eigenvalues of Wigner random matrices. The main new results of this  
paper are
an extension of the results in that paper from the bulk of the  
spectrum up to
the edge. In particular, we prove a variant of the universality  
results of
Soshnikov for the largest eigenvalues, assuming moment conditions  
rather than
symmetry conditions. The main new technical observation is that there  
is a
significant bias in the Cauchy interlacing law near the edge of the  
spectrum
which allows one to continue ensuring the delocalization of  
eigenvectors.


  http://arxiv.org/abs/0908.1982

---------------------------------------------------------------

8946. OPTIMAL CO-ADAPTED COUPLING FOR A RANDOM WALK ON THE HYPER- 
COMPLETE-GRAP

Stephen B. Connor

Let $G_d$ be the complete graph with d vertices, and let X and Y be two
simple symmetric continuous-time random walks on the vertices of $G_d^n 
$. When
d=2, X and Y are random walks on the hypercube, for which a  
stochastically
fastest co-adapted coupling is described by Connor & Jacka (2008).  
Here we
extend this result to random walks on $G_d^n$, once again producing a
stochastically optimal coupling: as d tends to infinity we show that  
this
optimal co-adapted coupling tends to a maximal coupling.


  http://arxiv.org/abs/0908.2038

---------------------------------------------------------------

8947. RECONSTRUCTION ON TREES: EXPONENTIAL MOMENT BOUNDS FOR LINEAR  
ESTIMATORS

Yuval Peres and Sebastien Roch

Consider a Markov chain $(\xi_v)_{v \in V} \in [k]^V$ on the infinite  
$b$-ary
tree $T = (V,E)$ with irreducible edge transition matrix $M$, where $b  
\geq 2$,
$k \geq 2$ and $[k] = \{1,...,k\}$. We denote by $L_n$ the level-$n$  
vertices
of $T$. Assume $M$ has a real second-largest (in absolute value)  
eigenvalue
$\lambda$ with corresponding real eigenvector $\nu \neq 0$. Letting $ 
\sigma_v =
\nu_{\xi_v}$, we consider the following root-state estimator, which was
introduced by Mossel and Peres (2003) in the context of the  
``recontruction
problem'' on trees: \begin{equation*} S_n = (b\lambda)^{-n} \sum_{x\in  
L_n}
\sigma_x. \end{equation*} As noted by Mossel and Peres, when $b 
\lambda^2 > 1$
(the so-called Kesten-Stigum reconstruction phase) the quantity $S_n$  
has
uniformly bounded variance. Here, we give bounds on the moment- 
generating
functions of $S_n$ and $S_n^2$ when $b\lambda^2 > 1$. Our results have
implications for the inference of evolutionary trees.


  http://arxiv.org/abs/0908.2056

---------------------------------------------------------------

8948. SEQUENCE-LENGTH REQUIREMENT OF DISTANCE-BASED PHYLOGENY  
RECONSTRUCTION:  BREAKING THE POLYNOMIAL BARRIER

Sebastien Roch

We introduce a new distance-based phylogeny reconstruction technique  
which
provably achieves, at sufficiently short branch lengths, a  
polylogarithmic
sequence-length requirement -- improving significantly over previous  
polynomial
bounds for distance-based methods. The technique is based on an  
averaging
procedure that implicitly reconstructs ancestral sequences.
   In the same token, we extend previous results on phase transitions in
phylogeny reconstruction to general time-reversible models. More  
precisely, we
show that in the so-called Kesten-Stigum zone (roughly, a region of the
parameter space where ancestral sequences are well approximated by  
``linear
combinations'' of the observed sequences) sequences of length $ 
\poly(\log n)$
suffice for reconstruction when branch lengths are discretized. Here $n 
$ is the
number of extant species.
   Our results challenge, to some extent, the conventional wisdom that  
estimates
of evolutionary distances alone carry significantly less information  
about
phylogenies than full sequence datasets.


  http://arxiv.org/abs/0908.2061

---------------------------------------------------------------

8949. SHARP APPROXIMATION FOR DENSITY DEPENDENT MARKOV CHAINS

Kamil Szczegot

Consider a sequence (indexed by n) of Markov chains Z^n in R^d  
characterized
by transition kernels that approximately (in n) depend only on the  
rescaled
state n^{-1} Z^n. Subject to a smoothness condition, such a family can  
be
closely coupled on short time intervals to a Brownian motion with  
quadratic
drift. This construction is used to determine the first two terms in the
asymptotic (in n) expansion of the probability that the rescaled chain  
exits a
convex polytope. The constant term and the first correction of size  
n^{-1/6}
admit sharp characterization by solutions to associated differential  
equations
and an absolute constant. The error is smaller than O(n^{-b}) for any  
b < 1/4.
   These results are directly applied to the analysis of randomized  
algorithms
at phase transitions. In particular, the `peeling' algorithm in large  
random
hypergraphs, or equivalently the iterative decoding scheme for low- 
density
parity-check codes over the binary erasure channel is studied to  
determine the
finite size scaling behavior for irregular hypergraph ensembles.


  http://arxiv.org/abs/0908.2088

---------------------------------------------------------------

8950. A SHARP ESTIMATE FOR DIVISORS OF BERNOULLI SUMS

Michel Weber

Let $S_n=\e_1+...+\e_n$, where $ \e_i $ are i.i.d. Bernoulli r.v.'s. Let
$0\le r_d(n)<2d$ be the least residue of $n$ mod$(2d)$, $\bar r_d(n)= 2d
-r_d(n)$ and $\b(n,d)=\max ({1\over d}, {1\over \sqrt n})[e^{-  
{r_d(n)^2/2 n}}
+e^{- {\bar r_d(n)^2/2 n}}]$. We show that $$\sup_{2\le d\le n}
\big|\P\big\{d|S_n\big\}- E(n,d) \big|= {\cal O}\big({\log^{5/2} n \over
n^{3/2}}\big), $$ where $E(n,d) $ verifies $c_1\b(n,d)\le E(n,d)\le  
c_2\b(n,d)
$ and $c_1,c_2 $ are numerical constants.


  http://arxiv.org/abs/0908.2047

---------------------------------------------------------------

8951. SIMPLE ERROR SCATTERING MODEL FOR IMPROVED INFORMATION  
RECONCILIATION

Stefan Rass

Implementations of quantum key distribution as available nowadays  
suffer from
inefficiencies due to post processing of the raw key that severely  
cuts down
the final secure key rate. We present a simple model for the error  
scattering
across the raw key and derive "closed form" expressions for the  
probability of
a parity check failure, or experiencing more than some fixed number of  
errors.
Our results can serve for improvement for key establishment, as  
information
reconciliation via interactive error correction and privacy  
amplification rests
on mostly unproven assumptions. We support those hypotheses on  
statistical
grounds.


  http://arxiv.org/abs/0908.2069

---------------------------------------------------------------

8952. PROBABILISTIC MODEL ASSOCIATED WITH THE PRESSURELESS GAS DYNAMICS

Sergio Albeverio and  Anastasia Korshunova and  Olga Rozanova

Using a method of stochastic perturbation of a Langevin system  
associated
with the non-viscous Burgers equation we construct a solution to the  
Riemann
problem for the pressureless gas dynamics describing sticky particles  
dynamics.
As a bridging step we consider a medium consisting of noninteracting  
particles.
We analyze the difference in the behavior of discontinuous solutions  
for these
two models and the relations between them. In our framework in 1D case  
we
obtain a unique entropy solution to the Riemann problem. Moreover, we  
describe
how starting from smooth data a $\delta$ - singularity arises in one  
component
of the solution.


  http://arxiv.org/abs/0908.2084

---------------------------------------------------------------

8953. THE MAHONIAN PROBABILITY DISTRIBUTION ON WORDS IS ASYMPTOTICALLY  
NORMAL

E. Rodney Canfield and  Svante Janson and  and Doron Zeilberger

The Mahonian statistic is the number of inversions in a permutation of a
multiset with $a_i$ elements of type $i$, $1\le i\le m$. The counting  
function
for this statistic is the $q$ analog of the multinomial coefficient
$\binom{a_1+...+a_m}{a_1,... a_m}$, and the probability generating  
function is
the normalization of the latter. We give two proofs that the  
distribution is
asymptotically normal. The first is {\it computer-assisted}, based on  
the
method of moments. The Maple package {\tt MahonianStat}, available  
from the
webpage of this article, can be used by the reader to perform  
experiments and
calculations. Our second proof uses characteristic functions. We then  
take up
the study of a local limit theorem to accompany our central limit  
theorem. Here
our result is less general, and we must be content with a conjecture  
about
further work. Our local limit theorem permits us to conclude that the
coeffiecients of the $q$-multinomial are log-concave, provided one  
stays near
the center (where the largest coefficients reside.)


  http://arxiv.org/abs/0908.2089

---------------------------------------------------------------

8954. RANDOM MATRICES: UNIVERSALITY OF LOCAL EIGENVALUE STATISTICS UP  
TO THE  EDGE

Terence Tao and  Van Vu

This is a continuation of our earlier paper on the universality of the
eigenvalues of Wigner random matrices. The main new results of this  
paper are
an extension of the results in that paper from the bulk of the  
spectrum up to
the edge. In particular, we prove a variant of the universality  
results of
Soshnikov for the largest eigenvalues, assuming moment conditions  
rather than
symmetry conditions. The main new technical observation is that there  
is a
significant bias in the Cauchy interlacing law near the edge of the  
spectrum
which allows one to continue ensuring the delocalization of  
eigenvectors.


  http://arxiv.org/abs/0908.1982

---------------------------------------------------------------

8955. OPTIMAL CO-ADAPTED COUPLING FOR A RANDOM WALK ON THE HYPER- 
COMPLETE-GRAP

Stephen B. Connor

Let $G_d$ be the complete graph with d vertices, and let X and Y be two
simple symmetric continuous-time random walks on the vertices of $G_d^n 
$. When
d=2, X and Y are random walks on the hypercube, for which a  
stochastically
fastest co-adapted coupling is described by Connor & Jacka (2008).  
Here we
extend this result to random walks on $G_d^n$, once again producing a
stochastically optimal coupling: as d tends to infinity we show that  
this
optimal co-adapted coupling tends to a maximal coupling.


  http://arxiv.org/abs/0908.2038

---------------------------------------------------------------

8956. RECONSTRUCTION ON TREES: EXPONENTIAL MOMENT BOUNDS FOR LINEAR  
ESTIMATORS

Yuval Peres and Sebastien Roch

Consider a Markov chain $(\xi_v)_{v \in V} \in [k]^V$ on the infinite  
$b$-ary
tree $T = (V,E)$ with irreducible edge transition matrix $M$, where $b  
\geq 2$,
$k \geq 2$ and $[k] = \{1,...,k\}$. We denote by $L_n$ the level-$n$  
vertices
of $T$. Assume $M$ has a real second-largest (in absolute value)  
eigenvalue
$\lambda$ with corresponding real eigenvector $\nu \neq 0$. Letting $ 
\sigma_v =
\nu_{\xi_v}$, we consider the following root-state estimator, which was
introduced by Mossel and Peres (2003) in the context of the  
``recontruction
problem'' on trees: \begin{equation*} S_n = (b\lambda)^{-n} \sum_{x\in  
L_n}
\sigma_x. \end{equation*} As noted by Mossel and Peres, when $b 
\lambda^2 > 1$
(the so-called Kesten-Stigum reconstruction phase) the quantity $S_n$  
has
uniformly bounded variance. Here, we give bounds on the moment- 
generating
functions of $S_n$ and $S_n^2$ when $b\lambda^2 > 1$. Our results have
implications for the inference of evolutionary trees.


  http://arxiv.org/abs/0908.2056

---------------------------------------------------------------

8957. SEQUENCE-LENGTH REQUIREMENT OF DISTANCE-BASED PHYLOGENY  
RECONSTRUCTION:  BREAKING THE POLYNOMIAL BARRIER

Sebastien Roch

We introduce a new distance-based phylogeny reconstruction technique  
which
provably achieves, at sufficiently short branch lengths, a  
polylogarithmic
sequence-length requirement -- improving significantly over previous  
polynomial
bounds for distance-based methods. The technique is based on an  
averaging
procedure that implicitly reconstructs ancestral sequences.
   In the same token, we extend previous results on phase transitions in
phylogeny reconstruction to general time-reversible models. More  
precisely, we
show that in the so-called Kesten-Stigum zone (roughly, a region of the
parameter space where ancestral sequences are well approximated by  
``linear
combinations'' of the observed sequences) sequences of length $ 
\poly(\log n)$
suffice for reconstruction when branch lengths are discretized. Here $n 
$ is the
number of extant species.
   Our results challenge, to some extent, the conventional wisdom that  
estimates
of evolutionary distances alone carry significantly less information  
about
phylogenies than full sequence datasets.


  http://arxiv.org/abs/0908.2061

---------------------------------------------------------------

8958. SHARP APPROXIMATION FOR DENSITY DEPENDENT MARKOV CHAINS

Kamil Szczegot

Consider a sequence (indexed by n) of Markov chains Z^n in R^d  
characterized
by transition kernels that approximately (in n) depend only on the  
rescaled
state n^{-1} Z^n. Subject to a smoothness condition, such a family can  
be
closely coupled on short time intervals to a Brownian motion with  
quadratic
drift. This construction is used to determine the first two terms in the
asymptotic (in n) expansion of the probability that the rescaled chain  
exits a
convex polytope. The constant term and the first correction of size  
n^{-1/6}
admit sharp characterization by solutions to associated differential  
equations
and an absolute constant. The error is smaller than O(n^{-b}) for any  
b < 1/4.
   These results are directly applied to the analysis of randomized  
algorithms
at phase transitions. In particular, the `peeling' algorithm in large  
random
hypergraphs, or equivalently the iterative decoding scheme for low- 
density
parity-check codes over the binary erasure channel is studied to  
determine the
finite size scaling behavior for irregular hypergraph ensembles.


  http://arxiv.org/abs/0908.2088

---------------------------------------------------------------

8959. A SHARP ESTIMATE FOR DIVISORS OF BERNOULLI SUMS

Michel Weber

Let $S_n=\e_1+...+\e_n$, where $ \e_i $ are i.i.d. Bernoulli r.v.'s. Let
$0\le r_d(n)<2d$ be the least residue of $n$ mod$(2d)$, $\bar r_d(n)= 2d
-r_d(n)$ and $\b(n,d)=\max ({1\over d}, {1\over \sqrt n})[e^{-  
{r_d(n)^2/2 n}}
+e^{- {\bar r_d(n)^2/2 n}}]$. We show that $$\sup_{2\le d\le n}
\big|\P\big\{d|S_n\big\}- E(n,d) \big|= {\cal O}\big({\log^{5/2} n \over
n^{3/2}}\big), $$ where $E(n,d) $ verifies $c_1\b(n,d)\le E(n,d)\le  
c_2\b(n,d)
$ and $c_1,c_2 $ are numerical constants.


  http://arxiv.org/abs/0908.2047

---------------------------------------------------------------

8960. SIMPLE ERROR SCATTERING MODEL FOR IMPROVED INFORMATION  
RECONCILIATION

Stefan Rass

Implementations of quantum key distribution as available nowadays  
suffer from
inefficiencies due to post processing of the raw key that severely  
cuts down
the final secure key rate. We present a simple model for the error  
scattering
across the raw key and derive "closed form" expressions for the  
probability of
a parity check failure, or experiencing more than some fixed number of  
errors.
Our results can serve for improvement for key establishment, as  
information
reconciliation via interactive error correction and privacy  
amplification rests
on mostly unproven assumptions. We support those hypotheses on  
statistical
grounds.


  http://arxiv.org/abs/0908.2069

---------------------------------------------------------------

8961. PROBABILISTIC MODEL ASSOCIATED WITH THE PRESSURELESS GAS DYNAMICS

Sergio Albeverio and  Anastasia Korshunova and  Olga Rozanova

Using a method of stochastic perturbation of a Langevin system  
associated
with the non-viscous Burgers equation we construct a solution to the  
Riemann
problem for the pressureless gas dynamics describing sticky particles  
dynamics.
As a bridging step we consider a medium consisting of noninteracting  
particles.
We analyze the difference in the behavior of discontinuous solutions  
for these
two models and the relations between them. In our framework in 1D case  
we
obtain a unique entropy solution to the Riemann problem. Moreover, we  
describe
how starting from smooth data a $\delta$ - singularity arises in one  
component
of the solution.


  http://arxiv.org/abs/0908.2084

---------------------------------------------------------------

8962. THE MAHONIAN PROBABILITY DISTRIBUTION ON WORDS IS ASYMPTOTICALLY  
NORMAL

E. Rodney Canfield and  Svante Janson and  and Doron Zeilberger

The Mahonian statistic is the number of inversions in a permutation of a
multiset with $a_i$ elements of type $i$, $1\le i\le m$. The counting  
function
for this statistic is the $q$ analog of the multinomial coefficient
$\binom{a_1+...+a_m}{a_1,... a_m}$, and the probability generating  
function is
the normalization of the latter. We give two proofs that the  
distribution is
asymptotically normal. The first is {\it computer-assisted}, based on  
the
method of moments. The Maple package {\tt MahonianStat}, available  
from the
webpage of this article, can be used by the reader to perform  
experiments and
calculations. Our second proof uses characteristic functions. We then  
take up
the study of a local limit theorem to accompany our central limit  
theorem. Here
our result is less general, and we must be content with a conjecture  
about
further work. Our local limit theorem permits us to conclude that the
coeffiecients of the $q$-multinomial are log-concave, provided one  
stays near
the center (where the largest coefficients reside.)


  http://arxiv.org/abs/0908.2089

---------------------------------------------------------------

8963. HIGH ORDER DISCRETIZATION SCHEMES FOR STOCHASTIC VOLATILITY MODELS

Benjamin Jourdain (CERMICS) and  Mohamed Sbai (CERMICS)

In usual stochastic volatility models, the process driving the  
volatility of
the asset price evolves according to an autonomous one-dimensional  
stochastic
differential equation. We assume that the coefficients of this  
equation are
smooth. Using It\^o's formula, we get rid, in the asset price  
dynamics, of the
stochastic integral with respect to the Brownian motion driving this  
SDE.
Taking advantage of this structure, we propose - a scheme, based on the
Milstein discretization of this SDE, with order one of weak trajectorial
convergence for the asset price, - a scheme, based on the Ninomiya- 
Victoir
discretization of this SDE, with order two of weak convergence for the  
asset
price. We also propose a specific scheme with improved convergence  
properties
when the volatility of the asset price is driven by an Orstein-Uhlenbeck
process. We confirm the theoretical rates of convergence by numerical
experiments and show that our schemes are well adapted to the  
multilevel Monte
Carlo method introduced by Giles [2008a,b].


  http://arxiv.org/abs/0908.1926

---------------------------------------------------------------

8964. HIGH ORDER DISCRETIZATION SCHEMES FOR STOCHASTIC VOLATILITY MODELS

Benjamin Jourdain (CERMICS) and  Mohamed Sbai (CERMICS)

In usual stochastic volatility models, the process driving the  
volatility of
the asset price evolves according to an autonomous one-dimensional  
stochastic
differential equation. We assume that the coefficients of this  
equation are
smooth. Using It\^o's formula, we get rid, in the asset price  
dynamics, of the
stochastic integral with respect to the Brownian motion driving this  
SDE.
Taking advantage of this structure, we propose - a scheme, based on the
Milstein discretization of this SDE, with order one of weak trajectorial
convergence for the asset price, - a scheme, based on the Ninomiya- 
Victoir
discretization of this SDE, with order two of weak convergence for the  
asset
price. We also propose a specific scheme with improved convergence  
properties
when the volatility of the asset price is driven by an Orstein-Uhlenbeck
process. We confirm the theoretical rates of convergence by numerical
experiments and show that our schemes are well adapted to the  
multilevel Monte
Carlo method introduced by Giles [2008a,b].


  http://arxiv.org/abs/0908.1926

---------------------------------------------------------------

8965. ON THE COPULA FOR MULTIVARIATE EXTREME VALUE DISTRIBUTIONS

Glauco Valle and Marco Aurelio Sanfins

We show that all multivariate Extreme Value distributions, which are the
possible weak limits of the $K$ largest order statistics of iid  
sequences, have
the same copula, the so called K-extremal copula. This copula is  
described
through exact expressions for its density and distribution functions.  
We also
study measures of dependence, we obtain a weak convergence result and we
propose a simulation algorithm for the K-extremal copula.


  http://arxiv.org/abs/0908.2144

---------------------------------------------------------------

8966. SIMULATION REDUCTIONS FOR THE ISING MODEL

Mark L. Huber

Polynomial time reductions between problems have long been used to  
delineate
problem classes. Simulation reductions also exist, where an oracle for
simulation from some probability distribution can be employed together  
with an
oracle for Bernoulli draws in order to obtain a draw from a different
distribution. Here linear time simulation reductions are given for:  
the Ising
spins world to the Ising subgraphs world and the Ising subgraphs world  
to the
Ising spins world. This answers a long standing question of whether  
such a
direct relationship between these two versions of the Ising model  
existed.
Moreover, these reductions result in the first method for perfect  
simulation
from the subgraphs world and a new Swendsen-Wang style Markov chain  
for the
Ising model. The method used is to write the desired distribution with  
set
parameters as a mixture of distributions where the parameters are at  
their
extreme values.


  http://arxiv.org/abs/0908.2151

---------------------------------------------------------------

8967. CONNECTIVITY BOUNDS FOR THE VACANT SET OF RANDOM INTERLACEMENTS

Vladas Sidoravicius and Alain-Sol Sznitman

The model of random interlacements on Z^d, d bigger or equal to 3, was
recently introduced in arXiv:0704.2560. A non-negative parameter u  
parametrizes
the density of random interlacements on Z^d. In the present note we  
investigate
the connectivity properties of the vacant set left by random  
interlacements at
level u, in the non-percolative regime, where u is bigger than the
non-degenerate critical parameter for percolation of the vacant set, see
arXiv:0704.2560, arXiv:0808.3344. We prove a stretched exponential  
decay of the
connectivity function for the vacant set at level u, when u is bigger  
than an
other critical parameter. It is presently an open problem whether  
these two
critical parameters actually coincide.


  http://arxiv.org/abs/0908.2206

---------------------------------------------------------------

8968. RANDOM PERMUTATIONS WITH CYCLE WEIGHTS

Volker Betz and  Daniel Ueltschi and  Yvan Velenik

We study the distribution of cycle lengths in models of nonuniform  
random
permutations with cycle weights. We identify several regimes.  
Depending on the
weights, the length of typical cycles grows like the total number n of
elements, or a fraction of n, or a logarithmic power of n.


  http://arxiv.org/abs/0908.2217

---------------------------------------------------------------

8969. THE TREE LENGTH OF AN EVOLVING COALESCENT

Peter Pfaffelhuber and  Anton Wakolbinger and  Heinz Weisshaupt

A well-established model for the genealogy of a large population in
equilibrium is Kingman's coalescent. For the population together with  
its
genealogy evolving in time, this gives rise to a time-stationary tree- 
valued
process. We study the sum of the branch lengths, briefly denoted as tree
length, and prove that the (suitably compensated) sequence of tree  
length
processes converges, as the population size tends to infinity, to a  
limit
process with cadlag paths, infinite infinitesimal variance, and a Gumbel
distribution as its equilibrium.


  http://arxiv.org/abs/0908.2444

---------------------------------------------------------------

8970. STOCHASTIC INTEGRAL REPRESENTATION OF THE $L^{2}$ MODULUS OF  
BROWNIAN  LOCAL TIME AND A CENTRAL LIMIT THEOREM

Yaozhong Hu and  David Nualart

The purpose of this note is to prove a central limit theorem for the
$L^2$-modulus of continuity of the Brownian local time obtained in  
\cite{CLMR},
using techniques of stochastic analysis. The main ingredients of the  
proof are
an asymptotic version of Knight's theorem and the Clark-Ocone formula  
for the
$L^2$-modulus of the Brownian local time.


  http://arxiv.org/abs/0908.2473

---------------------------------------------------------------

8971. ENVIRONMENTAL NOISE VARIABILITY IN POPULATION DYNAMICS MATRIX  
MODELS

Michel De Lara (CERMICS)

The impact of environmental variability on population size growth rate  
in
dynamic models is a recurrent issue in the theoretical ecology  
literature. In
the scalar case, R. Lande pointed out that results are ambiguous  
depending on
whether the noise is added at arithmetic or logarithmic scale, while  
the matrix
case has been investigated by S. Tuljapurkar. Our contribution  
consists first
in introducing another notion of variability than the widely used  
variance or
coefficient of variation, namely the so-called convex orders. Second, in
population dynamics matrix models, we focus on how matrix components  
depend
functionaly on uncertain environmental factors. In the log-convex  
case, we show
that, in a sense, environmental variability increases both mean  
population size
and mean log-population size and makes them more variable. Our main  
result is
that specific analytical dependence coupled with appropriate notion of
variability lead to wide generic results, valid for all times and not  
only
asymptotically, and requiring no assumptions of stationarity, of  
normality, of
independency, etc. Though the approach is different, our conclusions are
consistent with previous results in the literature. However, they make  
it clear
that the analytical dependence on environmental factors cannot be  
overlooked
when trying to tackle the influence of variability.


  http://arxiv.org/abs/0908.2499

---------------------------------------------------------------

8972. A BACKWARD PARTICLE INTERPRETATION OF FEYNMAN-KAC FORMULAE

Pierre Del Moral and  Arnaud Doucet and  Sumeetpal S. Singh

We design a particle interpretation of Feynman-Kac measures on path  
spaces
based on a backward Markovian representation combined with a  
traditional mean
field particle interpretation of the flow of their final time  
marginals. In
contrast to traditional genealogical tree based models, these new  
particle
algorithms can be used to compute normalized additive functionals "on- 
the-fly"
as well as their limiting occupation measures with a given precision  
degree
that does not depend on the final time horizon.
   We provide uniform convergence results w.r.t. the time horizon  
parameter as
well as functional central limit theorems and exponential concentration
estimates. We also illustrate these results in the context of  
computational
physics and imaginary time Schroedinger type partial differential  
equations,
with a special interest in the numerical approximation of the  
invariant measure
associated to $h$-processes.


  http://arxiv.org/abs/0908.2556

---------------------------------------------------------------

8973. THRESHOLD GRAPH LIMITS AND RANDOM THRESHOLD GRAPHS

Persi Diaconis and  Susan Holmes and  Svante Janson

We study the limit theory of large threshold graphs and apply this to a
variety of models for random threshold graphs. The results give a nice  
set of
examples for the emerging theory of graph limits.


  http://arxiv.org/abs/0908.2448

---------------------------------------------------------------

8974. PHASE TRANSITION FOR THE MIXING TIME OF THE GLAUBER DYNAMICS  
FOR  COLORING REGULAR TREES

Prasad Tetali and  Juan C. Vera and  Eric Vigoda and  Linji Yang

We prove that the mixing time of the Glauber dynamics for random
$k$-colorings of the complete tree with branching factor $b$ undergoes  
a phase
transition at $k=b(1+o_b(1))/\ln{b}$. Our main result shows nearly  
sharp bounds
on the mixing time of the dynamics on the complete tree with $n$  
vertices for
$k=Cb/\ln{b}$ colors with constant $C$. For $C\geq 1$ we prove the  
mixing time
is $O(n^{1+o_b(1)}\ln^2{n})$. On the other side, for $C< 1$ the mixing  
time
experiences a slowing down, in particular, we prove it is $O(n^{1/C +
o_b(1)}\ln^2{n})$ and $\Omega(n^{1/C-o_b(1)})$. The critical point C=1  
is
interesting since it coincides (at least up to first order) to the so- 
called
reconstruction threshold which was recently established by Sly. The
reconstruction threshold has been of considerable interest recently  
since it
appears to have close connections to the efficiency of certain local
algorithms, and this work was inspired by our attempt to understand  
these
connections in this particular setting.


  http://arxiv.org/abs/0908.2665

---------------------------------------------------------------

8975. STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED AND  
DEGENERATE  COEFFICIENTS

Xicheng Zhang

In this article, using DiPerna-Lions theory \cite{Di-Li}, we investigate
linear second order stochastic partial differential equations with  
unbounded
and degenerate non-smooth coefficients, and obtain several conditions  
for
existence and uniqueness. Moreover, we also prove the $L^1$- 
integrability and a
general maximal principle for generalized solutions of SPDEs. As  
applications,
we study nonlinear filtering problem and also obtain the existence and
uniqueness of generalized solutions for a degenerate nonlinear SPDE.


  http://arxiv.org/abs/0908.2695

---------------------------------------------------------------

8976. PROBABILISTIC REPRESENTATION FOR SOLUTIONS OF AN IRREGULAR  
POROUS MEDIA  TYPE EQUATION: THE DEGENERATE CASE

Viorel Barbu and  Michael Roeckner (SFB 701) and  Francesco Russo (LAGA)

We consider a possibly degenerate porous media type equation over all of
$\R^d$ with $d = 1$, with monotone discontinuous coefficients with  
linear
growth and prove a probabilistic representation of its solution in  
terms of an
associated microscopic diffusion. This equation is motivated by some  
singular
behaviour arising in complex self-organized critical systems. The main  
idea
consists in approximating the equation by equations with monotone
non-degenerate coefficients and deriving some new analytical  
properties of the
solution.


  http://arxiv.org/abs/0908.2701

---------------------------------------------------------------

8977. SHARP INTERFACE LIMIT FOR INVARIANT MEASURES OF A STOCHASTIC  
ALLEN-CAHN  EQUATION

Hendrik Weber

The invariant measure of a one-dimensional Allen-Cahn equation with an
additive space-time white noise is studied. This measure is absolutely
continuous with respect to a Brownian bridge with a density which can be
interpreted as a potential energy term. We consider the sharp  
interface limit
in this setup. In the right scaling this corresponds to a Gibbs type  
measure on
a growing interval with decreasing temperature. Our main result is  
that in the
limit we still see exponential convergence towards a curve of  
minimizers of the
energy if the interval does not grow too fast. In the original scaling  
the
limit measure is concentrated on configurations with precisely one  
jump. This
jump is distributed uniformly.


  http://arxiv.org/abs/0908.2717

---------------------------------------------------------------

8978. HYDRODYNAMIC LIMIT OF MOVE-TO-FRONT RULES AND SEARCH COST  
PROBABILITIES

Kumiko Hattori and  Tetsuya Hattori

We study a hydrodynamic limit approach to move-to-front rules, namely, a
scaling limit as the number of items tends to infinity, of the joint
distribution of jump rate and position of items. As an application of  
the limit
formula, we present asymptotic formulas on search cost probability
distributions, applicable for general jump rate distributions.


  http://arxiv.org/abs/0908.3222

---------------------------------------------------------------

8979. STOCHASTIC EVOLUTIONS OF POINT PROCESSES

Philippe Robert

The asymptotic behavior of birth and death processes of particles in a
compact space is analyzed. Births: Particles are created at rate $ 
\lambda_+$
and their location is independent of the current configuration. Deaths  
are due
to negative particles arriving at rate $\lambda_-$. The death of a  
particle
occurs when a negative particle arrives in its neighborhood and kills  
it.
Several killing schemes are considered. The arriving locations of  
positive and
negative particles are assumed to have the same distribution. By using a
combination of monotonicity properties and invariance relations it is  
shown
that the configurations of particles converge in distribution for  
several
models. The problems of uniqueness of invariant measures and of the  
existence
of accumulation points for the limiting configurations are also  
investigated.
It is shown for several natural models that if $\lambda_+<\lambda_-$  
then the
asymptotic configuration has a finite number of points with  
probability 1.
Examples with $\lambda_+<\lambda_-$ and an infinite number of  
particles in the
limit are also presented.


  http://arxiv.org/abs/0908.3256

---------------------------------------------------------------

8980. REFLECTED BROWNIAN MOTION IN WEYL CHAMBERS

Nizar Demni

We supply two different descriptions of the pushing process driving the
reflected Brownian motion in Weyl chambers, when the latter domains are
simplexes. The first one shows that a simple root lies in one and only  
one
orbit if and only if the pushing process in the direction of that  
simple root
increases as the sum of all the Brownian local times in the directions  
of the
orbit's positive elements. The last one shows that the pushing process  
may be
written as the sum of an inward normal vector at the chamber's  
boundary and an
inward normal vector at the origin, yielding a kind of a multivoque  
stochastic
differential equation for the reflected process. We finally give a  
particles
system interpretation of the reflected process and construct a  
multidimensional
skew Brownian motion.


  http://arxiv.org/abs/0908.3302

---------------------------------------------------------------

8981. A ZERO-ONE LAW FOR LINEAR TRANSFORMATIONS OF LEVY NOISE

Steven N. Evans

A L\'evy noise on $\mathbb{R}^d$ assigns a random real "mass" $\Pi(B)$  
to
each Borel subset $B$ of $\mathbb{R}^d$ with finite Lebesgue measure.  
The
distribution of $\Pi(B)$ only depends on the Lebesgue measure of $B$,  
and if
$B_1, ..., B_n$ is a finite collection of pairwise disjoint sets, then  
the
random variables $\Pi(B_1), ..., \Pi(B_n)$ are independent with $ 
\Pi(B_1 \cup
 >... \cup B_n) = \Pi(B_1) + ... + \Pi(B_n)$ almost surely. In  
particular, the
distribution of $\Pi \circ g$ is the same as that of $\Pi$ when $g$ is a
bijective transformation of $\mathbb{R}^d$ that preserves Lebesgue  
measure. It
follows from the Hewitt--Savage zero--one law that any event which is  
almost
surely invariant under the mappings $\Pi \mapsto \Pi \circ g$ for every
Lebesgue measure preserving bijection $g$ of $\mathbb{R}^d$ must have
probability 0 or 1. We investigate whether certain smaller groups of  
Lebesgue
measure preserving bijections also possess this property. We show that  
if $d
\ge 2$, the L\'evy noise is not purely deterministic, and the group  
consists of
linear transformations and is closed, then the invariant events all have
probability 0 or 1 if and only if the group is not compact.


  http://arxiv.org/abs/0908.3339

---------------------------------------------------------------

8982. FINITE-TIME BLOWUP AND EXISTENCE OF GLOBAL POSITIVE SOLUTIONS OF  
A  SEMI-LINEAR SPDE

Marco Dozzi (IECN) and  Jos\'e Alfredo Lopez

We consider stochastic equations of the prototype $du(t,x) =(\Delta
u(t,x)+u(t,x)^{1+\beta})dt+\kappa u(t,x) dW_{t}$ on a smooth domain $D 
\subset
\mathord{\rm I\mkern-3.6mu R\:}^d$, with Dirichlet boundary condition,  
where
$\beta$, $\kappa$ are positive constants and $\{W_t $, $t\ge0\}$ is a
one-dimensional standard Wiener process. We estimate the probability  
of finite
time blowup of positive solutions, as well as the probability of  
existence of
non-trivial positive global solutions.


  http://arxiv.org/abs/0908.3364

---------------------------------------------------------------

8983. LIMIT THEOREMS FOR RANDOM PROCESSES WITH RANDOM TIME SUBSTITUTION

Permyakova Elena

In this paper the sufficient conditions for convergence in Skorokhod  
space
$D[0,1]$ of sequence of random processes with random time substitution  
are
obtained.


  http://arxiv.org/abs/0908.3395

---------------------------------------------------------------

8984. POISSON SPLITTING BY FACTORS

Alexander E. Holroyd and  Russell Lyons and  and Terry Soo

Given a homogeneous Poisson process on R^d with intensity L, we prove  
that it
is possible to partition the points into two sets, as a deterministic  
function
of the process, and in an isometry-equivariant way, so that each set  
of points
forms a homogeneous Poisson process, with any given pair of  
intensities summing
to L. In particular, this answers a question of Ball, who proved that  
in d=1,
the Poisson points may be similarly partitioned (via a translation- 
equivariant
function) so that one set forms a Poisson process of lower intensity,  
and asked
whether the same was possible for all d. We do not know whether it is  
possible
similarly to add points (again chosen as a deterministic function of a  
Poisson
process) to obtain a Poisson process of higher intensity, but we prove  
that
this is not possible under an additional finitariness condition.


  http://arxiv.org/abs/0908.3409

---------------------------------------------------------------

8985. A RULE OF THUMB FOR RIFFLE SHUFFLING

Sami Assaf and  Persi Diaconis and  K. Soundararajan

We study how many riffle shuffles are required to mix n cards if only  
certain
features of the deck are of interest, e.g. suits disregarded or only  
the colors
of interest. For these features, the number of shuffles drops from 3/2  
log_2(n)
to log_2(n). We derive closed formulae and an asymptotic `rule of thumb'
formula which is remarkably accurate.


  http://arxiv.org/abs/0908.3462

---------------------------------------------------------------

8986. OPTIMAL TRANSPORTATION AND MONOTONIC QUANTITIES ON EVOLVING  
MANIFOLDS

Hong Huang

In this note we adapt Topping's $\mathcal{L}$-optimal transportation  
theory
for Ricci flow to a more general situation, i.e. to a closed manifold
$(M,g_{ij}(t)$ evolving by $\partial_tg_{ij}=-2S_{ij}$, where $S_{ij}$  
is a
symmetric tensor field of (2,0)-type on $M$. We extend some of  
Topping's and
Lott's recent results, generalize the monotonicity of List's (and  
hence also of
Perelman's) $\mathcal{W}$-entropy, and recover the monotonicity of
M$\ddot{u}$ller's (and hence also of Perelman's) reduced volume.


  http://arxiv.org/abs/0908.3293

---------------------------------------------------------------

8987. RANK-BASED ATTACHMENT LEADS TO POWER LAW GRAPHS

Jeannette Janssen and Pawel Pralat

We investigate the degree distribution resulting from graph generation  
models
based on rank-based attachment. In rank-based attachment, all vertices  
are
ranked according to a ranking scheme. The link probability of a given  
vertex is
proportional to its rank raised to the power -a, for some a in (0,1).  
Through a
rigorous analysis, we show that rank-based attachment models lead to  
graphs
with a power law degree distribution with exponent 1+1/a whenever  
vertices are
ranked according to their degree, their age, or a randomly chosen  
fitness
value. We also investigate the case where the ranking is based on the  
initial
rank of each vertex; the rank of existing vertices only changes to  
accommodate
the new vertex. Here, we obtain a sharp threshold for power law  
behaviour. Only
if initial ranks are biased towards lower ranks, or chosen uniformly  
at random,
we obtain a power law degree distribution with exponent 1+1/a. This  
indicates
that the power law degree distribution often observed in nature can be
explained by a rank-based attachment scheme, based on a ranking scheme  
that can
be derived from a number of different factors; the exponent of the  
power law
can be seen as a measure of the strength of the attachment.


  http://arxiv.org/abs/0908.3436

---------------------------------------------------------------

8988. NOTES ON FEIGE'S GUMBALL MACHINES PROBLEM

John H. Elton

We give a detailed proof, in the identically distributed case, of a
conjecture of Feige about the maximum probability that the sum of n  
independent
non-negative integer valued random variables, each of mean 1, exceeds  
n. The
general case is reduced to two-point distributions.


  http://arxiv.org/abs/0908.3528

---------------------------------------------------------------

8989. LIMIT THEOREMS FOR PROJECTIONS OF RANDOM WALK ON A HYPERSPHERE

Max Skipper

We show that almost any one-dimensional projection of a suitably scaled
random walk on a hypercube, inscribed in a hypersphere, converges  
weakly to an
Ornstein-Uhlenbeck process as the dimension of the sphere tends to  
infinity. We
also observe that the same result holds when the random walk is  
replaced with
spherical Brownian motion. This latter result can be viewed as a  
"functional"
generalisation of Poincar\'e's observation for projections of uniform  
measure
on high dimensional spheres; the former result is an analogous  
generalisation
of the Bernoulli-Laplace central limit theorem. Given the relation of  
these two
classic results to the central limit theorem for convex bodies, the  
modest
results provided here would appear to motivate a functional  
generalisation.


  http://arxiv.org/abs/0908.3536

---------------------------------------------------------------

8990. CAN AN INFINITE PRODUCT OF NONNEGATIVE MATRICES BE EXPRESSED IN  
TERMS OF  INFINITE PRODUCTS OF STOCHASTIC ONES?

Alain Thomas (LATP)

It is known that if the product $M_n... M_1$ converges to a nonnull  
limit
when $n\to\infty$ and if the $M_n$ belong to a finite set of complex  
matrices,
then the $M_n$ for $n\ge n_0$ have a common right eigenvector $V$ for  
the
eigenvalue 1. In case the $M_n$ are nonnegative and $V$ is positive,
$\Delta^{-1}M_{n_0}... M_n\Delta$ is the product of the stochastic  
matrices
$\Delta^{-1}M_n\Delta$, where the diagonal matrix $\Delta$ has on its  
diagonal
the same entries as $V$. In the last section we examine what happen  
when we
remove the hypothesis that $V$ is positive.


  http://arxiv.org/abs/0908.3538

---------------------------------------------------------------

8991. CRITICAL RANDOM GRAPHS: LIMITING CONSTRUCTIONS AND  
DISTRIBUTIONAL  PROPERTIES

L. Addario-Berry and  N. Broutin and C. Goldschmidt

We consider the Erdos--Renyi random graph G(n,p) inside the critical  
window,
where p = 1/n + lambda * n^{-4/3} for some lambda in R. We proved in a  
previous
paper (arXiv:0903.4730) that considering the connected components of  
G(n,p) as
a sequence of metric spaces with the graph distance rescaled by  
n^{-1/3} and
letting n go to infinity yields a non-trivial sequence of limit metric  
spaces C
= (C_1, C_2, ...). These limit metric spaces can be constructed from  
certain
random real trees with vertex-identifications. We give here equivalent
constructions using standard Brownian continuum random trees, their  
recursive
construction from inhomogeneous Poisson point processes, and Polya's urn
scheme. We also characterize the distributions of the masses and  
lengths in the
constituant parts of a limit component when it is decomposed according  
to its
cycle structure.


  http://arxiv.org/abs/0908.3629

---------------------------------------------------------------

8992. HARNACK INEQUALITIES AND APPLICATIONS FOR MULTIVALUED  
STOCHASTIC  EVOLUTION EQUATIONS

Shun-Xiang Ouyang

By the method of coupling and Girsanov transformation, Harnack  
inequalities
[F.-Y. Wang, 1997] and strong Feller property are proved for the  
transition
semigroup associated with the multivalued stochastic evolution  
equation on a
Gelfand triple. The concentration property of the invariant measure  
for the
semigroup is investigated. As applications of Harnack inequalities,  
explicit
upper bounds of the $L^p$-norm of the density, contractivity,  
compactness and
entropy-cost inequality for the semigroup are also presented.


  http://arxiv.org/abs/0908.3630

---------------------------------------------------------------

8993. APPLICATIONS OF WEAK CONVERGENCE FOR HEDGING OF AMERICAN AND  
GAME  OPTIONS

Yan Dolinsky

This paper studies stability of Dynkin's games value under weak  
convergence.
We use these results to approximate game options prices with path  
dependent
payoffs in continuous time models by sequence of game options prices in
discrete time models which can be calculated by dynamical programming
algorithms. We also show that shortfall risks of American options in a  
sequence
of multinomial approximations of the multidimensional BS market  
converge to the
corresponding quantities for similar American options in the  
multidimensional
BS market with path dependent payoffs. In comparison to previous  
papers we work
under more general convergence of underlying processes, as well, as  
weaker
condition on the payoffs.


  http://arxiv.org/abs/0908.3661

---------------------------------------------------------------

8994. ON THE MINIMAL PENALTY FOR MARKOV ORDER ESTIMATION

Ramon van Handel

We show that large-scale typicality of Markov sample paths implies  
that the
likelihood ratio statistic satisfies a law of iterated logarithm  
uniformly to
the same scale. As a consequence, the penalized likelihood Markov order
estimator is strongly consistent for penalties growing as slowly as  
log log n
when an upper bound is imposed on the order which may grow as rapidly  
as log n.
Our method of proof, using techniques from empirical process theory,  
does not
rely on the explicit expression for the maximum likelihood estimator  
in the
Markov case and could therefore be applicable in other settings.


  http://arxiv.org/abs/0908.3666

---------------------------------------------------------------

8995. ZERO-ONE LAWS FOR CONNECTIVITY IN RANDOM KEY GRAPHS

Osman Yagan and Armand M. Makowski

The random key graph is a random graph naturally associated with the  
random
key predistribution scheme of Eschenauer and Gligor for wireless sensor
networks. For this class of random graphs we establish a new version  
of a
conjectured zero-one law for graph connectivity as the number of nodes  
becomes
unboundedly large. The results reported here complement and strengthen  
recent
work on this conjecture by Blackburn and Gerke. In particular, the  
results are
given under conditions which are more realistic for applications to  
wireless
sensor networks.


  http://arxiv.org/abs/0908.3644

---------------------------------------------------------------

8996. RANDOMIZED SCHEDULING ALGORITHM FOR QUEUEING NETWORKS

Devavrat Shah and  Jinwoo Shin

There has recently been considerable interest in design of low- 
complexity,
myopic, distributed and stable scheduling policies for constrained  
queueing
network models that arise in the context of emerging communication  
networks.
Here, we consider two representative models. One, a model for the  
collection of
wireless nodes communicating through a shared medium, that represents  
randomly
varying number of packets in the queues at the nodes of networks. Two, a
buffered circuit switched network model for an optical core of future  
Internet,
to capture the randomness in calls or flows present in the network.  
The maximum
weight scheduling policy proposed by Tassiulas and Ephremide in 1992  
leads to a
myopic and stable policy for the packet-level wireless network model.  
But
computationally it is very expensive (NP-hard) and centralized. It is  
not
applicable to the buffered circuit switched network due to the  
requirement of
non-premption of the calls in the service. As the main contribution of  
this
paper, we present a stable scheduling algorithm for both of these  
models. The
algorithm is myopic, distributed and performs few logical operations  
at each
node per unit time.


  http://arxiv.org/abs/0908.3670

---------------------------------------------------------------

8997. ASYMPTOTIC REGIMES FOR THE PARTITION INTO COLONIES OF A  
BRANCHING  PROCESS WITH EMIGRATION

Jean Bertoin (PMA and  Dma)

We consider a spatial branching process with emigration in which  
children
either remain at the same site as their parents or migrate to new  
locations and
then found their own colonies. We are interested in asymptotics of the
partition of the total population into colonies for large populations  
with rare
migrations. Under appropriate regimes, we establish weak convergence  
of the
rescaled partition to some random measure that is constructed from the
restriction of a Poisson point measure to a certain random region, and  
whose
cumulant solves a simple integral equation.


  http://arxiv.org/abs/0908.3735

---------------------------------------------------------------

8998. ON THE ABSOLUTE CONTINUITY OF MULTIDIMENSIONAL ORNSTEIN- 
UHLENBECK  PROCESSES

Thomas Simon (LPP)

Let $X$ be a $n$-dimensional Ornstein-Uhlenbeck process, solution of the
S.D.E. $$\d X_t = AX_t \d t + \d B_t$$ where $A$ is a real $n\times n$  
matrix
and $B$ a L\'evy process without Gaussian part. We show that when $A$ is
non-singular, the law of $X_1$ is absolutely continuous in $\r^n$ if  
and only
if the jumping measure of $B$ fulfils a certain geometric condition with
respect to $A,$ which we call the exhaustion property. This optimal  
criterion
is much weaker than for the background driving L\'evy process $B$,  
which might
be very singular and sometimes even have a one-dimensional discrete  
jumping
measure. It also solves a difficult problem for a certain class of  
multivariate
Non-Gaussian infinitely divisible distributions.


  http://arxiv.org/abs/0908.3736

---------------------------------------------------------------

8999. EXTREMAL SUBGRAPHS OF RANDOM GRAPHS: AN EXTENDED VERSION

Graham Brightwell and  Konstantinos Panagiotou and  Angelika Steger

We prove that there is a constant $c >0$, such that whenever $p \ge  
n^{-c}$,
with probability tending to 1 when $n$ goes to infinity, every maximum
triangle-free subgraph of the random graph $G_{n,p}$ is bipartite.  
This answers
a question of Babai, Simonovits and Spencer (Journal of Graph Theory,  
1990).
   The proof is based on a tool of independent interest: we show, for  
instance,
that the maximum cut of almost all graphs with $M$ edges, where $M >> n 
$, is
``nearly unique''. More precisely, given a maximum cut $C$ of $G_{n,M} 
$, we can
obtain all maximum cuts by moving at most $O(\sqrt{n^3/M})$ vertices  
between
the parts of $C$.


  http://arxiv.org/abs/0908.3778

---------------------------------------------------------------

9000. MIXING TIME OF NEAR-CRITICAL RANDOM GRAPHS

Jian Ding and  Eyal Lubetzky and  Yuval Peres

Let $C_1$ be the largest component of the Erd\H{o}s-R\'enyi random graph
$G(n,p)$. The mixing time of random walk on $C_1$ in the strictly  
supercritical
regime, $p=c/n$ with fixed $c>1$, was shown to have order $\log^2 n$ by
Fountoulakis and Reed, and independently by Benjamini, Kozma and  
Wormald. In
the critical window, $p=(1+\epsilon)/n$ where $\lambda=\epsilon^3 n$ is
bounded, Nachmias and Peres proved that the mixing time on $C_1$ is of  
order
$n$. However, it was unclear how to interpolate between these results,  
and
estimate the mixing time as the giant component emerges from the  
critical
window. Indeed, even the asymptotics of the diameter of $C_1$ in this  
regime
were only recently obtained by Riordan and Wormald, as well as the  
present
authors and Kim.
   In this paper we show that for $p=(1+\epsilon)/n$ with $\lambda= 
\epsilon^3
n\to\infty$ and $\lambda=o(n)$, the mixing time on $C_1$ is with high
probability of order $(n/\lambda)\log^2 \lambda$. In addition, we show  
that
this is the order of the largest mixing time over all components, both  
in the
slightly supercritical and in the slightly subcritical regime (i.e.,
$p=(1-\epsilon)/n$ with $\lambda$ as above).


  http://arxiv.org/abs/0908.3870

---------------------------------------------------------------

9001. UTILITY OPTIMIZATION IN CONGESTED QUEUEING NETWORKS

Neil Stuart Walton

We consider a multi-class single server queueing network as a model of a
packet switching network. The rates packets are sent into this network  
are
controlled by queues which act as congestion windows. By considering a  
sequence
of such congestion windows we allow the network to become congested.  
We show
the stationary throughput of routes on this sequence of networks  
converges to
an allocation that maximizes aggregate utility subject to the network's
capacity constraints. To perform this analysis we require that our  
utility
functions satisfy an exponential concavity condition. This family of  
utilities
includes weighted $\alpha$-fair utilities for $\alpha >1$.


  http://arxiv.org/abs/0908.3787

---------------------------------------------------------------

9002. DISTRIBUTED AVERAGING VIA LIFTED MARKOV CHAINS

Kyomin Jung and  Devavrat Shah and  Jinwoo Shin

Motivated by applications of distributed linear estimation, distributed
control and distributed optimization, we consider the question of  
designing
linear iterative algorithms for computing the average of numbers in a  
network.
Specifically, our interest is in designing such an algorithm with the  
fastest
rate of convergence given the topological constraints of the network.  
As the
main result of this paper, we design an algorithm with the fastest  
possible
rate of convergence using a non-reversible Markov chain on the given  
network
graph. We construct such a Markov chain by transforming the standard  
Markov
chain, which is obtained using the Metropolis-Hastings method. We call  
this
novel transformation pseudo-lifting. We apply our method to graphs with
geometry, or graphs with doubling dimension. Specifically, the  
convergence time
of our algorithm (equivalently, the mixing time of our Markov chain) is
proportional to the diameter of the network graph and hence optimal.  
As a
byproduct, our result provides the fastest mixing Markov chain given the
network topological constraints, and should naturally find their  
applications
in the context of distributed optimization, estimation and control.


  http://arxiv.org/abs/0908.4073

---------------------------------------------------------------

9003. HYDRODYNAMIC LIMIT OF THE EXCLUSION PROCESS IN INHOMOGENEOUS MEDIA

Milton Jara

We obtain the hydrodynamic limit of a simple exclusion process in an
inhomogeneous environment of divergence form. Our main assumption is a  
suitable
version of Gamma-convergence for the environment. In this way we  
obtain an
unified approach to recent works on the field.


  http://arxiv.org/abs/0908.4120

---------------------------------------------------------------

9004. CONTACT PROCESS IN A WEDGE

J. Theodore Cox and  Nevena Maric and  Rinaldo B. Schinazi

We prove that the supercritical one-dimensional contact process  
survives in
certain wedge-like space-time regions, and that when it survives it  
couples
with the unrestricted contact process started from its upper invariant  
measure.
As an application we show that a type of weak coexistence is possible  
in the
nearest-neighbor ``grass-bushes-trees'' successional model introduced in
Durrett and Swindle (1991).


  http://arxiv.org/abs/0908.4125

---------------------------------------------------------------

9005. KOLMOGOROV EQUATION ASSOCIATED TO THE STOCHASTIC REFLECTION  
PROBLEM ON A  SMOOTH CONVEX SET OF A HILBERT SPACE

Viorel Barbu and  Giuseppe Da Prato and  Luciano Tubaro

We consider the stochastic reflection problem associated with a self- 
adjoint
operator $A$ and a cylindrical Wiener process on a convex set $K$ with  
nonempty
interior and regular boundary $\Sigma$ in a Hilbert space $H$. We  
prove the
existence and uniqueness of a smooth solution for the corresponding  
elliptic
infinite-dimensional Kolmogorov equation with Neumann boundary  
condition on
$\Sigma$.


  http://arxiv.org/abs/0908.4139

---------------------------------------------------------------

9006. THE SURVIVAL OF LARGE DIMENSIONAL THRESHOLD CONTACT PROCESSES

Thomas Mountford and  Roberto H. Schonmann

We study the threshold $\theta$ contact process on $\mathbb{Z}^d$ with
infection parameter $\lambda$. We show that the critical point
$\lambda_{\mathrm{c}}$, defined as the threshold for survival starting  
from
every site occupied, vanishes as $d\to\infty$. This implies that the  
threshold
$\theta$ voter model on $\mathbb{Z}^d$ has a nondegenerate extremal  
invariant
measure, when $d$ is large.


  http://arxiv.org/abs/0908.4146

---------------------------------------------------------------

9007. ON THE EXTENDIBILITY OF PARTIALLY AND MARKOV EXCHANGEABLE  
BINARY  SEQUENCES

Davide Di Cecco

In [Fortini et al., Stoch. Proc. Appl. 100 (2002), 147--165] it is
demonstrated that a recurrent Markov exchangeable process in the sense  
of
Diaconis and Freedman is essentially a partially exchangeable process  
in the
sense of de Finetti. In case of finite sequences there is not such an
equivalence. We analyze both finite partially exchangeable and finite  
Markov
exchangeable binary sequences and formulate necessary and sufficient  
conditions
for extendibility in both cases.


  http://arxiv.org/abs/0908.4158

---------------------------------------------------------------

9008. ASYMPTOTIC PROPERTIES OF THE COLUMNS IN THE PRODUCTS OF  
NONNEGATIVE  MATRICES

\'Eric Olivier (LATP) and  Alain Thomas (LATP)

We consider the sequence of column-vectors $R_n=A_1... A_nR$  
associated to a
sequence $(A_n)$ of nonnegative $d\times d$ matrices and to a positive
$d$-dimensional column-vector $R$. The problem to know the necessary and
sufficient conditions -- on the sequence $(A_n)$ -- for
$\displaystyle{R_n\over\Vert R_n\Vert}$ to converge is yet not solved.
Nevertheless we prove this convergence in case the $A_n$ are -- in a  
sense --
echeloned and fulfill certain boundness conditions. If the $A_n$ do  
not fulfill
the conditions and even if they are sparse, it may exist a sequence of  
integers
$(n_k)$ such that the matrices $A'_k=A_{n_k+1}... A_{n_{k+1}}$ do: we  
see in
some other paper how to proceed in one example, and how to use the  
obtained
result to study some continuous singular measure.


  http://arxiv.org/abs/0908.4171

---------------------------------------------------------------

9009. ON THE INVERSE FIRST-PASSAGE-TIME PROBLEM FOR A WIENER PROCESS

Cristina Zucca and  Laura Sacerdote

The inverse first-passage problem for a Wiener process $(W_t)_{t\ge0}$  
seeks
to determine a function $b{}:{}\mathbb{R}_+\to\mathbb{R}$ such that
\[\tau=\inf\{t>0| W_t\ge b(t)\}\] has a given law. In this paper two  
methods
for approximating the unknown function $b$ are presented. The errors  
of the two
methods are studied. A set of examples illustrates the methods. Possible
applications are enlighted.


  http://arxiv.org/abs/0908.4213

---------------------------------------------------------------

9010. EXTREMAL SHOT NOISES, HEAVY TAILS AND MAX-STABLE RANDOM FIELDS

Cl\'ement Dombry (LMA)

Extremal shot noises naturally appear in extreme value theory as a  
model for
spatial extremes and serve as basic models for annual maxima of  
rainfall or for
coverage field in telecommunication. In this work, we examine their  
properties
such as boundedness, regularity, ergodicity ... Connexions with max- 
stable
random fields are established: we prove a limit theorem when the  
distribution
of the weights is heavy tailed and the intensity of points goes to  
infinity. We
use a point process approach strongly connected to the Peak Over  
Threshold
method used by hydrologists. Properties of the limit max-stable random  
fields
are also investigated.


  http://arxiv.org/abs/0908.4221

---------------------------------------------------------------

9011. STOCHASTIC COMPLETENESS AND VOLUME GROWTH

Christian Baer and  G. Pacelli Bessa

It has been suggested in 1999 that a certain volume growth condition for
geodesically complete Riemannian manifolds might imply that the  
manifold is
stochastically complete. This is motivated by a large class of  
examples and by
a known analogous criterion for recurrence of Brownian motion. We show  
that the
suggested implication is not true in general. We also give counter- 
examples to
a converse implication.


  http://arxiv.org/abs/0908.4222

---------------------------------------------------------------

9012. MATRIX FACTORIZATION IDENTITY FOR ALMOST SEMI-CONTINUOUS  
PROCESSES ON A  MARKOV CHAIN

D.V. Gusak and  E.V. Karnaukh

In this article almost semi-continuous processes with stationary  
independent
increments on a finite irreducible Markov chain are considered. For  
these
processes the components of matrix factorization identity are concretely
defined. On the basis of this concrete definition the relations for the
distributions of extrema and distributions of their complements for  
the almost
upper semi-continuous processes are established.


  http://arxiv.org/abs/0908.4326

---------------------------------------------------------------

9013. LIMIT LAWS OF TRANSIENT EXCITED RANDOM WALKS ON INTEGERS

Elena Kosygina and  Thomas Mountford

We consider excited random walks (ERWs) on integers with a bounded  
number of
i.i.d. cookies per site without the non-negativity assumption on the  
drifts
induced by the "cookies". E. Kosygina and M.P.W. Zerner have shown  
that when
the total expected drift per site, delta, is larger than 1 then ERW is
transient to the right and, moreover, for delta>4 under the averaged  
measure it
obeys the Central Limit Theorem. We show that when delta is in (2,4] the
limiting behavior of an appropriately centered and scaled excited  
random walk
is described by a strictly stable law with parameter delta/2. Our  
method also
extends the results obtained by A.-L. Basdevant and A. Singh for delta  
in (1,2]
under the non-negativity assumption to the setting which allows both  
positive
and negative cookies.


  http://arxiv.org/abs/0908.4356

---------------------------------------------------------------

9014. POISSON DIRICHLET$(\ALPHA,\THETA)$-BRIDGE EQUATIONS AND   
COAGULATION-FRAGMENTATION DUALITY

Lancelot F. James

This paper derives distributional properties of a class of exchangeable
bridges closely related to the Poisson-Dirichlet $(\alpha,\theta)$  
family of
bridges. We then show that various stochastic equations derived for  
these
bridges lead to constructions of a new large class of coagulation and
fragmentation operators that satisfy a duality property, and are  
otherwise
easily manipulated. This class, builds on, and includes the duality  
relations
developed in Pitman (1999), Bertoin and Goldschmidt (2004), and Dong,
Goldschmidt and Martin (2006),which we can treat in a unified way. Our
exposition also suggests an approach to obtain other dualities and  
related
results.


  http://arxiv.org/abs/0908.4436

---------------------------------------------------------------

9015. CONVERGENCE OF NUMERICAL TIME-AVERAGING AND STATIONARY MEASURES  
VIA  POISSON EQUATIONS

Jonathan C. Mattingly and  Andrew M. Stuart and  M.V. Tretyakov

Numerical approximation of the long time behavior of a stochastic
differential equation (SDE) is considered. Error estimates for time- 
averaging
estimators are obtained and then used to show that the stationary  
behavior of
the numerical method converges to that of the SDE. The error analysis  
is based
on using an associated Poisson equation for the underlying SDE. The main
advantage of this approach is its simplicity and universality. It  
works equally
well for a range of explicit and implicit schemes including those with  
simple
simulation of random variables, and for general hypoelliptic SDEs. An  
analogy
between this approach and Stein's method is indicated. Some practical
implications of the results are discussed.


  http://arxiv.org/abs/0908.4450

---------------------------------------------------------------

9016. TIME AVERAGES, RECURRENCE AND TRANSIENCE IN THE STOCHASTIC  
REPLICATOR  DYNAMICS

Josef Hofbauer and  Lorens A. Imhof

We investigate the long-run behavior of a stochastic replicator process,
which describes game dynamics for a symmetric two-player game under  
aggregate
shocks. We establish an averaging principle that relates time averages  
of the
process and Nash equilibria of a suitably modified game. Furthermore, a
sufficient condition for transience is given in terms of mixed  
equilibria and
definiteness of the payoff matrix. We also present necessary and  
sufficient
conditions for stochastic stability of pure equilibria.


  http://arxiv.org/abs/0908.4467

---------------------------------------------------------------

9017. BUBBLES, CONVEXITY AND THE BLACK--SCHOLES EQUATION

Erik Ekstr\"{o}m and  Johan Tysk

A bubble is characterized by the presence of an underlying asset whose
discounted price process is a strict local martingale under the pricing
measure. In such markets, many standard results from option pricing  
theory do
not hold, and in this paper we address some of these issues. In  
particular, we
derive existence and uniqueness results for the Black--Scholes  
equation, and we
provide convexity theory for option pricing and derive related  
ordering results
with respect to volatility. We show that American options are convexity
preserving, whereas European options preserve concavity for general  
payoffs and
convexity only for bounded contracts.


  http://arxiv.org/abs/0908.4468

---------------------------------------------------------------

9018. ON CONVERGENCE TO STATIONARITY OF FRACTIONAL BROWNIAN STORAGE

Michel Mandjes and  Ilkka Norros and  Peter Glynn

With $M(t):=\sup_{s\in[0,t]}A(s)-s$ denoting the running maximum of a
fractional Brownian motion $A(\cdot)$ with negative drift, this paper  
studies
the rate of convergence of $\mathbb {P}(M(t)>x)$ to $\mathbb{P}(M>x)$.  
We
define two metrics that measure the distance between the (complementary)
distribution functions $\mathbb{P}(M(t)>\cdot)$ and $\mathbb{P}(M> 
\cdot)$. Our
main result states that both metrics roughly decay as $\exp(-\vartheta
t^{2-2H})$, where $\vartheta$ is the decay rate corresponding to the  
tail
distribution of the busy period in an fBm-driven queue, which was  
computed
recently [Stochastic Process. Appl. (2006) 116 1269--1293]. The proofs
extensively rely on application of the well-known large deviations  
theorem for
Gaussian processes. We also show that the identified relation between  
the decay
of the convergence metrics and busy-period asymptotics holds in other  
settings
as well, most notably when G\"artner--Ellis-type conditions are  
fulfilled.


  http://arxiv.org/abs/0908.4472

---------------------------------------------------------------

9019. RANDOM RECURRENCE EQUATIONS AND RUIN IN A MARKOV-DEPENDENT  
STOCHASTIC  ECONOMIC ENVIRONMENT

Jeffrey F. Collamore

We develop sharp large deviation asymptotics for the probability of  
ruin in a
Markov-dependent stochastic economic environment and study the  
extremes for
some related Markovian processes which arise in financial and insurance
mathematics, related to perpetuities and the $\operatorname {ARCH}(1)$  
and
$\operatorname {GARCH}(1,1)$ time series models. Our results build  
upon work of
Goldie [Ann. Appl. Probab. 1 (1991) 126--166], who has developed tail
asymptotics applicable for independent sequences of random variables  
subject to
a random recurrence equation. In contrast, we adopt a general approach  
based on
the theory of Harris recurrent Markov chains and the associated theory  
of
nonnegative operators, and meanwhile develop certain recurrence  
properties for
these operators under a nonstandard "G\"artner--Ellis" assumption on the
driving process.


  http://arxiv.org/abs/0908.4479

---------------------------------------------------------------

9020. NON-MARKOV PROPERTY OF CERTAIN EIGENVALUE PROCESSES ANALOGOUS TO  
DYSON'S  MODEL

Ryoki Fukushima and  Atsushi Tanida and  Kouji Yano

It is proven that the eigenvalue process of Dyson's random matrix  
process of
size two becomes non-Markov if the common coefficient $1/\sqrt{2}$ in  
the
non-diagonal entries is replaced by a different positive number.


  http://arxiv.org/abs/0908.4481

---------------------------------------------------------------

9021. OPTIMAL REINSURANCE/INVESTMENT PROBLEMS FOR GENERAL INSURANCE  
MODELS

Yuping Liu and  Jin Ma

In this paper the utility optimization problem for a general insurance  
model
is studied. The reserve process of the insurance company is described  
by a
stochastic differential equation driven by a Brownian motion and a  
Poisson
random measure, representing the randomness from the financial market  
and the
insurance claims, respectively. The random safety loading and stochastic
interest rates are allowed in the model so that the reserve process is
non-Markovian in general. The insurance company can manage the  
reserves through
both portfolios of the investment and a reinsurance policy to optimize a
certain utility function, defined in a generic way. The main feature  
of the
problem lies in the intrinsic constraint on the part of reinsurance  
policy,
which is only proportional to the claim-size instead of the current  
level of
reserve, and hence it is quite different from the optimal
investment/consumption problem with constraints in finance. Necessary  
and
sufficient conditions for both well posedness and solvability will be  
given by
modifying the ``duality method'' in finance and with the help of the
solvability of a special type of backward stochastic differential  
equations.


  http://arxiv.org/abs/0908.4538

---------------------------------------------------------------

9022. RECURSIVE ESTIMATION OF TIME-AVERAGE VARIANCE CONSTANTS

Wei Biao Wu

For statistical inference of means of stationary processes, one needs to
estimate their time-average variance constants (TAVC) or long-run  
variances.
For a stationary process, its TAVC is the sum of all its covariances  
and it is
a multiple of the spectral density at zero. The classical TAVC  
estimate which
is based on batched means does not allow recursive updates and the  
required
memory complexity is O(n). We propose a faster algorithm which  
recursively
computes the TAVC, thus having memory complexity of order O(1) and the
computational complexity scales linearly in $n$. Under short-range  
dependence
conditions, we establish moment and almost sure convergence of the  
recursive
TAVC estimate. Convergence rates are also obtained.


  http://arxiv.org/abs/0908.4540

---------------------------------------------------------------

9023. ASYMPTOTIC BEHAVIOR OF UNSTABLE INAR(P) PROCESSES

Matyas Barczy and  Marton Ispany and  Gyula Pap

In this paper the asymptotic behavior of an unstable integer-valued
autoregressive model of order p (INAR(p)) is described. Under a natural
assumption it is proved that the sequence of appropriately scaled  
random step
functions formed from an unstable INAR(p) process converges weakly  
towards a
squared Bessel process. We note that this limit behavior is quite  
different
from that of familiar unstable autoregressive processes of order p.


  http://arxiv.org/abs/0908.4560

---------------------------------------------------------------

9024. ANALYSIS OF A STOCHASTIC PREDATOR-PREY MODEL WITH APPLICATIONS  
TO  INTRAHOST HIV GENETIC DIVERSITY

Sivan Leviyang

During an infection, HIV experiences strong selection by immune system T
cells. Recent experimental work has shown that MHC escape mutations  
form an
important pathway for HIV to avoid such selection. In this paper, we  
study a
model of MHC escape mutation. The model is a predator-prey model with  
two prey,
composed of two HIV variants, and one predator, the immune system CD8  
cells. We
assume that one HIV variant is visible to CD8 cells and one is not.  
The model
takes the form of a system of stochastic differential equations.  
Motivated by
well-known results concerning the short life-cycle of HIV intrahost,  
we assume
that HIV population dynamics occur on a faster time scale then CD8  
population
dynamics. This separation of time scales allows us to analyze our  
model using
an asymptotic approach.
   Using this model we study the impact of an MHC escape mutation on the
population dynamics and genetic evolution of the intrahost HIV  
population. From
the perspective of population dynamics, we show that the competition  
between
the visible and invisible HIV variants can reach steady states in  
which either
a single variant exists or in which coexistence occurs depending on the
parameter regime. We show that in some parameter regimes the end state  
of the
system is stochastic. From a genetics perspective, we study the impact  
of the
population dynamics on the lineages of HIV samples taken after an escape
mutation occurs. We show that the lineages go through severe  
bottlenecks and
that the lineage distribution can be characterized by a Kingman  
coalescent.


  http://arxiv.org/abs/0908.4569

---------------------------------------------------------------

9025. STABILITY OF A SPATIAL POLLING SYSTEM WITH GREEDY MYOPIC SERVICE

Lasse Leskel\"a and  Falk Unger

This paper studies a spatial queueing system on a circle, polled at  
random
locations by a myopic server that can only observe customers in a  
bounded
neighborhood. The server operates according to a greedy policy, always  
serving
the nearest customer in its neighborhood, and leaving the system  
unchanged at
polling instants where the neighborhood is empty. This system is  
modeled as a
measure-valued random process, which is shown to be positive recurrent  
under a
natural stability condition that does not depend on the server's scan  
range.
When the interpolling times are light-tailed, the stable system is  
shown to be
geometrically ergodic. We also briefly discuss how the stationary mean  
number
of customers behaves in light and heavy traffic.


  http://arxiv.org/abs/0908.4585

---------------------------------------------------------------

9026. STRICT POSITIVITY OF THE DENSITY FOR NON-LINEAR SPATIALLY  
HOMOGENEOUS  SPDES

Eulalia Nualart

In this paper, we consider a system of $k$ second order non-linear  
stochastic
differential equations with spatial dimension $d \geq 1$, driven by a
$k$-dimensional Gaussian noise, which is white in time and with some  
spatially
homogeneous covariance. We prove existence, smoothness, and strict  
positivity
of the density of the law of the solution of this system of equations,  
on the
set where the diffusion matrix is invertible, under sufficient  
conditions on
the fundamental solution $\Gamma$ of the deterministic equation. For  
this, we
apply techniques of Malliavin calculus. We apply this result to the  
case of the
stochastic heat equation in any space dimension and the the stochastic  
wave
equation in dimension $d\in \{1,2,3\}$, with a spatial covariance  
given by a
Riesz kernel. We then study the strict positivity of the density for  
the case
of a single equation ($k=1$), and apply it to the stochastic heat  
equation in
any space dimension, and the stochastic wave equation in dimension $d\in
\{1,2,3\}$, with a general spatial covariance.


  http://arxiv.org/abs/0908.4587

---------------------------------------------------------------

9027. ON THE SPECTRAL DIMENSION OF CAUSAL TRIANGULATIONS

Bergfinnur Durhuus and  Thordur Jonsson and  John F. Wheater

We introduce an ensemble of infinite causal triangulations, called the
uniform infinite causal triangulation, and show that it is equivalent  
to an
ensemble of infinite trees, the uniform infinite planar tree. It is  
proved that
in both cases the Hausdorff dimension almost surely equals 2. The  
infinite
causal triangulations are shown to be almost surely recurrent or,  
equivalently,
their spectral dimension is almost surely less than or equal to 2. We  
also
establish that for certain reduced versions of the infinite causal
triangulations the spectral dimension equals 2 both for the ensemble  
average
and almost surely. The triangulation ensemble we consider is  
equivalent to the
causal dynamical triangulation model of two-dimensional quantum  
gravity and
therefore our results apply to that model.


  http://arxiv.org/abs/0908.3643

---------------------------------------------------------------

9028. STOCHASTIC CAHN-HILLIARD EQUATION WITH DOUBLE SINGULAR  
NONLINEARITIES  AND TWO REFLECTIONS

Arnaud Debussche (IRMAR) and  Ludovic Gouden\`ege (IRMAR)

We consider a stochastic partial differential equation with two  
logarithmic
nonlinearities, with two reflections at 1 and -1 and with a constraint  
of
conservation of the space average. The equation, driven by the  
derivative in
space of a space-time white noise, contains a bi-Laplacian in the  
drift. The
lack of the maximum principle for the bi-Laplacian generates  
difficulties for
the classical penalization method, which uses a crucial monotonicity  
property.
Being inspired by the works of Debussche, Gouden\`ege and Zambotti, we  
obtain
existence and uniqueness of solution for initial conditions in the  
interval
$(-1,1)$. Finally, we prove that the unique invariant measure is  
ergodic, and
we give a result of exponential mixing.


  http://arxiv.org/abs/0908.4295


-----------------------------------
Stefano M. Iacus
Department of Economics,
Business and Statistics
University of Milan
Via Conservatorio, 7
I-20123 Milan - Italy
Ph.: +39 02 50321 461
Fax: +39 02 50321 505
http://www.economia.unimi.it/iacus
------------------------------------------------------------------------------------
Please don't send me Word or PowerPoint attachments if not
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From pas at lists.imstat.org  Wed Nov  4 02:43:17 2009
From: pas at lists.imstat.org (Probability Abstract Service)
Date: Wed, 04 Nov 2009 09:43:17 +0100
Subject: [PAS] Probability Abstracts 112
Message-ID: <2925F44D-A08D-4676-9B06-22CE72F84BF0@unimi.it>

Probability Abstracts 112
This document contains abstracts 9029-9332
from Sep-1-2009 to October-31-2009.
They have been mailed on Nov 4th, 2009.

9029. Multivariate Log-Concave Distributions as a Nearly Parametric  
Model
Author(s): Dominic Schuhmacher and Andre Huesler and Lutz Duembgen

Abstract: In this paper we show that the family P_d of probability  
distributions on R^d with log-concave densities satisfies a strong  
continuity condition. In particular, it turns out that weak  
convergence within this family entails (i) convergence in total  
variation distance, (ii) convergence of arbitrary moments, and (iii)  
pointwise convergence of Laplace transforms. Hence the nonparametric  
model P_d has similar properties as parametric models such as, for  
instance, the family of all d-variate Gaussian distributions.

http://arxiv.org/abs/0907.0250


9030. SDEs driven by a time-changed L\'evy process and their  
associated time-fractional order pseudo-differential equations
Author(s): Marjorie G. Hahn and Kei Kobayashi and Sabir Umarov

Abstract: It is known that if a stochastic process is a solution to a  
classical Ito stochastic differential equation (SDE), then its  
transition probabilities satisfy in the weak sense the associated  
Cauchy problem for the forward Kolmogorov equation. The forward  
Kolmogorov equation is a parabolic partial differential equation with  
coefficients determined by the corresponding SDE. Stochastic processes  
which are scaling limits of continuous time random walks have been  
connected with time-fractional differential equations. However, the  
class of SDEs that is associated with time-fractional Kolmogorov type  
equations is unknown. The present paper shows that in the cases of  
either time-fractional order or more general time-distributed order  
differential equations, the associated class of SDEs can be described  
within the framework of SDEs driven by semimartingales. These  
semimartingales are time-changed Levy processes where the independent  
time-change is given respectively by the inverse of a stable  
subordinator or the inverse of a mixture of independent stable  
subordinators.

http://arxiv.org/abs/0907.0253


9031. Brownian and fractional Brownian stochastic currents via  
Malliavin calculus
Author(s): Franco Flandoli and Ciprian Tudor (CES and SAMOS)

Abstract: By using Malliavin calculus and multiple Wiener-It\^o  
integrals, we study the existence and the regularity of stochastic  
currents defined as Skorohod (divergence) integrals with respect to  
the Brownian motion and to the fractional Brownian motion. We consider  
also the multidimensional multiparameter case and we compare the  
regularity of the current as a distribution in negative Sobolev spaces  
with its regularity in Watanabe space.

http://arxiv.org/abs/0907.0292


9032. A min-type stochastic fixed-point equation related to the  
smoothing transformation
Author(s): Gerold Alsmeyer and Matthias Meiners

Abstract: This paper is devoted to the study of the stochastic fixed- 
point equation X \stackrel{d}{=} \inf_{i \geq 1: T_i > 0} X_i/T_i and  
the connection with its additive counterpart $X \stackrel{d}{=} \sum_{i 
\ge 1}T_{i}X_{i}$ associated with the smoothing transformation. Here $ 
\stackrel{d}{=}$ means equality in distribution, $T := (T_i)_{i \geq  
1}$ is a given sequence of nonnegative random variables and $X,  
X_1, ...$ is a sequence of nonnegative i.i.d. random variables  
independent of $T$. We draw attention to the question of the existence  
of nontrivial solutions and, in particular, of special solutions named  
$\alpha$-regular solutions $(\alpha>0)$. We give a complete answer to  
the question of when $\alpha$-regular solutions exist and prove that  
they are always mixtures of Weibull distributions or certain periodic  
variants. We also give a complete characterization of all fixed points  
of this kind. A disintegration method which leads to the study of  
certain multiplicative martingales and a pathwise renewal equation  
after a suitable transform are the key tools for our analysis.  
Finally, we provide corresponding results for the fixed points of the  
related additive equation mentioned above. To some extent, these  
results have been obtained earlier by Iksanov.

http://arxiv.org/abs/0907.0300


9033. Interlacement percolation on transient weighted graphs
Author(s): Augusto Teixeira

Abstract: In this article, we first extend the construction of random  
interlacements, introduced by A.S. Sznitman in [arXiv:0704.2560], to  
the more general setting of transient weighted graphs. We prove the  
Harris-FKG inequality for this model and analyze some of its  
properties on specific classes of graphs. For the case of non-amenable  
graphs, we prove that the critical value u_* for the percolation of  
the vacant set is finite. We also prove that, once G satisfies the  
isoperimetric inequality IS_6 (see (1.5)), u_* is positive for the  
product GxZ (where we endow Z with unit weights). When the graph under  
consideration is a tree, we are able to characterize the vacant  
cluster containing some fixed point in terms of a Bernoulli  
independent percolation process. For the specific case of regular  
trees, we obtain an explicit formula for the critical value u_*.

http://arxiv.org/abs/0907.0316


9034. A functional combinatorial central limit theorem
Author(s): A. D. Barbour and Svante Janson

Abstract: The paper establishes a functional version of the Hoeffding  
combinatorial central limit theorem. First, a pre-limiting Gaussian  
process approximation is defined, and is shown to be at a distance of  
the order of the Lyapounov ratio from the original random process.  
Distance is measured by comparison of expectations of smooth  
functionals of the processes, and the argument is by way of Stein's  
method. The pre-limiting process is then shown, under weak conditions,  
to converge to a Gaussian limit process. The theorem is used to  
describe the shape of random permutation tableaux.

http://arxiv.org/abs/0907.0347


9035. Stability Properties of Linear File-Sharing Networks
Author(s): L. Leskel\"a (MSA) and Philippe Robert (INRIA) and Florian  
Simatos

Abstract: File-sharing networks are distributed systems used to  
disseminate files among a subset of the nodes of the Internet. A file  
is split into several pieces called chunks, the general simple  
principle is that once a node of the system has retrieved a chunk, it  
may become a server for this chunk. A stochastic model is considered  
for arrival times and durations of time to download chunks. One  
investigates the maximal arrival rate that such a network can  
accommodate, i.e., the conditions under which the Markov process  
describing this network is ergodic. Technical estimates related to the  
survival of interacting branching processes are key ingredients to  
establish the stability of these systems. Several cases are  
considered: networks with one and two chunks where a complete  
classification is obtained and several cases of a network with $n$  
chunks.

http://arxiv.org/abs/0907.0375


9036. Semimartingale decomposition of convex functions of continuous  
semimartingales by Brownian perturbation
Author(s): Nastasiya F Grinberg

Abstract: In this note we prove that the local martingale part of a  
convex function f of a d-dimensional semimartingale X=M+A can be  
wrtitten in terms of an Ito stochastic intergral of H(x), some  
measurable choice of subgradient of fat x, against M, the martingale  
part of X. This result was first proved by Bouleau in [2]. Here we  
present a new treatment of the problem.

http://arxiv.org/abs/0907.0382


9037. Majority dynamics on trees and the dynamic cavity method
Author(s): Yashodhan Kanoria and Andrea Montanari

Abstract: An elector sits on each vertex of an infinite tree of degree  
$k$, and has to decide between two alternatives. At each time step,  
each elector switches to the opinion of the majority of her neighbors.  
We analyze this majority process when opinions are initialized to  
independent and identically distributed random variables. In  
particular, we bound the threshold value of the initial bias such that  
the process converges to consensus. In order to prove an upper bound,  
we characterize the process of a single node in the large $k$-limit.  
This approach is inspired by the theory of mean field spin-glass and  
can potentially be generalized to a wider class of models. We also  
derive a lower bound that is non-trivial for small, odd values of $k$.

http://arxiv.org/abs/0907.0449


9038. A strong log-concavity property for measures on Boolean algebras
Author(s): Jeff Kahn and Michael Neiman

Abstract: We introduce the antipodal pairs property for probability  
measures on finite Boolean algebras and prove that conditional  
versions imply strong forms of log-concavity. We give several  
applications of this fact, including improvements of some results of  
Wagner; a new proof of a theorem of Liggett stating that ultra-log- 
concavity of sequences is preserved by convolutions; and some progress  
on a well-known log-concavity conjecture of J. Mason.

http://arxiv.org/abs/0907.0243


9039. A Cut-off Phenomenon in Location Based Random Access Games with  
Imperfect Information
Author(s): Hazer Inaltekin and Mung Chiang and H. Vincent Poor

Abstract: This paper analyzes the behavior of selfish transmitters  
under imperfect location information. The scenario considered is that  
of a wireless network consisting of selfish nodes that are randomly  
distributed over the network domain according to a known probability  
distribution, and that are interested in communicating with a common  
sink node using common radio resources. In this scenario, the wireless  
nodes do not know the exact locations of their competitors but rather  
have belief distributions about these locations. Firstly, properties  
of the packet success probability curve as a function of the node-sink  
separation are obtained for such networks. Secondly, a monotonicity  
property for the best-response strategies of selfish nodes is  
identified. That is, for any given strategies of competitors of a  
node, there exists a critical node-sink separation for this node such  
that its best-response is to transmit when its distance to the sink  
node is smaller than this critical threshold, and to back off  
otherwise. Finally, necessary and sufficient conditions for a given  
strategy profile to be a Nash equilibrium are provided.

http://arxiv.org/abs/0907.0255


9040. Self-Intersections of Random Geodesics on Negatively Curved  
Surfaces
Author(s): Steven P. Lalley

Abstract: We study the fluctuations of self-intersection counts of  
random geodesic segments of length $t$ on a compact, negatively curved  
surface in the limit of large $t$. If the initial direction vector of  
the geodesic is chosen according to the \emph{Liouville measure}, then  
it is not difficult to show that the number $N (t)$ of self- 
intersections by time $t$ grows like $\kappa t^{2}$, where $\kappa = 
\kappa_{M}$ is a positive constant depending on the surface $M$. We  
show that (for a smooth modification of $N (t)$) the fluctuations are  
of size $t$, and the limit distribution is a weak limit of Gaussian  
quadratic forms. We also show that the fluctuations of \emph 
{localized} self-intersection counts (that is, only self-intersections  
in a fixed subset of $M$ are counted) are typically of size $t^{3/2}$,  
and the limit distribution is Gaussian.

http://arxiv.org/abs/0907.0259


9041. Reducing the Ising model to matchings
Author(s): Mark Huber (Claremont McKenna College) and Jenny Law (Duke  
University)

Abstract: Canonical paths is one of the most powerful tools available  
to show that a Markov chain is rapidly mixing, thereby enabling  
approximate sampling from complex high dimensional distributions. Two  
success stories for the canonical paths method are chains for drawing  
matchings in a graph, and a chain for a version of the Ising model  
called the subgraphs world. In this paper, it is shown that a  
subgraphs world draw can be obtained by taking a draw from matchings  
on a graph that is linear in the size of the original graph. This  
provides a partial answer to why canonical paths works so well for  
both problems, as well as providing a new source of algorithms for the  
Ising model. For instance, this new reduction immediately yields a  
fully polynomial time approximation scheme for the Ising model on a  
bounded degree graph when the magnitization is bounded away from 0.

http://arxiv.org/abs/0907.0477


9042. Zeros of a two-parameter random walk
Author(s): Davar Khoshnevisan and Pal Revesz

Abstract: We prove that the number gamma(N) of the zeros of a two- 
parameter simple random walk in its first N-by-N time steps is almost  
surely equal to N to the power 1+o(1) as N goes to infinity. This is  
in contrast with our earlier joint effort with Z. Shi [4]; that work  
shows that the number of zero crossings in the first N-by-N time steps  
is N to the power (3/2)+o(1) as N goes to infinity. We prove also that  
the number of zeros on the diagonal in the first N time steps is (c+o 
(1)) log N as N goes to infinity, where c is 2\pi.

http://arxiv.org/abs/0907.0487


9043. Branching Random Walks in Space-Time Random Environment:  
Survival Probability, Global and Local Growth Rates
Author(s): Francis Comets and Nobuo Yoshida

Abstract: We study the survival probability and the growth rate for  
branching random walks in random environment (BRWRE). The particles  
perform simple symmetric random walks on the $d$-dimensional integer  
lattice, while at each time unit, they split into independent copies  
according to time-space i.i.d. offspring distributions. The BRWRE is  
naturally associated with the directed polymers in random environment  
(DPRE), for which the quantity called the free energy is well studied.  
We discuss the survival probability (both global and local) for BRWRE  
and give a criterion for its positivity in terms of the free energy of  
the associated DPRE. We also show that the global growth rate for the  
number of particles in BRWRE is given by the free energy of the  
associated DPRE, though the local growth rateis given by the  
directional free energy.

http://arxiv.org/abs/0907.0509


9044. Uniform estimates for metastable transition times in a coupled  
bistable system
Author(s): Florent Barret (CMAP) and Anton Bovier (IAM) and Sylvie M 
\'el\'eard (CMAP)

Abstract: We consider a coupled bistable N-particle system driven by a  
Brownian noise, with a strong coupling corresponding to the  
synchronised regime. Our aim is to obtain sharp estimates on the  
metastable transition times betwen the two stable states, both for  
fixed N and in the limit when N tends to infinity. These estimates  
would be the main step for a rigorous understanding of the metastable  
behavior of infinite dimensional systems, as the stochastically  
perturbed Ginzburg-Landau equation. The quantities of interest are  
objects of potential theory, as capacities and equilibrium measure. We  
prove estimates with error bounds that are uniform in the dimension of  
the system.

http://arxiv.org/abs/0907.0537


9045. Upper large deviations for maximal flows through a tilted cylinder
Author(s): Marie Theret

Abstract: We consider the standard first passage percolation model in $ 
\ZZ^d$ for $d\geq 2$ and we study the maximal flow from the upper half  
part to the lower half part (respectively from the top to the bottom)  
of a cylinder whose basis is a hyperrectangle of sidelength  
proportional to $n$ and whose height is $h(n)$ for a certain height  
function $h$. We denote this maximal flow by $\tau_n$ (respectively $ 
\phi_n$). We emphasize the fact that the cylinder may be tilted. We  
look at the probability that these flows, rescaled by the surface of  
the basis of the cylinder, are greater than $\nu(\vec{v})+\eps$ for  
some positive $\eps$, where $\nu(\vec{v})$ is the almost sure limit of  
the rescaled variable $\tau_n$ when $n$ goes to infinity. On one hand,  
we prove that the speed of decay of this probability in the case of  
the variable $\tau_n$ depends on the tail of the distribution of the  
capacities of the edges: it can decays exponentially fast with $n^ 
{d-1}$, or with $n^{d-1} \min(n,h(n))$, or at an intermediate regime.  
On the other hand, we prove that this probability in the case of the  
variable $\phi_n$ decays exponentially fast with the volume of the  
cylinder as soon as the law of the capacity of the edges admits one  
exponential moment; the importance of this result is however limited  
by the fact that $\nu(\vec{v})$ is not in general the almost sure  
limit of the rescaled maximal flow $\phi_n$, but it is the case at  
least when the height $h(n)$ of the cylinder is negligible compared to  
$n$.

http://arxiv.org/abs/0907.0614


9046. Central Limit Theorems for Multicolor Urns with Dominated Colors
Author(s): Patrizia Berti (Dip. di Matematica and Univ. Modena and  
Italy) and Irene Crimaldi (Dip. Matematica, Univ. Bologna, Italy),  
Luca Pratelli (Accademia Navale, Livorno, Italy), Pietro Rigo (Dip.  
Economia politica e Metodi quantitativi, Univ. Pavia, Italy)

Abstract: An urn contains balls of d colors. At each time, a ball is  
drawn and then replaced together with a random number of balls of the  
same color. Assuming that some colors are dominated by others, we  
prove central limit theorems. Some statistical applications are  
discussed.

http://arxiv.org/abs/0907.0676


9047. D\'eviations mod\'er\'ees de la distance chimique
Author(s): Olivier Garet (IECN) and R\'egine Marchand (IECN)

Abstract: In this paper, we establish moderate deviations for the  
chemical distance in Bernoulli percolation. The chemical distance  
between two points is the length of the shortest open path between  
these two points. Thus, we study the size of random fluctuations  
around the mean value, and also the asymptotic behavior of this mean  
value. The estimates we obtain improve our knowledge of the  
convergence to the asymptotic shape. Our proofs rely on concentration  
inequalities proved by Boucheron, Lugosi and Massart, and also on the  
approximation theory of subadditive functions initiated by Alexander.

http://arxiv.org/abs/0907.0697


9048. Moderate deviations for the chemical distance in Bernoulli  
percolation
Author(s): Olivier Garet (IECN) and R\'egine Marchand (IECN)

Abstract: In this paper, we establish moderate deviations for the  
chemical distance in Bernoulli percolation. The chemical distance  
between two points is the length of the shortest open path between  
these two points. Thus, we study the size of random fluctuations  
around the mean value, and also the asymptotic behavior of this mean  
value. The estimates we obtain improve our knowledge of the  
convergence to the asymptotic shape. Our proofs rely on concentration  
inequalities proved by Boucheron, Lugosi and Massart, and also on the  
approximation theory of subadditive functions initiated by Alexander.

http://arxiv.org/abs/0907.0698


9049. On the preservation of Gibbsianness under symbol amalgamation
Author(s): Jean-Rene Chazottes and Edgardo Ugalde

Abstract: Starting from the full-shift on a finite alphabet $A$,  
suppose we confound some symbols of $A$. This gives a new full shift  
on a new alphabet $B$. The amalgamation map, call it $\pi$, defines a  
`factor map', that is, a continuous transformation between $(A^\nn,T_A) 
$ and $(B^\nn,T_B)$ with the property that $\pi\circ T_A=T_B\circ \pi 
$, where $T_A$, resp. $T_B$, is the shift map on $A^\nn$, resp. $B^\nn 
$. Given a regular function $\psi:A^\nn\to\rr$ (a `potential'), there  
is a unique Gibbs measure $\mu_\psi$. In this article, we prove that,  
for a large class of potentials, the pushforward measure $\mu_\psi\circ 
\pi^{-1}$ is still Gibbsian for a potential $\phi:B^\nn\to\rr$ having  
a `bit less' regularity than $\psi$. In the special case where $\psi$  
is a `2-symbol' potential, the Gibbs measure $\mu_\psi$ is none other  
than a Markov measure and the amalgamation $\pi$ defines a hidden  
Markov chain. In that special case, our theorem can be recast by  
saying that a hidden Markov chain is a Gibbs measure (for a H\"older  
potential).

http://arxiv.org/abs/0907.0528


9050. Poincar\'e inequality and exponential integrability of hitting  
times for linear diffusions
Author(s): D. Loukianova and O. Loukianov and Sh. Song

Abstract: Let $X$ be a regular linear continuous positively recurrent  
Markov process with state space $\R$, scale function $S$ and speed  
measure $m$. For $a\in \R$ denote B^+_a&=\sup_{x\geq a} \m(]x,+\infty[) 
(S(x)-S(a)) B^-_a&=\sup_{x\leq a} \m(]-\infty;x[)(S(a)-S(x)) We study  
some characteristic relations between $B^+_a$, $B^-_a$, the  
exponential moments of the hitting times $T_a$ of $X$, the Hardy and  
Poincar\'e inequalities for the Dirichlet form associated with $X$. As  
a corollary, we establish the equivalence between the existence of  
exponential moments of the hitting times and the spectral gap of the  
generator of $X$.

http://arxiv.org/abs/0907.0762


9051. Boundary Harnack Inequality for alpha-harmonic functions on the  
Sierpi\'nski triangle
Author(s): Kamil Kaleta and Mateusz Kwa\'snicki

Abstract: We prove an uniform boundary Harnack inequality for  
nonnegative functions harmonic with respect to $\alpha$-stable process  
on the Sierpi{\'n}ski triangle, where $\alpha \in (0, 1)$. Our result  
requires no regularity assumptions on the domain of harmonicity.

http://arxiv.org/abs/0907.0793


9052. Duality and Intertwining for discrete Markov kernels: a relation  
and examples
Author(s): Thierry Huillet (LPTM) and Servet Martinez

Abstract: We work out some relations between duality and intertwining  
in the context of discrete Markov chains, fixing up the background of  
previous relations first established for birth and death chains and  
their Siegmund duals. In view of the results, the monotone properties  
resulting from the Siegmund dual of birth and death chains are  
revisited in some detail, with emphasis on the non neutral Moran  
model. We also introduce an ultrametric type dual extending the  
Siegmund kernel. Finally we discuss the sharp dual, following closely  
the Diaconis-Fill study.

http://arxiv.org/abs/0907.0840


9053. Diffusion approximation for the components in critical  
inhomogeneous random graphs of rank 1
Author(s): Tatyana S. Turova

Abstract: Consider the random graph on $n$ vertices $1, ..., n$. Each  
vertex $i$ is assigned a type $X_i$ with $X_1, ..., X_n$ being  
independent identically distributed as a nonnegative discrete random  
variable $X$. We assume that ${\bf E} X^3<\infty$. Given types of all  
vertices, an edge exists between vertices $i$ and $j$ independent of  
anything else and with probability $\min \{1, \frac{X_iX_j}{n}(1+\frac 
{a}{n^{1/3}}) \}$. We study the critical phase, which is known to take  
place when ${\bf E} X^2=1$. We prove that normalized by $n^{-2/3}$ the  
asymptotic joint distributions of component sizes of the graph equals  
the joint distribution of the excursions of a reflecting Brownian  
motion $B^a(s)$ with diffusion coefficient $\sqrt{{\bf E}X{\bf E}X^3}$  
and drift $a-\frac{1}{({\bf E}X)^2}s$. This shows that finiteness of $ 
{\bf E}X^3$ is the necessary condition for the diffusion limit. In  
particular, we conclude that the size of the largest connected  
component is of order $n^{2/3}$.

http://arxiv.org/abs/0907.0897


9054. Differentiability of quadratic BSDE generated by continuous  
martingales and hedging in incomplete markets
Author(s): Peter Imkeller and Anthony Reveillac and Anja Richter

Abstract: In this paper we consider a class of BSDE with drivers of  
quadratic growth, on a stochastic basis generated by continuous local  
martingales. We first derive the Markov property of a forward-backward  
system (FBSDE) if the generating martingale is a strong Markov  
process. Then we establish the differentiability of a FBSDE with  
respect to the initial value of its forward component. This enables us  
to obtain the main result of this article which from the perspective  
of a utility optimization interpretation of the underlying control  
problem on a financial market takes the following form. The control  
process of the BSDE steers the system into a random liability  
depending on a market external uncertainty and this way describes the  
optimal derivative hedge of the liability by investment in a capital  
market the dynamics of which is described by the forward component.  
This delta hedge is described in a key formula in terms of a  
derivative functional of the solution process and the correlation  
structure of the internal uncertainty captured by the forward process  
and the external uncertainty responsible for the market  
incompleteness. The formula largely extends the scope of validity of  
the results obtained by several authors in the Brownian setting,  
designed to give a genuinely stochastic representation of the optimal  
delta hedge in the context of cross hedging insurance derivatives  
generalizing the derivative hedge in the Black-Scholes model. Of  
course, Malliavin's calculus needed in the Brownian setting is not  
available in the general local martingale framework. We replace it by  
new tools based on stochastic calculus techniques.

http://arxiv.org/abs/0907.0941


9055. On the orthogonal component of BSDEs in a Markovian setting
Author(s): Anthony R\'eveillac

Abstract: In this Note we consider a quadratic backward stochastic  
differential equation (BSDE) driven by a continuous martingale $M$ and  
whose generator is a deterministic function. We prove (in Theorem \ref 
{theorem:main}) that if $M$ is a strong homogeneous Markov process and  
if the BSDE has the form \eqref{BSDE} then the unique solution $(Y,Z,N) 
$ of the BSDE is reduced to $(Y,Z)$, \textit{i.e.} the orthogonal  
martingale $N$ is equal to zero showing that in a Markovian setting  
the "usual" solution $(Y,Z)$ has not to be completed by a strongly  
orthogonal even if $M$ does not enjoy the martingale representation  
property.

http://arxiv.org/abs/0907.1071


9056. A constructive approach to the Monge-Kantorovich problem for  
chains of infinite order
Author(s): Antonio Galves and Nancy L. Garcia and Clementine Prieur

Abstract: We propose a constructive approach to solve the Monge- 
Kantorovich problem for chains of infinite order on a finite alphabet  
with an additive cost function. From this constructive description of  
the Kantorovich coupling we obtain, for any $\epsilon > 0$, a perfect  
simulation algorithm for sampling from an $\epsilon$-approximating  
coupling which assigns to the cost function an expectation which is $ 
\epsilon$-close to the minimum cost. Our approach is based on a  
regenerative scheme which enable us to construct the Kantorovich  
coupling as a mixture of product measures.

http://arxiv.org/abs/0907.1113


9057. Hsu-Robbins and Spitzer's theorems for the variations of  
fractional Brownian motion
Author(s): Ciprian Tudor (CES and Samos)

Abstract: Using recent results on the behavior of multiple Wiener-It 
\^o integrals based on Stein's method, we prove Hsu-Robbins and  
Spitzer's theorems for sequences of correlated random variables  
related to the increments of the fractional Brownian motion.

http://arxiv.org/abs/0907.1116


9058. Convergence to L\'evy stable processes under strong mixing  
conditions
Author(s): Marta Tyran-Kaminska

Abstract: For a strictly stationary sequence of random vectors in $ 
\mathbb{R}^d$ we study convergence of partial sums processes to L\'evy  
stable process in the Skorohod space with $J_1$-topology. We identify  
necessary and sufficient conditions for such convergence and provide  
sufficient conditions when the stationary sequence is strongly mixing.

http://arxiv.org/abs/0907.1185


9059. An application to credit risk of a hybrid Monte Carlo-Optimal  
quantization method
Author(s): Giorgia Callegaro and Abass Sagna (PMA)

Abstract: In this paper we use a hybrid Monte Carlo-Optimal  
quantization method to approximate the conditional survival  
probabilities of a firm, given a structural model for its credit  
defaul, under partial information. We consider the case when the  
firm's value is a non-observable stochastic process $(V_t)_{t \geq 0}$  
and inverstors in the market have access to a process $(S_t)_{t \geq  
0}$, whose value at each time t is related to $(V_s, s \leq t)$. We  
are interested in the computation of the conditional survival  
probabilities of the firm given the "investor information". As a  
application, we analyse the shape of the credit spread curve for zero  
coupon bonds in two examples.

http://arxiv.org/abs/0907.0645


9060. Perimeter and Area of the Convex Hull of N Planar Brownian Motions
Author(s): Julien Randon-Furling and Satya N. Majumdar and Alain Comtet

Abstract: We compute exactly the mean perimeter and area of the convex  
hull of N independent planar Brownian paths each of duration T, both  
for open and closed paths. We show that the mean perimeter < L_N > =  
\alpha_N, \sqrt{T} and the mean area = \beta_N T for all T. The  
prefactors \alpha_N and \beta_N, computed exactly for all N, increase  
very slowly (logarithmically) with increasing N. This slow growth is a  
consequence of extreme value statistics and has interesting  
implication in ecological context in estimating the home range of a  
herd of animals with population size N.

http://arxiv.org/abs/0907.0921


9061. Distributed Random Access Algorithm: Scheduling and Congesion  
Control
Author(s): Libin Jiang and Devavrat Shah and Jinwoo Shin and Jean  
Walrand

Abstract: This paper provides proofs of the rate stability, Harris  
recurrence, and epsilon-optimality of CSMA algorithms where the  
backoff parameter of each node is based on its backlog. These  
algorithms require only local information and are easy to implement.  
The setup is a network of wireless nodes with a fixed conflict graph  
that identifies pairs of nodes whose simultaneous transmissions  
conflict. The paper studies two algorithms. The first algorithm  
schedules transmissions to keep up with given arrival rates of  
packets. The second algorithm controls the arrivals in addition to the  
scheduling and attempts to maximize the sum of the utilities of the  
flows of packets at the different nodes. For the first algorithm, the  
paper proves rate stability for strictly feasible arrival rates and  
also Harris recurrence of the queues. For the second algorithm, the  
paper proves the epsilon-optimality. Both algorithms operate with  
strictly local information in the case of decreasing step sizes, and  
operate with the additional information of the number of nodes in the  
network in the case of constant step size.

http://arxiv.org/abs/0907.1266


9062. Dynkin's isomorphism theorem and the stochastic heat equation
Author(s): Nathalie Eisenbaum and Mohammud Foondun and Davar  
Khoshnevisan

Abstract: Consider the stochastic heat equation $\partial_t u = \sL u  
+ \dot{W}$, where $\sL$ is the generator of a [Borel right] Markov  
process in duality. We show that the solution is locally mutually  
absolutely continuous with respect to a smooth perturbation of the  
Gaussian process that is associated, via Dynkin's isomorphism theorem,  
to the local times of the replica-symmetric process that corresponds  
to $\sL$.In the case that $\sL$ is the generator of a L\'evy process  
on $\R^d$, our result gives a probabilistic explanation of the recent  
findings of Foondun et al.

http://arxiv.org/abs/0907.1316


9063. On the discretization of backward doubly stochastic differential  
equations
Author(s): Omar Aboura (CES and Samos)

Abstract: In this paper, we are dealing with the approximation of the  
process (Y,Z) solution to the backward doubly stochastic differential  
equation with the forward process X . After proving the L2-regularity  
of Z, we use the Euler scheme to discretize X and the Zhang approach  
in order to give a discretization scheme of the process (Y,Z).

http://arxiv.org/abs/0907.1406


9064. Existence and uniqueness of solutions for Fokker-Planck  
equations on Hilbert spaces
Author(s): Vladimir Bogachev and Giuseppe Da Prato and Michael Roeckner

Abstract: We consider a stochastic differential equation in a Hilbert  
space with time-dependent coefficients for which no general existence  
and uniqueness results are known. We prove, under suitable  
assumptions, existence and uniqueness of a measure valued solution,  
for the corresponding Fokker--Planck equation. In particular, we  
verify the Chapman--Kolmogorov equations and get an evolution system  
of transition probabilities for the stochastic dynamics informally  
given by the stochastic differential equation.

http://arxiv.org/abs/0907.1431


9065. Limit distributions for large P\'olya urns
Author(s): Brigitte Chauvin and Nicolas Pouyanne and R\'eda Sahnoun

Abstract: We consider a two colors P\'olya urn with balance $S$.  
Assume it is a \emph{large} urn \emph{i.e.} the second eigenvalue $m$  
of the replacement matrix satisfies $1/2

http://arxiv.org/abs/0907.1477


9066. Inhomogeneity and universality: off-critical behavior of  
interfaces
Author(s): Pierre Nolin

Abstract: We further study the interfaces arising in a situation of  
inhomogeneity. More precisely, we identify a characteristic length for  
the gradient percolation model, that enables us to tighten previous  
estimates established for it. This allows to construct non-trivial  
scaling limits: the limiting objects share some properties with  
critical percolation interfaces, but locally, they rather behave like  
off-critical percolation interfaces.

http://arxiv.org/abs/0907.1495


9067. Les Probabilit\'es D\'efaillance comme Indicateurs de  
Performance des Barri\`eres Techniques de S\'ecurit\'e ? Approche  
Analytique
Author(s): Florent Brissaud (INERIS and UTT) and Brice Lanternier  
(INERIS)

Abstract: French environmental laws require industrialists to include  
probability criteria in risk assessments, especially to define  
confidence levels for risk management measures. This paper presents  
the failure probabilities as efficient indicators for technical safety  
barrier performances. Generic formulas are proposed to evaluate these  
probabilities, including failure rate, barrier architecture, full and  
partial proof tests. In many cases, these results can be directly used  
to assess safety barrier confidence levels.

http://arxiv.org/abs/0907.1516


9068. A Remark on Zeros of Brownian Motion
Author(s): Weber Michel

Abstract: Let $ \{W(t), t\ge 0\}$ be a standard Brownian motion. If $I 
$ is a bounded interval on which $W $ has no zero, an almost sure  
lower bound to $\inf\{|W(t)|, t\in I\}$ can be provided, when $I$ is  
taken from a given countable family of intervals covering the positive  
half-line.

http://arxiv.org/abs/0907.1572


9069. Symmetrization of L\'evy processes and applications
Author(s): Rodrigo Banuelos and Pedro J. Mendez-Hernandez

Abstract: It is shown that many of the classical generalized  
isoperimetric inequalities for the Laplacian when viewed in terms of  
Brownian motion extend to a wide class of Levy processes. The results  
are derived from the multiple integral inequalities of Brascamp, Lieb  
and Luttinger but the probabilistic structure of the processes plays a  
crucial role in the proofs.

http://arxiv.org/abs/0907.1598


9070. Are fractional Brownian motions predictable?
Author(s): Adam Jakubowski

Abstract: We provide a device, called the local predictor, which  
extends the idea of the predictable compensator. It is shown that a  
fBm with the Hurst index greater than 1/2 coincides with its local  
predictor while fBm with the Hurst index smaller than 1/2 does not  
admit any local predictor. The local predictor of a martingale (in  
particular: Brownian motion) trivially exists and equals 0.

http://arxiv.org/abs/0907.1618


9071. Random walks on discrete cylinders with large bases and random  
interlacements
Author(s): David Windisch

Abstract: Following the recent work of Sznitman (arXiv:0805.4516), we  
investigate the microscopic picture induced by a random walk  
trajectory on a cylinder of the form G_N x Z, where G_N is a large  
finite connected weighted graph, and relate it to the model of random  
interlacements on infinite transient weighted graphs. Under suitable  
assumptions, the set of points not visited by the random walk until a  
time of order |G_N|^2 in a neighborhood of a point with Z-component of  
order |G_N| converges in distribution to the law of the vacant set of  
a random interlacement on a certain limit model describing the  
structure of the graph in the neighborhood of the point. The level of  
the random interlacement depends on the local time of a Brownian  
motion. The result also describes the limit behavior of the joint  
distribution of the local pictures in the neighborhood of several  
distant points with possibly different limit models. As examples of  
G_N, we treat the d-dimensional box of side length N, the Sierpinski  
graph of depth N and the d-ary tree of depth N, where d >= 2.

http://arxiv.org/abs/0907.1627


9072. Fluctuations of the nodal length of random spherical harmonics
Author(s): Igor Wigman

Abstract: Using the multiplicities of the Laplace eigenspace on the  
sphere (the space of spherical harmonics) we endow the space with  
Gaussian probability measure. This induces a notion of random Gaussian  
spherical harmonics of degree $n$ having Laplace eigenvalue $E=n(n 
+1)$. We study the length distribution of the nodal lines of random  
spherical harmonics. It is known that the expected length is of order  
$n$. It is natural to conjecture that the variance should be of order  
$n$, due to the natural scaling. Our principal result is that, due to  
an unexpected cancelation, the variance of the nodal length of random  
spherical harmonics is of order $\log{n}$. This behaviour is  
consistent to the one predicted by Berry for nodal lines on chaotic  
billiards (Random Wave Model). In addition we find that a similar  
result is applicable for "generic" linear statistics of the nodal lines.

http://arxiv.org/abs/0907.1648


9073. Some almost sure results for unbounded functions of intermittent  
maps and their associated Markov chains
Author(s): Jerome Dedecker (LSTA) and Sebastien Gouezel (IRMAR) and  
Florence Merlevede (LAMA)

Abstract: We consider a large class of piecewise expanding maps T of  
[0,1] with a neutral fixed point, and their associated Markov chain  
Y_i whose transition kernel is the Perron-Frobenius operator of T with  
respect to the absolutely continuous invariant probability measure. We  
give a large class of unbounded functions f for which the partial sums  
of f\circ T^i satisfy both a central limit theorem and a bounded law  
of the iterated logarithm. For the same class, we prove that the  
partial sums of f(Y_i) satisfy a strong invariance principle. When the  
class is larger, so that the partial sums of f\circ T^i may belong to  
the domain of normal attraction of a stable law of index p\in (1, 2),  
we show that the almost sure rates of convergence in the strong law of  
large numbers are the same as in the corresponding i.i.d. case.

http://arxiv.org/abs/0907.1403


9074. Almost sure invariance principle for dynamical systems by  
spectral methods
Author(s): Sebastien Gouezel (IRMAR)

Abstract: We prove the almost sure invariance principle for stationary  
R^d--valued processes (with dimension-independent very precise error  
terms), solely under a strong assumption on the characteristic  
functions of these processes. This assumption is easy to check for  
large classes of dynamical systems or Markov chains, using strong or  
weak spectral perturbation arguments.

http://arxiv.org/abs/0907.1404


9075. Forest fires on $\Z_+$ with ignition only at 0
Author(s): Stanislav Volkov

Abstract: We consider a version of the forest fire model on graph $G$,  
where each vertex of a graph becomes occupied with rate one. A fixed  
vertex $v_0$ is hit by lightning with the same rate, and when this  
occurs, the whole cluster of occupied vertices containing $v_0$ is  
burnt out. We show that when $G=Z_{+}$, the times between consecutive  
burnouts at vertex $n$, divided by $\log n$, converge weakly as $n\to 
\infty$ to a random variable which distribution is $1-\rho(x)$ where $ 
\rho(x)$ is the Dickman function. We also show that on transitive  
graphs with a non-trivial site percolation threshold and one infinite  
cluster at most, the distributions of the time till the first burnout  
of {\it any} vertex have exponential tails. Finally, we give an  
elementary proof of an interesting limit: $\lim_{n\to\infty} \sum_{k=1} 
^n {n \choose k} (-1)^k \log k -\log\log n=\gamma$.

http://arxiv.org/abs/0907.1821


9076. Queueing with neighbours
Author(s): Vadim Shcherbakov and Stanislav Volkov

Abstract: In this paper we study asymptotic behaviour of a growth  
process generated by a semi-deterministic variant of cooperative  
sequential adsorption model (CSA). This model can also be viewed as a  
particular queueing system with local interactions. We show that quite  
limited randomness of the model still generates a rich collection of  
possible limiting behaviours.

http://arxiv.org/abs/0907.1826


9077. Estimates on the speedup and slowdown for a diffusion in a  
drifted brownian potential
Author(s): Gabriel Faraud

Abstract: We study a model of diffusion in a brownian potential. This  
model was firstly introduced by T. Brox (1986) as a continuous time  
analogue of random walk in random environment. We estimate the  
deviations of this process above or under its typical behavior. Our  
results rely on different tools such as a representation introduced by  
Y. Hu, Z. Shi and M. Yor, Kotani's lemma, introduced at first by K.  
Kawazu and H. Tanaka (1997), and a decomposition of hitting times  
developed in a recent article by A. Fribergh, N. Gantert and S. Popov  
(2008). Our results are in agreement with their results in the  
discrete case.

http://arxiv.org/abs/0907.1864


9078. Hidden Markov processes in the context of symbolic dynamics
Author(s): Mike Boyle (University of Maryland) and Karl Petersen  
(University of North Carolina)

Abstract: In an effort to aid communication among different fields and  
perhaps facilitate progress on problems common to all of them, this  
article discusses hidden Markov processes from several viewpoints,  
especially that of symbolic dynamics, where they are known as sofic  
measures, or continuous shift-commuting images of Markov measures. It  
provides background, describes known tools and methods, surveys some  
of the literature, and proposes several open problems.

http://arxiv.org/abs/0907.1858


9079. The triangle and the open triangle
Author(s): Gady Kozma

Abstract: We show that for percolation on any transitive graph, the  
triangle condition implies the open triangle condition.

http://arxiv.org/abs/0907.1959


9080. Lp-solution of backward doubly stochastic differential equations
Author(s): Auguste Aman (LMAI)

Abstract: In this paper, our goal is solving backward doubly  
stochastic differential equation (BDSDE for short) under weak  
assumptions on the data. The first part of the paper is devoted to the  
development of some new technical aspects of stochastic calculus  
related to BDSDEs. Then we derive a priori estimates and prove  
existence and uniqueness of solutions, extending the results of  
Pardoux and Peng \cite{PP1} to the case where the solution is taked in  
$L^{p}, p>1$ and the monotonicity conditions are satisfied. This study  
is limited to deterministic terminal time.

http://arxiv.org/abs/0907.1983


9081. On the Optimal Amount of Experimentation in Sequential Decision  
Problems
Author(s): Dinah Rosenberg and Eilon Solan and Nicolas Vieille

Abstract: We provide a tight bound on the amount of experimentation  
under the optimal strategy in sequential decision problems. We show  
the applicability of the result by providing a bound on the cut-off in  
a one-arm bandit problem.

http://arxiv.org/abs/0907.2002


9082. New rates for exponential approximation and the theorems of R 
\'enyi and Yaglom
Author(s): Erol Pek\"oz and Adrian R\"ollin

Abstract: We introduce two abstract theorems that reduce a variety of  
complex exponential distributional approximation problems to the  
construction of couplings. These are applied to obtain rates of  
convergence with respect to the Wasserstein and Kolmogorov metrics for  
the theorem of R\'enyi on random sums and generalizations of it,  
hitting times for Markov chains, and to obtain a new rate for the  
classical theorem of Yaglom on the exponential asymptotic behavior of  
a critical Galton-Watson process conditioned on non-extinction. The  
primary tools are an adaptation of Stein's method, Stein couplings, as  
well as the equilibrium distributional transformation from renewal  
theory.

http://arxiv.org/abs/0907.2009


9083. L$^{p}$-solution of reflected generalized BSDEs with non- 
Lipschitz coefficients
Author(s): Auguste Aman (LMAI)

Abstract: In this paper, we continue in solving reflected generalized  
backward stochastic differential equations (RGBSDE for short) and  
fixed terminal time with use some new technical aspects of the  
stochastic calculus related to the reflected generalized BSDE. Here,  
existence and uniqueness of solution is proved under a non-Lipschitz  
condition on the coefficients.

http://arxiv.org/abs/0907.2032


9084. Numerical scheme for backward doubly stochastic differential  
equations
Author(s): Auguste Aman (LMAI)

Abstract: We study a discrete-time approximation for solutions of  
systems of decoupled forward-backward doubly stochastic differential  
equations (FBDSDEs). Assuming that the coefficients are Lipschitz- 
continuous, we prove the convergence of the scheme when the step of  
time discretization, $|\pi|$ goes to zero. The rate of convergence is  
exactly equal to $|\pi|^{1/2}$. The proof is based on a generalization  
of a remarkable result on the $^{2}$-regularity of the solution of the  
backward equation derived by J. Zhang

http://arxiv.org/abs/0907.2035


9085. Homeomorphism of solutions to backward doubly SDEs and  
applications
Author(s): Auguste Aman (LMAI)

Abstract: In this paper we study the homeomorphic properties of the  
solutions to one dimensional backward doubly stochastic differential  
equations under suitable assumptions, where the terminal values depend  
on a real parameter. Then, we apply them to the solutions for a class  
of second order quasilinear parabolic stochastic partial differential  
equations.

http://arxiv.org/abs/0907.2036


9086. Reflected generalized backward doubly SDEs driven by L\'evy  
processes and Applications
Author(s): Auguste Aman (LMAI)

Abstract: In this paper, a class of reflected generalized backward  
doubly stochastic differential equations (reflected GBDSDEs in short)  
driven by Teugels martingales associated with L\'{e}vy process and the  
integral with respect to an adapted continuous increasing process is  
investigated. We obtain the existence and uniqueness of solutions to  
these equations. A probabilistic interpretation for solutions to a  
class of reflected stochastic partial differential integral equations  
(PDIEs in short) with a nonlinear Neumann boundary condition is given.

http://arxiv.org/abs/0907.2037


9087. Stochastic 2D hydrodynamical systems: Support theorem
Author(s): Igor Chueshov and Annie Millet (SAMOS and Ces and Pma)

Abstract: We deal with a class of abstract nonlinear stochastic models  
with multiplicative noise, which covers many 2D hydrodynamical models  
including the 2D Navier-Stokes equation, 2D MHD models and 2D magnetic  
B\'enard problems as well as some shell models of turbulence. Our main  
result describes the support of the distribution of solutions. Both  
inclusions are proved by means of a general result of convergence in  
probability for non linear stochastic PDEs driven by a Hilbert-valued  
Brownian motion and some adapted finite dimensional approximation of  
this process.

http://arxiv.org/abs/0907.2100


9088. Perfect simulation for stochastic chains with unbounded variable  
length memory
Author(s): Alexsandro Gallo

Abstract: We present a new perfect simulation algorithm for stationary  
chains (indexed by $\mathbb{Z}$) having unbounded variable length  
memory. This is the class of infinite memory chains for which the  
family of transition probabilities is represented through the form of  
a \emph{probabilistic context tree}. Our condition is expressed in  
terms of the structure of the context tree. In particular, we do not  
assume the continuity of the family of transition probabilities. We  
give an explicit construction of the chain using a sequence of i.i.d.  
random variables uniformly distributed in $[0,1[$.

http://arxiv.org/abs/0907.2150


9089. On the Domination of Random Walk on a Discrete Cylinder by  
Random Interlacements
Author(s): Alain-Sol Sznitman

Abstract: We consider simple random walk on a discrete cylinder with  
base a large d-dimensional torus of side-length N, when d is two or  
more. We develop a stochastic domination control on the local picture  
left by the random walk in boxes of side-length almost of order N, at  
certain random times comparable to the square of the number of sites  
in the base. We show a domination control in terms of the trace left  
in similar boxes by random interlacements in the infinite (d+1)- 
dimensional cubic lattice at a suitably adjusted level. As an  
application we derive a lower bound on the disconnection time of the  
discrete cylinder, which as a by-product shows the tightness of the  
laws of the ratio of the square of the number of sites in the base to  
the disconnection time. This fact had previously only been established  
when d is at least 17, in arXiv: math/0701414.

http://arxiv.org/abs/0907.2184


9090. A Path Guessing Game with Wagering
Author(s): Marcus Pendergrass

Abstract: We consider a two-player game in which the first player (the  
Guesser) tries to guess, edge-by-edge, the path that second player  
(the Chooser) takes through a directed graph. At each step, the  
Guesser makes a wager as to the correctness of her guess, and receives  
a payoff proportional to her wager if she is correct. We derive  
optimal strategies for both players for various classes of graphs, and  
describe the Markov-chain dynamics of the game under optimal play.  
These results are applied to the infinite-duration Lying Oracle Game,  
in which the Guesser must use information provided by an unreliable  
Oracle to predict the outcome of a coin toss.

http://arxiv.org/abs/0907.2196


9091. On the philosophy of Cram\'er-Rao-Bhattacharya Inequalities in  
Quantum Statistics
Author(s): K. R. Parthasarathy

Abstract: To any parametric family of states of a finite level quantum  
system we associate a space of Fisher maps and introduce the natural  
notions of Cram\'er-Rao-Bhattacharya tensor and Fisher information  
form. This leads us to an abstract Cram\'er-Rao-Bhattacharya lower  
bound for the covariance matrix of any finite number of unbiased  
estimators of parameteric functions. A number of illustrative examples  
is included. Modulo technical assumptions of various kinds our methods  
can be applied to infinite level quantum systems as well as parametric  
families of classical probability distributions on Borel spaces.

http://arxiv.org/abs/0907.2210


9092. Optimal investment on finite horizon with random discrete order  
flow in illiquid markets
Author(s): Paul Gassiat (PMA) and Huyen Pham (PMA and CREST) and Mihai  
Sirbu

Abstract: We study the problem of optimal portfolio selection in an  
illiquid market with discrete order flow. In this market, bids and  
offers are not available at any time but trading occurs more  
frequently near a terminal horizon. The investor can observe and trade  
the risky asset only at exogenous random times corresponding to the  
order flow given by an inhomogenous Poisson process. By using a direct  
dynamic programming approach, we first derive and solve the fixed  
point dynamic programming equation satisfied by the value function,  
and then perform a verification argument which provides the existence  
and characterization of optimal trading strategies. We prove the  
convergence of the optimal performance, when the deterministic  
intensity of the order flow approaches infinity at any time, to the  
optimal expected utility for an investor trading continuously in a  
perfectly liquid market model with no-short sale constraints.

http://arxiv.org/abs/0907.2203


9093. A Shape Theorem for Riemannian First-Passage Percolation
Author(s): Tom LaGatta and Jan Wehr

Abstract: Riemannian first-passage percolation (FPP) is a continuum  
analogue of standard FPP on the lattice, where the discrete passage  
times of standard FPP are replaced by a random Riemannian metric. We  
prove a shape theorem for this model--that balls in this metric grow  
linearly in time--and from this conclude that the metric is complete.

http://arxiv.org/abs/0907.2228


9094. Heavy tail phenomenon and convergence to stable laws iterated  
Lipschitz maps
Author(s): Mariusz Mirek

Abstract: We consider the Markov chain $\{X_n^x\}_{n=0}^\infty$ on $ 
\R^d$ defined by the stochastic recursion $X_{n}^{x}=\p_{\theta_{n}}(X_ 
{n-1}^{x})$, starting at $x\in\R^d$, where $\theta_{1}, \theta_{2}, ... 
$ are i.i.d. random variables taking their values in a matric space $ 
(\Theta, d)$ and $\p_{\theta_{n}}:\R^d\mapsto\R^d$ are Lipschitz maps.  
Assume that the Markov chain has a unique stationary measure $\nu$.  
Under appropriate assumptions on $\p_{\theta_n}$ we will show that the  
measure $\nu$ has a heavy tail with the exponent $\alpha>0$ i.e. $\nu(\ 
{x\in\R^d: |x|>t\})\asymp t^{-\alpha}$. Using this result we show that  
properly normalized Birkhoff sums $S_n^x=\sum_{k=0}^n X_k^x$, converge  
in law to an $\alpha$--stable laws for $\alpha\in(0, 2]$.

http://arxiv.org/abs/0907.2261


9095. Uniform Modulus of Continuity of Random Fields
Author(s): Yimin Xiao

Abstract: A sufficient condition for the uniform modulus of continuity  
of a random field $X = \{X(t), t \in \R^N\}$ is provided. The result  
is applicable to random fields with heavy-tailed distribution such as  
stable random fields.

http://arxiv.org/abs/0907.2291


9096. Spectral Analysis of Multi-dimensional Self-similar Markov  
Processes
Author(s): N. Modarresi and S. Rezakhah

Abstract: In this paper we consider a wide sense discrete scale  
invariant process with scale $l>1$. We consider to have $T$ samples at  
each scale, and choose $\alpha$ by the equality $l=\alpha^T$. Our  
special scheme of sampling is to choose our samples at discrete points  
$\alpha^k, k\in W$. So we provide a discrete time wide sense scale  
invariant(DT-SI) process. We find the spectral representation of the  
covariance function of such DT-SI process. By providing harmonic like  
representation of multi-dimensional self-similar processes, spectral  
density function of them are presented. We also consider a discrete  
time scale invariance Markov(DT-SIM) process with the above scheme of  
sampling at points $\alpha^k, k\in {\bf W}$ and show that the spectral  
density matrix of DT-SIM process and its associated $T$-dimensional  
self-similar Markov process is fully specified by $\{R_{j}^H(1),R_{j}^H 
(0),j=0, 1, ..., T-1\}$ where $R_{j}^H(\tau)=\mathrm{Cov}\big(X(\alpha^ 
{j+\tau}),X(\alpha^j)\big)$

http://arxiv.org/abs/0907.2295


9097. Heat Kernel Upper Bounds on Long Range Percolation Clusters
Author(s): Nicholas Crawford and Allan Sly

Abstract: In this paper, we derive upper bounds for the heat kernel of  
the simple random walk on the infinite cluster of a supercritical long  
range percolation process. For any $d \geq 1$ and for any exponent $s  
\in (d, (d+2) \wedge 2d)$ giving the rate of decay of the percolation  
process, we show that the return probability decays like $t^{-\ffrac{d} 
{s-d}}$ up to logarithmic corrections, where $t$ denotes the time the  
walk is run. Moreover, our methods also yield generalized bounds on  
the spectral gap of the dynamics and on the diameter of the largest  
component in a box. Besides its intrinsic interest, the main result is  
needed for a companion paper studying the scaling limit of simple  
random walk on the infinite cluster.

http://arxiv.org/abs/0907.2434


9098. A graph-based equilibrium problem for the limiting distribution  
of non-intersecting Brownian motions at low temperature
Author(s): Steven Delvaux and Arno B.J. Kuijlaars

Abstract: We consider n non-intersecting Brownian motion paths with p  
prescribed starting positions at time t=0 and q prescribed ending  
positions at time t=1. The positions of the paths at any intermediate  
time are a determinantal point process, which in the case p=1 is  
equivalent to the eigenvalue distribution of a random matrix from the  
Gaussian unitary ensemble with external source. For general p and q,  
we show that if a temperature parameter is sufficiently small, then  
the distribution of the Brownian paths is characterized in the large n  
limit by a vector equilibrium problem with an interaction matrix that  
is based on a bipartite planar graph. Our proof is based on a steepest  
descent analysis of an associated (p+q) by (p+q) matrix valued Riemann- 
Hilbert problem whose solution is built out of multiple orthogonal  
polynomials. A new feature of the steepest descent analysis is a  
systematic opening of a large number of global lenses.

http://arxiv.org/abs/0907.2310


9099. 3-Connected Cores In Random Planar Graphs
Author(s): Nikolaos Fountoulakis and Konstantinos Panagiotou

Abstract: The study of the structural properties of large random  
planar graphs has become in recent years a field of intense research  
in computer science and discrete mathematics. Nowadays, a random  
planar graph is an important and challenging model for evaluating  
methods that are developed to study properties of random graphs from  
classes with structural side constraints. In this paper we focus on  
the structure of random biconnected planar graphs regarding the sizes  
of their 3-connected building blocks, which we call cores. In fact, we  
prove a general theorem regarding random biconnected graphs. If B_n is  
a graph drawn uniformly at random from a class B of labeled  
biconnected graphs, then we show that with probability 1-o(1) B_n  
belongs to exactly one of the following categories: (i) Either there  
is a unique giant core in B_n, that is, there is a 0 < c < 1 such that  
the largest core contains ~ cn vertices, and every other core contains  
at most n^a vertices, where 0 < a < 1; (ii) or all cores of B_n  
contain O(log n) vertices. Moreover, we find the critical condition  
that determines the category to which B_n belongs, and also provide  
sharp concentration results for the counts of cores of all sizes  
between 1 and n. As a corollary, we obtain that a random biconnected  
planar graph belongs to category (i), where in particular c = 0.765...  
and a = 2/3.

http://arxiv.org/abs/0907.2326


9100. On divergence form SPDEs with growing coefficients in $W^{1}_ 
{2}$ spaces without weights
Author(s): N.V. Krylov

Abstract: We consider divergence form uniformly parabolic SPDEs with  
bounded and measurable leading coefficients and possibly growing lower- 
order coefficients in the deterministic part of the equations. We look  
for solutions which are summable to the second power with respect to  
the usual Lebesgue measure along with their first derivatives with  
respect to the spatial variable.

http://arxiv.org/abs/0907.2467


9101. On the structure of Gaussian random variables
Author(s): Ciprian Tudor (CES and Samos)

Abstract: We study when a given Gaussian random variable on a given  
probability space $(\Omega, {\cal{F}}, P) $ is equal almost surely to $ 
\beta_{1}$ where $\beta $ is a Brownian motion defined on the same (or  
possibly extended) probability space. As a consequences of this  
result, we prove that the distribution of a random variable  
(satisfying in addition a certain property) in a finite sum of Wiener  
chaoses cannot be normal. This result also allows to understand better  
some characterization of the Gaussian variables obtained via Malliavin  
calculus.

http://arxiv.org/abs/0907.2501


9102. Weak convergence for the stochastic heat equation driven by  
Gaussian white noise
Author(s): Xavier Bardina and Maria Jolis and Lluis Quer-Sardanyons

Abstract: In this paper, we consider a quasi-linear stochastic heat  
equation on $[0,1]$, with Dirichlet boundary conditions and controlled  
by the space-time white noise. We formally replace the random  
perturbation by a family of noisy inputs depending on a parameter $n 
\in \mathbb{N}$ such that approximate the white noise in some sense.  
Then, we provide sufficient conditions ensuring that the real-valued  
{\it mild} solution of the SPDE perturbed by this family of noises  
converges in law, in the space $\mathcal{C}([0,T]\times [0,1])$ of  
continuous functions, to the solution of the white noise driven SPDE.  
Making use of a suitable continuous functional of the stochastic  
convolution term, we show that it suffices to tackle the linear  
problem. For this, we prove that the corresponding family of laws is  
tight and we identify the limit law by showing the convergence of the  
finite dimensional distributions. We have also considered two  
particular families of noises to that our result applies. The first  
one involves a Poisson process in the plane and has been motivated by  
a one-dimensional result of Stroock, which states that the family of  
processes $n \int_0^t (-1)^{N(n^2 s)} ds$, where $N$ is a standard  
Poisson process, converges in law to a Brownian motion. The second one  
is constructed in terms of the kernels associated to the extension of  
Donsker's theorem to the plane.

http://arxiv.org/abs/0907.2508


9103. The diversity of a distributed genome in bacterial populations
Author(s): F. Baumdicker and W. R. Hess and P. Pfaffelhuber

Abstract: The distributed genome hypothesis states that the set of  
genes in a population of bacteria is distributed over all individuals  
that belong to the specific taxon. It implies that certain genes can  
be gained and lost from generation to generation. We use the random  
genealogy given by a Kingman coalescent in order to superimpose events  
of gene gain and loss along ancestral lines. Gene gains occur at  
constant rate along ancestral lines. We assume that gained genes have  
never been present in the population before. Gene losses occur at a  
rate proportional to the number of genes present along the ancestral  
line. In this "infinitely many genes model" we derive moments for  
several statistics within a sample: the average number of genes per  
individual, the average number of genes differing between individuals,  
the number of incongruent pairs of genes, the total number of  
different genes in the sample and the gene frequency spectrum. We  
demonstrate that the model gives a reasonable fit with gene frequency  
data from marine cyanobacteria.

http://arxiv.org/abs/0907.2572


9104. Extremal solutions for stochastic equations indexed by negative  
integers and taking values in compact groups
Author(s): Takao Hirayama and Kouji Yano

Abstract: Stochastic equations indexed by negative integers and taking  
values in compact groups are studied. Extremal solutions of the  
equations are characterized in terms of infinite products of  
independent random variables. This result is applied to characterize  
several properties of the set of all solutions in terms of the law of  
the driving noise.

http://arxiv.org/abs/0907.2587


9105. On a zero-one law for the norm process of transient random walk
Author(s): Ayako Matsumoto and Kouji Yano

Abstract: A zero-one law of Engelbert--Schmidt type is proven for the  
norm process of a transient random walk. An invariance principle for  
random walk local times and a limit version of Jeulin's lemma play key  
roles.

http://arxiv.org/abs/0907.2588


9106. Local limit of packable graphs
Author(s): Itai Benjamini and Nicolas Curien

Abstract: We adapt some of the planar results into higher dimensions.  
In particular, it is shown that every unbiased local limit of graphs  
sphere packed in R^d is d-parabolic (under some additional boundedness  
assumptions). We then extend parts of the circle packing theory into  
higher dimensions and derive few geometric corollaries. E.g. every  
infinite graph ``well'' packed in R^d has either strictly positive  
isoperimetric (Cheeger) constant or admits arbitrarily large finite  
sets W with boundary size which satisfies |\partial W| < |W|^{(d-1)/d  
+ o(1)}, were "well" is a local bounded geometry assumption. Some open  
problems and conjectures are gathered at the end.

http://arxiv.org/abs/0907.2609


9107. A CLT for the third integrated moment of Brownian local time  
increments
Author(s): Jay Rosen

Abstract: Let $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote  
the local time of Brownian motion. Our main result is to show that for  
each fixed $t$ $${\int (L^{x+h}_t- L^x_t)^3 dx-12h\int (L^{x+h}_t -  
L^x_t)L^x_t dx\over h^2} \stackrel{\mathcal{L}}{\Longrightarrow}\sqrt 
{192}(\int (L^x_t)^3dx)^{1/2}\eta$$ as $h\to 0$, where $\eta$ is a  
normal random variable with mean zero and variance one that is  
independent of $L^{x}_{t}$. This generalizes our previous result for  
the second moment. We also explain why our approach will not work for  
higher moments

http://arxiv.org/abs/0907.2693


9108. Stochastic Taylor expansions and heat kernel asymptotics
Author(s): Fabrice Baudoin

Abstract: These notes focus on the applications of the stochastic  
Taylor expansion of solutions of stochastic differential equations to  
the study of heat kernels in small times. As an illustration of these  
methods we provide a new heat kernel proof of the Chern-Gauss-Bonnet  
theorem.

http://arxiv.org/abs/0907.2711


9109. Explicit solutions of G-heat equation with a class of initial  
conditions by G-Brownian motion
Author(s): Mingshang Hu

Abstract: We obtain the viscosity solution of G-heat equation with the  
initial condition $\phi(x)=x^{n}$ for each integer $n\geq1$ using the  
method of G-Brownian motion.

http://arxiv.org/abs/0907.2748


9110. Generalized backward doubly stochastic differential equations  
driven by L\'evy processes with non-Lipschitz coefficients
Author(s): Auguste Aman (LMAI) and Jean Marc Owo (LMAI)

Abstract: We prove an existence and uniqueness result for generalized  
backward doubly stochastic differential equations driven by L\'evy  
processes with non-Lipschitz assumptions.

http://arxiv.org/abs/0907.2785


9111. Sharpness of the percolation transition in the two-dimensional  
contact process
Author(s): Jacob van den Berg

Abstract: For ordinary (independent) percolation on a large class of  
lattices it is well-known that below the critical percolation  
parameter the cluster size distribution has exponential decay, and  
that power-law behaviour of this distribution can only occur at the  
critical value. This behaviour is often called `sharpness of the  
percolation transition'. For theoretical reasons as well as motivated  
by applied research, there is an increasing interest in percolation  
models with (weak) dependencies. For instance, biologists and  
agricultural researchers have used (stationary distributions of)  
certain two-dimensional contact-like processes to model vegetation  
patterns in an arid landscape. In that context, occupied clusters are  
interpreted as patches of vegetation. For some of these models it has  
been reported in the literature that computer simulations indicate  
power-law behaviour in some interval of positive length of a model  
parameter. This would mean that in these models the percolation  
transition is not sharp. This motivated us to investigate similar  
questions for the ordinary ('basic') two-dimensional contact process  
with parameter the infection rate. We show, using techniques from  
papers on Voronoi and Johnson-Mehl tessellations by Bollob\'as and  
Riordan, that for the upper invariant measure of the contact process  
the percolation transition is sharp.

http://arxiv.org/abs/0907.2843


9112. Conditional limit theorems for ordered random walks
Author(s): D. Denisov and V. Wachtel

Abstract: In a recent paper of Eichelsbacher and Koenig (2008) the  
model of ordered random walks has been considered. There it has been  
shown that, under certain moment conditions, one can construct a k- 
dimensional random walk conditioned to stay in a strict order at all  
times. Moreover, they have shown that the rescaled random walk  
converges to the Dyson Brownian motion. In the present paper we find  
the optimal moment assumptions for the construction of the conditional  
random walk and generalise the limit theorem for this conditional  
process.

http://arxiv.org/abs/0907.2854


9113. On Sojourn Times in the Finite Capacity $M/M/1$ Queue with  
Processor Sharing
Author(s): Qiang Zhen and Charles Knessl

Abstract: We consider a processor shared $M/M/1$ queue that can  
accommodate at most a finite number $K$ of customers. We give an exact  
expression for the sojourn time distribution in the finite capacity  
model, in terms of a Laplace transform. We then give the tail  
behavior, for the limit $K\to\infty$, by locating the dominant  
singularity of the Laplace transform.

http://arxiv.org/abs/0907.2908


9114. Correlation and Brascamp-Lieb inequalities for Markov semigroups
Author(s): F. Barthe and D. Cordero-Erausquin and M. Ledoux and B.  
Maurey

Abstract: This paper builds upon several recent works, where semigroup  
proofs of Brascamp-Lieb inequalities are provided in various settings  
(Euclidean space, spheres and symmetric groups). Our aim is twofold.  
Firstly, we provide a general, unifying, framework based on Markov  
generators, in order to cover a variety of examples of interest going  
beyond previous investigations. Secondly, we put forward the  
combinatorial reasons for which unexpected exponents occur in these  
inequalities.

http://arxiv.org/abs/0907.2858


9115. The geometry of Euclidean convolution inequalities and entropy
Author(s): Dario Cordero-Erausquin and Michel Ledoux

Abstract: The goal of this note is to show that some convolution type  
inequalities from Harmonic Analysis and Information Theory, such as  
Young's convolution inequality (with sharp constant), Nelson's  
hypercontractivity of the Hermite semi-group or Shannon's inequality,  
can be reduced to a simple geometric study of frames of $\R^2$. We  
shall derive directly entropic inequalities, which were recently  
proved to be dual to the Brascamp-Lieb convolution type inequalities.

http://arxiv.org/abs/0907.2861


9116. Asymptotic Expansions for the Conditional Sojourn Time  
Distribution in the $M/M/1$-PS Queue
Author(s): Qiang Zhen and Charles Knessl

Abstract: We consider the $M/M/1$ queue with processor sharing. We  
study the conditional sojourn time distribution, conditioned on the  
customer's service requirement, in various asymptotic limits. These  
include large time and/or large service request, and heavy traffic,  
where the arrival rate is only slightly less than the service rate.  
The asymptotic formulas relate to, and extend, some results of  
Morrison \cite{MO} and Flatto \cite{FL}.

http://arxiv.org/abs/0907.2910


9117. Weak approximation of fractional SDES: The Donsker setting
Author(s): Xavier Bardina and Samy Tindel and Carles Rovira

Abstract: In this note, we take up the study of weak convergence for  
stochastic differential equations driven by a (Liouville) fractional  
Brownian motion $B$ with Hurst parameter $H\in(1/3,1/2)$. In the  
current paper, we approximate the $d$-dimensional fBm by the  
convolution of a rescaled random walk with Liouville's kernel. We then  
show that the corresponding differential equation converges in law to  
a fractional SDE driven by $B$.

http://arxiv.org/abs/0907.3030


9118. Bootstrap percolation in high dimensions
Author(s): Jozsef Balogh and Bela Bollobas and Robert Morris

Abstract: In r-neighbour bootstrap percolation on a graph G, a set of  
initially infected vertices A \subset V(G) is chosen independently at  
random, with density p, and new vertices are subsequently infected if  
they have at least r infected neighbours. The set A is said to  
percolate if eventually all vertices are infected. Our aim is to  
understand this process on the grid, [n]^d, for arbitrary functions n  
= n(t), d = d(t) and r = r(t), as t -> infinity. The main question is  
to determine the critical probability p_c([n]^d,r) at which  
percolation becomes likely, and to give bounds on the size of the  
critical window. In this paper we study this problem when r = 2, for  
all functions n and d satisfying d \gg log n. The bootstrap process  
has been extensively studied on [n]^d when d is a fixed constant and 2  
\le r \le d, and in these cases p_c([n]^d,r) has recently been  
determined up to a factor of 1 + o(1) as n -> infinity. At the other  
end of the scale, Balogh and Bollobas determined p_c([2]^d,2) up to a  
constant factor, and Balogh, Bollobas and Morris determined p_c([n] 
^d,d) asymptotically if d > (log log n)^{2+\eps}, and gave much  
sharper bounds for the hypercube. Here we prove the following result:  
let x be the smallest positive root of the equation \sum_{k=0}^\infty  
(-1)^k x^k / (2^{k^2-k} k!) = 0, so x \approx 1.166. Then (16x/d^2 +  
(log d)/d^{5/2) 2^{-2\sqrt{d}} < p_c([2]^d,2) < (16x/d^2 + 5(log d)^2/ 
d^{5/2}) 2^{-2\sqrt{d}} if d is sufficiently large, and moreover we  
determine a sharp threshold for the critical probability p_c([n]^d,2)  
for every function n = n(d) with d \gg log n.

http://arxiv.org/abs/0907.3097


9119. The Maxwell-Boltzmann Distribution is not the Equilibrium on a  
Hyperboloid
Author(s): S. G. Rajeev

Abstract: We give a geometric formulation of the Fokker-Planck-Kramer  
equations for a particle moving on a Lie algebra under the influence  
of a dissipative and a random force. Special cases of interest are  
fluid mechanics, the Stochastic Loewner Equation and the rigid body.  
We find that the Boltzmann distribution, although a static solution,  
is not normalizable when the algebra is not unimodular. This is  
because the invariant measure of integration in momentum space is not  
the standard one. We solve the special case of the upper half-plane  
(hyperboloid) explicitly: there is another equilibrium solution to the  
Fokker-Planck equation, which is integrable. It breaks rotation  
invariance; moreover, the most likely value for velocity is not zero.

http://arxiv.org/abs/0907.2401


9120. From a dichotomy for images to Haagerup's inequality
Author(s): Iosif Pinelis

Abstract: Let X be a compact topological space, and let D be a subset  
of X. Let Y be a Hausdorff topological space. Let f be a continuous  
map of the closure of D to Y such that f(D) is open. Let E be any  
connected subset of the complement (to Y) of the boundary of D. Then f 
(D) either contains E or is contained in the complement of E.  
Applications of this dichotomy principle are given, in particular for  
holomorphic maps, including maximum and minimum modulus principles, an  
inverse boundary correspondence, and a proof of Haagerup's inequality  
for the absolute power moments of linear combinations of independent  
Rademacher random variables.

http://arxiv.org/abs/0907.2960


9121. Large deviations for flows of interacting Brownian motions
Author(s): A.A.Dorogovtsev and O.V.Ostapenko

Abstract: We establish the large deviation principle (LDP) for  
stochastic flows of interacting Brownian motions. In particular, we  
consider smoothly correlated flows, coalescing flows and Brownian  
motion stopped at a hitting moment.

http://arxiv.org/abs/0907.3207


9122. Scaling limits of random planar maps with large faces
Author(s): Jean-Fran\c{c}ois Le Gall (LM-Orsay) and Gr\'egory Miermont  
(DMA)

Abstract: We discuss asymptotics for large random planar maps under  
the assumption that the distribution of the degree of a typical face  
is in the domain of attraction of a stable distribution with index $ 
\alpha\in(1,2)$. When the number $n$ of vertices of the map tends to  
infinity, the asymptotic behavior of distances from a distinguished  
vertex is described by a random process called the continuous distance  
process, which can be constructed from a centered stable process with  
no negative jumps and index $\alpha$. In particular, the profile of  
distances in the map, rescaled by the factor $n^{?1/2\alpha}$,  
converges to a random measure defined in terms of the distance  
process. With the same rescaling of distances, the vertex set viewed  
as a metric space converges in distribution as $n\to\infty$, at least  
along suitable subsequences, towards a limiting random compact metric  
space whose Hausdorff dimension is equal to $2\alpha$.

http://arxiv.org/abs/0907.3262


9123. q-Exchangeability via quasi-invariance
Author(s): Alexander Gnedin and Grigori Olshanski

Abstract: For positive q, the q-exchangeability is introduced as quasi- 
invariance under permutations, with a special cocycle. This allows us  
to extend the q-analogue of de Finetti's theorem for binary sequences  
(arXiv:0905.0367) to the general real-valued sequences. In contrast to  
the classical case with q=1, the order on the reals plays for the q- 
analogues a significant role. An explicit construction of ergodic q- 
exchangeable measures involves a random shuffling of the set N= 
{1,2,..} by iteration of the geometric choice. For q distinct from 1,  
the shuffling yields a probability measure Q that is supported by the  
group of bijections of N, and has the property of quasi-invariance  
under both left and right multiplications by finite permutations. We  
establish connections of the q-exchangeability to certain transient  
Markov chains on the q-Pascal pyramids and to invariant random flags  
over the Galois fields.

http://arxiv.org/abs/0907.3275


9124. High level excursion set geometry for non-Gaussian infinitely  
divisible random fields
Author(s): Robert J Adler and Gennady Samorodnitsky and Jonathan E  
Taylor

Abstract: We consider smooth, infinitely divisible random fields $X 
(t), t\in M)$, $M\subset \real^d$, with regularly varying L\'evy  
measure, and are interested in the geometric characteristics of the  
excursion sets \begin{eqnarray*} A_u = \{t\in M: X(t) >u\} \end 
{eqnarray*} over high levels $u$. For a large class of such random  
fields we compute the $u\to\infty$ asymptotic joint distribution of  
the numbers of critical points, of various types, of $X$ in $A_u$,  
conditional on $A_u$ being non-empty. This allows us, for example, to  
obtain the asymptotic conditional distribution of the Euler  
characteristic of the excursion set. In a significant departure from  
the Gaussian situation, the high level excursion sets for these random  
fields can have quite a complicated geometry. Whereas in the Gaussian  
case non-empty excursion sets are, with high probability, roughly  
ellipsoidal, in the more general infinitely divisible setting almost  
any shape is possible.

http://arxiv.org/abs/0907.3359


9125. Disorder chaos and multiple valleys in spin glasses
Author(s): Sourav Chatterjee

Abstract: We prove that the Sherrington-Kirkpatrick model of spin  
glasses is chaotic under small perturbations of the couplings at any  
temperature in the absence of an external field. The result is proved  
for two kinds of perturbations: (a) distorting the couplings via  
Ornstein-Uhlenbeck flows, and (b) replacing a small fraction of the  
couplings by independent copies. We further prove that the S-K model  
exhibits multiple valleys in its energy landscape, i.e. there are many  
states with near-minimal energy that are mutually nearly orthogonal.  
We show that the variance of the free energy of the S-K model is  
unusually small at any temperature. (By `unusually small' we mean that  
it is much smaller than the number of sites; in other words, it beats  
the classical Gaussian concentration inequality, a phenomenon that we  
call `superconcentration'.) We prove that the bond overlap in the  
Edwards-Anderson model of spin glasses is not chaotic under  
perturbations of the couplings, even large perturbations. Lastly, we  
obtain sharp lower bounds on the variance of the free energy in the E- 
A model on any bounded degree graph, generalizing a result of Wehr and  
Aizenman and establishing the absence of superconcentration in this  
class of models. Our techniques apply for the p-spin models and the  
Random Field Ising Model as well, although we do not work out the  
details in these cases.

http://arxiv.org/abs/0907.3381


9126. Spin Needlets Spectral Estimation
Author(s): Daryl Geller and Xiaohong Lan and Domenico Marinucci

Abstract: We consider the statistical analysis of random sections of a  
spin fibre bundle over the sphere. These may be thought of as random  
fields that at each point p in $S^2$ take as a value a curve (e.g. an  
ellipse) living in the tangent plane at that point $T_{p}S^2$, rather  
than a number as in ordinary situations. The analysis of such fields  
is strongly motivated by applications, for instance polarization  
experiments in Cosmology. To investigate such fields, spin needlets  
were recently introduced by Geller and Marinucci (2008) and Geller et  
al. (2008). We consider the use of spin needlets for spin angular  
power spectrum estimation, in the presence of noise and missing  
observations, and we provide Central Limit Theorem results, in the  
high frequency sense; we discuss also tests for bias and asymmetries  
with an asymptotic justification.

http://arxiv.org/abs/0907.3369


9127. A bijection theorem for domino tiling with diagonal impurities
Author(s): Fumihiko Nakano and Taizo Sadahiro

Abstract: We consider the dimer problem on a non-bipartite graph $G$,  
where there are two types of dimers one of which we regard impurities.  
Results of simulations using Markov chain seem to indicate that  
impurities are tend to distribute on the boundary, which we set as a  
conjecture. We first show that there is a bijection between the set of  
dimer coverings on $G$ and the set of spanning forests on two graphs  
which are made from $G$, with configuration of impurities satisfying a  
pairing condition. This bijection can be regarded as a extension of  
the Temperley bijection. We consider local move consisting of two  
operations, and by using the bijection mentioned above, we prove local  
move connectedness. We further obtained some bound of the number of  
dimer coverings and the probability finding an impurity at given edge,  
by extending the argument in our previous result.

http://arxiv.org/abs/0907.3252


9128. Optimal Execution Problem with Market Impact
Author(s): Takashi Kato

Abstract: We study the optimal execution problem in the market model  
in consideration of market impact. First we study the discrete-time  
model and describe the value function with respect to the trader's  
optimization problem. Then, by shortening the intervals of execution  
times, we derive the value function of the continuous-time model and  
study some properties of them (continuity, semi-group property and the  
characterization as the viscosity solution of HJB.) We show that the  
properties of the continuous-time value function vary by the strength  
of market impact. Moreover we introduce some examples of this model,  
which tell us that the forms of the optimal execution strategies  
entirely change according to the amount of the security holding.

http://arxiv.org/abs/0907.3282


9129. De Finetti theorems for easy quantum groups
Author(s): Teodor Banica and Stephen Curran and Roland Speicher

Abstract: We study sequences of noncommutative random variables which  
are invariant under ``quantum transformations'' coming from an  
orthogonal quantum group satisfying the ``easiness'' condition  
axiomatized in our previous paper. For 10 easy quantum groups, we  
obtain de Finetti type theorems characterizing the joint distribution  
of any infinite, quantum invariant sequence. In particular, we give a  
new and unified proof of the classical results of de Finetti and  
Freedman for the easy groups S_n, O_n, which is based on the  
combinatorial theory of cumulants. We also recover the free de Finetti  
theorem of K\"ostler and Speicher, and the characterization of  
operator-valued free semicircular families due to Curran. We consider  
also finite sequences, and prove an approximation result in the spirit  
of Diaconis and Freedman.

http://arxiv.org/abs/0907.3314


9130. SRB Measures For Certain Markov Processes
Author(s): Wael Bahsoun and Pawel Gora

Abstract: We study Markov processes generated by iterated function  
systems (IFS). The constituent maps of the IFS are monotonic  
transformations of the interval with common fixed points at 0 and 1.  
We first obtain an upper bound on the number of SRB (Sinai-Ruelle- 
Bowen) measures for the IFS. Then theorems are given to analyze  
properties of the ergodic invariant measures $\delta_0$ and $ 
\delta_1$. In particular, sufficient conditions for $\delta_0$ and/or $ 
\delta_1$ to be SRB measures are given. We apply our results to asset  
market games.

http://arxiv.org/abs/0907.3372


9131. Optimal Execution Problem with Market Impact
Author(s): Takashi Kato

Abstract: We study the optimal execution problem in the market model  
in consideration of market impact. First we study the discrete-time  
model and describe the value function with respect to the trader's  
optimization problem. Then, by shortening the intervals of execution  
times, we derive the value function of the continuous-time model and  
study some properties of them (continuity, semi-group property and the  
characterization as the viscosity solution of HJB.) We show that the  
properties of the continuous-time value function vary by the strength  
of market impact. Moreover we introduce some examples of this model,  
which tell us that the forms of the optimal execution strategies  
entirely change according to the amount of the security holding.

http://arxiv.org/abs/0907.3282


9132. Fractional Normal Inverse Gaussian Process
Author(s): Arun Kumar and P. Vellaisamy

Abstract: Normal inverse Gaussian (NIG) process was introduced by  
Barndorff-Nielsen (1997) by subordinating Brownian motion with drift  
to an inverse Gaussian process. Increments of NIG process are  
independent and stationary. In this paper, we introduce dependence  
between the increments of NIG process, by subordinating fractional  
Brownian motion to an inverse Gaussian process and call it fractional  
normal inverse Gaussian (FNIG) process. The basic properties of this  
process are discussed. Its marginal distributions are scale mixtures  
of normal laws, infinitely divisible for the Hurst parameter 1/2<=H< 1  
and are heavy tailed. First order increments of the process are  
stationary and possess long-range dependence (LRD) property. It is  
shown that they have persistence of signs LRD property also. A  
generalization of the FNIG process called n-FNIG process is also  
discussed which allows Hurst parameter H in the interval (n-1, n).  
Possible applications to mathematical finance and hydraulics are also  
pointed out

http://arxiv.org/abs/0907.3637


9133. Flow of diffeomorphisms for SDEs with unbounded H\"older  
continuous drift
Author(s): F. Flandoli and M. Gubinelli and E. Priola

Abstract: We consider a SDE with a smooth multiplicative non- 
degenerate noise and a possibly unbounded Holder continuous drift  
term. We prove existence of a global flow of diffeomorphisms by means  
of a special transformation of the drift of Ito-Tanaka type. The proof  
requires non-standard elliptic estimates in Holder spaces. As an  
application of the stochastic flow, we obtain a Bismut-Elworthy-Li  
type formula for the first derivatives of the associated diffusion  
semigroup.

http://arxiv.org/abs/0907.3668


9134. Systems of one-dimensional random walks in a common random  
environment
Author(s): Jonathon Peterson

Abstract: We consider a system of independent one-dimensional random  
walks in a common random environment under the condition that the  
random walks are transient with positive speed $v_P$. We give upper  
bounds on the quenched probability that at least one of the random  
walks started in the interval $[An, Bn]$ has traveled a distance of  
less than $(v_P - \epsilon)n$. This leads to both a uniform law of  
large numbers and a hydrodynamic limit. We also identify a family of  
distributions on the configuration of particles (parameterized by  
particle density) which are stationary under the (quenched) dynamics  
of the random walks and show that these are the limiting distributions  
for the system when started from a certain natural collection of  
distributions.

http://arxiv.org/abs/0907.3680


9135. A Spectral Analysis of the Sequence of Firing Phases in  
Stochastic Integrate-and-Fire Oscillators
Author(s): Peter Baxendale and John Mayberry

Abstract: Integrate and fire oscillators are widely used to model the  
generation of action potentials in neurons. In this paper, we discuss  
small noise asymptotic results for a class of stochastic integrate and  
fire oscillators (SIFs) in which the buildup of membrane potential in  
the neuron is governed by a Gaussian diffusion process. To analyze  
this model, we study the asymptotic behavior of the spectrum of the  
firing phase transition operator. We begin by proving strong versions  
of a law of large numbers and central limit theorem for the first  
passage-time of the underlying diffusion process across a general time  
dependent boundary. Using these results, we obtain asymptotic  
approximations of the transition operator's eigenvalues. We also  
discuss connections between our results and earlier numerical  
investigations of SIFs.

http://arxiv.org/abs/0907.3700


9136. Evolution in predator-prey systems
Author(s): Rick Durrett and John Mayberry

Abstract: We study the adaptive dynamics of predator prey systems  
modeled by a dynamical system in which the characteristics are allowed  
to evolve by small mutations. When only the prey are allowed to  
evolve, and the size of the mutational change tends to 0, the system  
does not exhibit long term prey coexistence and the parameters of the  
resident prey type converges to the solution of an ODE. When only the  
predators are allowed to evolve, coexistence of predators occurs. In  
this case, depending on the parameters being varied we see (i) the  
number of coexisting predators remains tight and the differences of  
the parameters from a reference species converge in distribution to a  
limit, or (ii) the number of coexisting predators tends to infinity,  
and we conjecture that the differences converge to a deterministic  
limit.

http://arxiv.org/abs/0907.3702


9137. High Moments of Large Wigner Random MAtrices and Asymptotic  
Properties of the Spectral Norm
Author(s): O. Khorunzhiy

Abstract: We further modify the method proposed by Ya. Sinai and A.  
Soshnikov and developed by A. Ruzmaikina to study the high moments of  
large Wigner random matrices. Our result concern the asymptotic  
estimates of the high moments of n-dimensional real symmetric random  
matrices whose elements have symmetric distribution such that the  
12+delta-th moment exists.

http://arxiv.org/abs/0907.3743


9138. On the One Dimensional Critical "Learning from Neighbours" Model
Author(s): Antar Bandyopadhyay and Rahul Roy and Anish Sarkar

Abstract: We consider a model of a discrete time "interacting particle  
system" on the integer line where infinitely many changes are allowed  
at each instance of time. We describe the model using chameleons of  
two different colours, {\it viz}., red ($R$) and blue ($B$). At each  
instance of time each chameleon performs an independent but identical  
coin toss experiment with probability $\alpha$ to decide whether to  
change its colour or not. If the coin lands head then the creature  
retains its colour (this is to be interpreted as a "success"),  
otherwise it observes the colours and coin tosses of its two nearest  
neighbours and changes its colour only if, among its neighbors and  
including itself, the proportion of successes of the other colour is  
larger than the proportion of successes of its own colour. This  
produces a Markov chain with infinite state space ${R, B}^{\Zbold}$.  
This model was first studied by Chatterjee and Xu (2004) where  
different colours had different success probabilities. In this work we  
consider the "critical" case where the success probability, $\alpha$,  
is the same irrespective of the colour of the chameleon. We show that  
starting from any initial translation invariant distribution of  
colours the Markov chain converges to a limit of a single colour,  
i.e., even at the critical case there is no "coexistence" of the two  
colours at the limit. Moreover we show that starting with an i.i.d.  
colour distribution the limiting distribution gives some advantage to  
the "underdog".

http://arxiv.org/abs/0907.3828


9139. On Hele-Shaw problems arising as scaling limits
Author(s): Pavel Etingof

Abstract: We discuss conjectural scaling limits of discrete 2- 
dimensional aggregation models conditioned on a semi-axis considered  
by Levine and Peres in arXiv:0712.3378. These are certain problems  
about Hele-Show flows. We study moment properties of their solutions,  
and solve some of them using conformal mappings. In particular, we  
predict the exact formula for the computer-generated shape on the left  
side of Fig. 4 in arXiv:0712.3378.

http://arxiv.org/abs/0907.3856


9140. Wright-Fisher Diffusion in One Dimension
Author(s): Charles L. Epstein and Rafe Mazzeo

Abstract: We analyze the diffusion processes associated to equations  
of Wright-Fisher type in one spatial dimension. These are defined by a  
degenerate second order operator on the interval [0, 1], where the  
coefficient of the second order term vanishes simply at the endpoints,  
and the first order term is an inward-pointing vector field. We  
consider various aspects of this problem, motivated by applications in  
population genetics, including a sharp regularity theory for the zero  
flux boundary conditions, as well as a derivation of the precise  
asymptotics for solutions of this equation, both as t goes to 0 and  
infinity, and as x goes to 0, 1.

http://arxiv.org/abs/0907.3881


9141. Hard Core entropy: lower bounds
Author(s): Kari Eloranta

Abstract: We establish lower bounds for the entropy of the Hard Core  
Model on a few 2d lattices $\scriptstyle {\rm {\bf L}}.$ In this model  
the allowed configurations inside $\scriptstyle \{0,1\}^{{\rm {\bf L}}} 
$ are the one's in which the nearest neighbor $\scriptstyle 1$'s are  
forbidden. Our method which is based on a sequential fill-in scheme is  
unbiassed and thereby yields in principle arbitrarily good estimates  
for the topological entropy. The procedure also gives some detailed  
information on the support of the measure of maximal entropy.

http://arxiv.org/abs/0907.4035


9142. Binomial Approximations for Barrier Options of Israeli Style
Author(s): Yan Dolinsky and Yuri Kifer

Abstract: We show that prices and shortfall risks of game (Israeli)  
barrier options in a sequence of binomial approximations of the Black-- 
Scholes (BS) market converge to the corresponding quantities for  
similar game barrier options in the BS market with path dependent  
payoffs and the speed of convergence is estimated, as well. The  
results are new also for usual American style options and they are  
interesting from the computational point of view, as well, since in  
binomial markets these quantities can be obtained via dynamical  
programming algorithms. The paper continues the study of [11]and [7]  
but requires substantial additional arguments in view of pecularities  
of barrier options which, in particular, destroy the regularity of  
payoffs needed in the above papers.

http://arxiv.org/abs/0907.4136


9143. An Introduction to Stochastic PDEs
Author(s): Martin Hairer

Abstract: These notes are based on a series of lectures given first at  
the University of Warwick in spring 2008 and then at the Courant  
Institute in spring 2009. It is an attempt to give a reasonably self- 
contained presentation of the basic theory of stochastic partial  
differential equations, taking for granted basic measure theory,  
functional analysis and probability theory, but nothing else. The  
approach taken in these notes is to focus on semilinear parabolic  
problems driven by additive noise. These can be treated as stochastic  
evolution equations in some infinite-dimensional Banach or Hilbert  
space that usually have nice regularising properties and they already  
form a very rich class of problems with many interesting properties.  
Furthermore, this class of problems has the advantage of allowing to  
completely pass under silence many subtle problems arising from  
stochastic integration in infinite-dimensional spaces.

http://arxiv.org/abs/0907.4178


9144. Localization for a Class of Linear Systems
Author(s): Yukio Nagahata and Nobuo Yoshida

Abstract: We consider a class of continuous-time stochastic growth  
models on $d$-dimensional lattice with non-negative real numbers as  
possible values per site. The class contains examples such as binary  
contact path process and potlatch process. We show the equivalence  
between the slow population growth and localization property that the  
time integral of the replica overlap diverges. We also prove, under  
reasonable assumptions, a localization property in a stronger form  
that the spatial distribution of the population does not decay  
uniformly in space.

http://arxiv.org/abs/0907.4200


9145. The rank of diluted random graphs
Author(s): Charles Bordenave and Marc Lelarge

Abstract: We investigate the rank of the adjacency matrix of large  
diluted random graphs: for a sequence of graphs converging locally to  
a tree, we give new formulas for the asymptotic of the multiplicity of  
the eigenvalue 0. In particular, the result depends only on the  
limiting tree structure, showing that the normalized rank is  
'continuous at infinity'. Our work also gives a new formula for the  
mass at zero of the spectral measure of a Galton-Watson tree. Our  
techniques of proofs borrow ideas from analysis of algorithms, random  
matrix theory, statistical physics and analysis of Schrodinger  
operators on trees.

http://arxiv.org/abs/0907.4244


9146. Hausdorff measure of arcs and Brownian motion on Brownian  
spatial trees
Author(s): David A. Croydon

Abstract: A Brownian spatial tree is defined to be a pair $(\mathcal 
{T},\phi)$, where $\mathcal{T}$ is the rooted real tree naturally  
associated with a Brownian excursion and $\phi$ is a random continuous  
function from $\mathcal{T}$ into $\mathbb{R}^d$ such that, conditional  
on $\mathcal{T}$, $\phi$ maps each arc of $\mathcal{T}$ to the image  
of a Brownian motion path in $\mathbb{R}^d$ run for a time equal to  
the arc length. It is shown that, in high dimensions, the Hausdorff  
measure of arcs can be used to define an intrinsic metric $d_{\mathcal 
{S}}$ on the set $\mathcal{S}:=\phi(\mathcal{T})$. Applications of  
this result include the recovery of the spatial tree $(\mathcal{T}, 
\phi)$ from the set $\mathcal{S}$ alone, which implies in turn that a  
Dawson--Watanabe super-process can be recovered from its range.  
Furthermore, $d_{\mathcal{S}}$ can be used to construct a Brownian  
motion on $\mathcal{S}$, which is proved to be the scaling limit of  
simple random walks on related discrete structures. In particular, a  
limiting result for the simple random walk on the branching random  
walk is obtained.

http://arxiv.org/abs/0907.4260


9147. Scaling limits for critical inhomogeneous random graphs with  
finite third moments
Author(s): Shankar Bhamidi and Remco van der Hofstad and Johan van  
Leeuwaarden

Abstract: We identify the scaling limits for the sizes of the largest  
components at criticality for inhomogeneous random graphs when the  
degree exponent $\tau$ satisfies $\tau>4$. We see that the sizes of  
the (rescaled) components converge to the excursion lengths of an  
inhomogeneous Brownian motion, extending results of \cite{Aldo97}. We  
rely heavily on martingale convergence techniques, and concentration  
properties of (super)martingales. This paper is part of a programme to  
study the critical behavior in inhomogeneous random graphs of so- 
called rank-1 initiated in \cite{Hofs09a}.

http://arxiv.org/abs/0907.4279


9148. Time-reversal and elliptic boundary value problems
Author(s): Zhen-Qing Chen and Tusheng Zhang

Abstract: In this paper, we prove that there exists a unique, bounded  
continuous weak solution to the Dirichlet boundary value problem for a  
general class of second-order elliptic operators with singular  
coefficients, which does not necessarily have the maximum principle.  
Our method is probabilistic. The time reversal of symmetric Markov  
processes and the theory of Dirichlet forms play a crucial role in our  
approach.

http://arxiv.org/abs/0907.4301


9149. Notes on Using Control Variates for Estimation with Reversible  
MCMC Samplers
Author(s): Ioannis Kontoyiannis and Petros Dellaportas

Abstract: A general methodology is presented for the construction and  
effective use of control variates for reversible MCMC samplers. The  
values of the coefficients of the optimal linear combination of the  
control variates are computed, and adaptive, consistent MCMC  
estimators are derived for these optimal coefficients. All  
methodological and asymptotic arguments are rigorously justified.  
Numerous MCMC simulation examples from Bayesian inference applications  
demonstrate that the resulting variance reduction can be quite dramatic.

http://arxiv.org/abs/0907.4160


9150. The scaling window for a random graph with a given degree sequence
Author(s): Hamed Hatami and Michael Molloy

Abstract: We consider a random graph on a given degree sequence ${\cal  
D}$, satisfying certain conditions. We focus on two parameters $Q=Q 
({\cal D}), R=R({\cal D})$. Molloy and Reed proved that Q=0 is the  
threshold for the random graph to have a giant component. We prove  
that if $|Q|=O(n^{-1/3} R^{2/3})$ then, with high probability, the  
size of the largest component of the random graph will be of order $ 
\Theta(n^{2/3}R^{-1/3})$. If $|Q|$ is asymptotically larger than $n^ 
{-1/3}R^{2/3}$ then the size of the largest component is  
asymptotically smaller or larger than $n^{2/3}R^{-1/3}$. Thus, we  
establish that the scaling window is $|Q|=O(n^{-1/3} R^{2/3})$.

http://arxiv.org/abs/0907.4211


9151. Dense packing on uniform lattices
Author(s): Kari Eloranta

Abstract: We study the Hard Core Model on the graphs ${\rm {\bf  
\scriptstyle G}}$ obtained from Archimedean tilings i.e.  
configurations in $\scriptstyle \{0,1\}^{{\rm {\bf G}}}$ with the  
nearest neighbor 1's forbidden. Our particular aim in choosing these  
graphs is to obtain insight to the geometry of the densest packings in  
a uniform discrete set-up. We establish density bounds, optimal  
configurations reaching them in all cases, and introduce a  
probabilistic cellular automaton that generates the legal  
configurations. Its rule involves a parameter which can be naturally  
characterized as packing pressure. It can have a critical value but  
from packing point of view just as interesting are the noncritical  
cases. These phenomena are related to the exponential size of the set  
of densest packings and more specifically whether these packings are  
maximally symmetric, simple laminated or essentially random packings.

http://arxiv.org/abs/0907.4247


9152. On the Distribution of a Second Class Particle in the Asymmetric  
Simple Exclusion Process
Author(s): Craig A. Tracy and Harold Widom

Abstract: We give an exact expression for the distribution of the  
position X(t) of a single second class particle in the asymmetric  
simple exclusion process (ASEP) where initially the second class  
particle is located at the origin and the first class particles occupy  
the sites {1,2,...}.

http://arxiv.org/abs/0907.4395


9153. Stein's Method of Exchangeable Pairs with Application to the  
Curie-Weiss Model
Author(s): Sourav Chatterjee and Qi-Man Shao

Abstract: Let $(W, W')$ be an exchangeable pair. Assume that $$E(W-W'  
| W) = g(W)+r(W),$$ where $g(W)$ is a dominated term and $r(W)$ is  
negligible. Let $G(t) = \int_0^t g(s) ds$ and define $p(t) = c_1 e^{- 
c_0 G(t)}$, where $c_0$ is a properly chosen constant and $c_1 =  
1/\int_{-\infty}^\infty p(t) dt$. Let $Y$ be a random variable with  
the probability density function $p$. It is proved that $W$ converges  
to $Y$ in distribution when the conditional second moment of $(W-W')$  
given $W$ satisfies a law of large numbers. A Berry-Esseen type bound  
is also given. We use this technique to obtain a Berry-Esseen error  
bound of order $1/\sqrt{n}$ in the non-central limit theorem for the  
magnetization in the Curie-Weiss ferromagnet at the critical  
temperature.

http://arxiv.org/abs/0907.4450


9154. Large Deviation in Harnack type Dirichlet spaces
Author(s): Ann-Kathrin Jarecki

Abstract: In the framework of Harnack type Dirichlet forms, we prove a  
large deviation principle for the asymptotics of reversible Markov  
processes with rate function given by the energy of the paths.

http://arxiv.org/abs/0907.4479


9155. Upper Bound for Large Deviations of Reversible Diffusion Processes
Author(s): Ann-Kathrin Jarecki

Abstract: For a Markov process associated with a diffusion type  
Dirichlet form an upper bound is shown for the law of the finite  
dimensional distributions of the process. Under some more assumptions  
on the underlaying space this is also shown for the law of the Markov  
process itself. In the last section we want to give an application to  
the Wasserstein diffusion.

http://arxiv.org/abs/0907.4483


9156. Bounding relative entropy by the relative entropy of local  
specifications in product spaces
Author(s): Katalin Marton

Abstract: For a class of density functions $q^n(x^n)$ on $\Bbb R^n$ we  
prove an inequality between relative entropy and the sum of average  
conditional relative entropies of the following form: For any density  
function $p^n(x^n)$ on $\Bbb R^n$, $D(p^n||q^n)\leq Const. \sum_{i=1} 
^n \Bbb E D(p_i(\cdot|Y_1,..., Y_{i-1},Y_{i+1},..., Y_n) || Q_i(\cdot| 
Y_1,..., Y_{i-1},Y_{i+1},..., Y_n)),$ where $p_i(\cdot|y_1,..., y_ 
{i-1},y_{i+1},..., y_n)$ and $Q_i(\cdot|x_1,..., x_{i-1},x_{i+1},...,  
x_n)$ denote the local specifications for $p^n$ resp. $q^n$, i.e., the  
conditional density functions of the $i$'th coordinate, given the  
other coordinates. The constant depends on the properties of the local  
specifications of $q^n$. The above inequality implies a logarithmic  
Sobolev inequality for $q^n$. We get an explicit lower bound for the  
logarithmic Sobolev constant of $q^n$ under the assumptions that: (i)  
the local specifications of $q^n$ satisfy logarithmic Sobolev  
inequalities with constants $\rho_i$, and (ii) they also satisfy some  
condition expressing that the mixed partial derivatives of the  
Hamiltonian of $q^n$ are not too large relative to the logarithmic  
Sobolev constants $\rho_i$. Condition (ii) may be weaker than that  
used in Otto and Reznikoff's recent paper on the estimation of  
logarithmic Sobolev constants of spin systems.

http://arxiv.org/abs/0907.4491


9157. On Markov chains induced by partitioned transition probability  
matrices
Author(s): Thomas Kaijser

Abstract: Let S be a denumerable state space and let P be a transition  
probability matrix on S. If a denumerable set M of nonnegative  
matrices is such that the sum of the matrices is equal to P, then we  
call M a partition of P. Let K denote the set of probability vectors  
on S. To every partition M of P we can associate a transition  
probability function on K defined in such a way that if p in K and m  
in M are such that ||pm|| > 0, then, with probability ||pm|| the  
vector p is transferred to the vector pm/||pm||. Here ||.|| denotes  
the l_1-norm. In this paper we investigate convergence in distribution  
for Markov chains generated by transition probability functions  
induced by partitions of transition probability matrices. An important  
application of the convergence results obtained is to filtering  
processes of partially observed Markov chains.

http://arxiv.org/abs/0907.4502


9158. Return probabilities of random walks among polynomial lower tail  
random conductances
Author(s): Omar Boukhadra

Abstract: We study models of continuous-time, symmetric, $\Z^{d}$- 
valued random walks in random environments, driven by a field of  
i.i.d. random nearest-neighbor conductances $\omega_{xy}\in[0,1]$ with  
a power law with an exponent $\gamma$ near 0. We are interested in  
estimating the quenched decay of the return probability $P_\omega^{t} 
(0,0)$, as $t$ tends to $+\infty$. We show that for $\gamma> \frac{d} 
{2}$, the standard bound turns out to be of the correct logarithmic  
order. As an expected concequence, the same result holds for the  
discrete-time case.

http://arxiv.org/abs/0907.4525


9159. Recurrence and transience of branching random walks are  
dynamically stable
Author(s): Sebastian M\"uller

Abstract: Consider a sequence of i.i.d. random variables $X_n$ where  
each random variable is refreshed independently according to a Poisson  
clock. At any fixed time $t$ the law of the sequence is the same as  
for the sequence at time 0 but at random times almost sure properties  
of the sequence may be violated. If there are such \emph{exceptional  
times} we say that the property is \emph{dynamically sensitive},  
otherwise we call it \emph{dynamically stable}. In this note we  
consider branching random walks on Cayley graphs and prove that  
recurrence and transience are dynamically stable. Our proof combines  
techniques from the theory of branching random walks with those of  
dynamical percolation.

http://arxiv.org/abs/0907.4557


9160. The t-Martin boundary of reflected random walks on a half-space
Author(s): Irina Ignatiouk-Robert

Abstract: The t-Martin boundary of a random walk on a half-space with  
reflected boundary conditions is identified. It is shown in particular  
that the t-Martin boundary of such a random walk is not stable in the  
following sense : for different values of t, the t-Martin  
compactifications are not homeomorphic to each other.

http://arxiv.org/abs/0907.4592


9161. Invariant random fields in vector bundles and application to  
cosmology
Author(s): Anatoliy Malyarenko

Abstract: We develop the theory of invariant random fields in vector  
bundles. The spectral decomposition of an invariant random field in a  
homogeneous vector bundle generated by an induced representation of a  
compact connected Lie group $G$ is obtained. We discuss an application  
to the theory of cosmic microwave background, where $G=SO(3)$. A  
theorem about equivalence of two different groups of assumptions in  
cosmological theories is proved.

http://arxiv.org/abs/0907.4620


9162. Disjoint Hamilton cycles in the random geometric graph
Author(s): Xavier P\'erez-Gim\'enez and Nicholas C. Wormald

Abstract: We prove a conjecture of Penrose about the standard random  
geometric graph process, in which n vertices are placed at random on  
the unit square and edges are sequentially added in increasing order  
of lengths taken in the l_p norm. We show that the first edge that  
makes the random geometric graph Hamiltonian is a.a.s. exactly the  
same one that gives 2-connectivity. We also extend this result to  
arbitrary connectivity, by proving that the first edge in the process  
that creates a k-connected graph coincides a.a.s. with the first edge  
that causes the graph to contain k/2 pairwise edge-disjoint Hamilton  
cycles (for even k), or (k-1)/2 Hamilton cycles plus one perfect  
matching, all of them pairwise edge-disjoint (for odd k).

http://arxiv.org/abs/0907.4459


9163. Limit theorems for vertex-reinforced jump processes on regular  
trees
Author(s): Andrea Collevecchio

Abstract: Consider a vertex-reinforced jump process defined on a  
regular tree, where each vertex has exactly $b$ children, with $b \ge  
3$. We prove the strong law of large numbers and the central limit  
theorem for the distance of the process from the root. Notice that it  
is still unknown if vertex-reinforced jump process is transient on the  
binary tree.

http://arxiv.org/abs/0907.4854


9164. Stochastic Flows of SDEs with Irregular Drifts and Stochastic  
Transport Equations
Author(s): Xicheng Zhang

Abstract: In this article we study (possibly degenerate) stochastic  
differential equations (SDE) with irregular (or discontiuous) drifts,  
and prove that under certain conditions on the coefficients, there  
exists a unique almost everywhere stochastic invertible flow  
associated with the SDE in the sense of Lebesgue measure. In the case  
of constant diffusions and BV drifts, we obtain such a result by  
studying the related stochastic transport equation. In the case of non- 
constant diffusions and Sobolev drifts, we use a direct method. In  
particular, we extend the recent results on ODEs with non-smooth  
vector fields to SDEs.

http://arxiv.org/abs/0907.4866


9165. The Monotone Cumulants
Author(s): Takahiro Hasebe and Hayato Saigo

Abstract: In the present paper we define the notion of generalized  
cumulants which gives a universal framework for commutative, free,  
Boolean, and especially, monotone probability theories. The uniqueness  
of generalized cumulants holds for each independence, and hence,  
generalized cumulants are equal to the usual cumulants in commutative,  
free and Boolean cases. The way we define (generalized) cumulants is  
so elementary that we need neither partition lattices nor generating  
functions. This new approach open the way to introduce monotone  
cumulants and we obtain quite simple proof of central limit theorem  
and Poisson's law of small numbers in monotone probability theory.

http://arxiv.org/abs/0907.4896


9166. Invariant Measures and Decay of Correlations of a Class of  
Ergodic Probabilistic Cellular Automata
Author(s): Cristian Coletti (CMCC) and Pierre Tisseur (CMCC)

Abstract: Using an extended version of the duality concept between two  
stochastic processes, we give new ergodicity conditions for two states  
probabilistic cellular automata (PCA) of any dimensions and any  
radius. Under these assumptions, in the one dimensional case, we study  
some properties of the unique invariant measure and show that it is  
shift mixing. Also, the decay of correlation is studied in detail. In  
this sense, the extended concept of duality gives exponential decay of  
correlation. When the extended concept of duality can not be applied  
we are able to get, once again, exponential decay of correlation using  
well known results from the theory of branching processes.

http://arxiv.org/abs/0907.4841


9167. Bayesian estimate of the zero-density frequency of a Cs fountain
Author(s): D Calonico and F Levi and L Lorini and G Mana

Abstract: Caesium fountain frequency-standards realize the second in  
the International System of Units with a relative uncertainty  
approaching 10^-16. Among the main contributions to the accuracy  
budget, cold collisions play an important role because of the atomic  
density shift of the reference atomic transition. This paper describes  
an application of the Bayesian analysis of the clock frequency to  
estimate the density shift and describes how the Bayes theorem allows  
the a priori knowledge of the sign of the collisional coefficient to  
be rigourously embedded into the analysis. As an application, data  
from the INRIM caesium fountain are used and the Bayesian and orthodox  
analyses are compared. The Bayes theorem allows the orthodox  
uncertainty to be reduced by 28% and demonstrates to be an important  
tool in primary frequency-metrology.

http://arxiv.org/abs/0907.4849


9168. Dirichlet polynomials: some old and recent results, and their  
interplay in number theory
Author(s): Michel Weber

Abstract: In the first part of the paper, we present and discuss the  
interplay of Dirichlet polynomials in some classical problems of  
number theory, notably the Lindel\"of Hypothesis. We review some  
typical properties of their means and continue with some  
investigations concerning their supremum properties. Their random  
counterpart is next considered in the second part of the paper. An  
analysis of their supremum properties, which is entirely based on  
methods of stochastic processes, is presented. Some complementary  
results and related questions are included in the last section of the  
paper.

http://arxiv.org/abs/0907.4931


9169. An Analogue of the L\'Evy-Cram\'Er Theorem for Multi-Dimensional  
Rayleigh Distributions
Author(s): Thu Nguyen

Abstract: In the present paper we prove that every k-dimensional  
Cartesian product of Kingman convolutions can be embedded into a k- 
dimensional symmetric convolution (k=1, 2, ...) and obtain an analogue  
of the Cram\'er-L\'evy theorem for multi-dimensional Rayleigh  
distributions. A new and more general class of multi-dimensional  
Rayleygh distributions and associated higher dimensional Bessel  
processes are introduced and studied. This class of processes inherits  
the well-known characteristics of Brownian motions: They are  
independent stationary "increments" processes with continuous sample  
paths.

http://arxiv.org/abs/0907.5035


9170. The weak coupling limit of disordered copolymer models
Author(s): Francesco Caravenna and Giambattista Giacomin

Abstract: A copolymer is a chain of repetitive units (monomers) that  
are almost identical, but they differ in their degree of affinity for  
certain solvents. This difference leads to striking phenomena when the  
polymer fluctuates in a non-homogeneous medium, for example made up by  
two solvents separated by an interface. One may observe, for instance,  
the localization of the polymer at the interface between the two  
solvents. A discrete model of such system, based on the simple  
symmetric random walk on Z, has been investigated in [Bolthausen and  
den Hollander, Ann. Probab. 25 (1997), 1334-1366], notably in the weak  
polymer-solvent coupling limit, where the convergence of the discrete  
model toward a continuum model, based on Brownian motion, has been  
established. This result is remarkable because it strongly suggests a  
universal feature of copolymer models. In this work we prove that this  
is indeed the case. More precisely, we determine the weak coupling  
limit for a general class of discrete copolymer models, obtaining as  
limits a one-parameter (\alpha \in (0,1)) family of continuum models,  
based on \alpha-stable regenerative sets.

http://arxiv.org/abs/0907.5076


9171. Law of large numbers for the maximal flow through tilted  
cylinders in two-dimensional first passage percolation
Author(s): Rapha\"el Rossignol and Marie Th\'eret

Abstract: Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d.  
random capacities. We prove a law of large numbers for the maximal  
flow crossing a rectangle in $\mathbb{R}^2$ when the side lengths of  
the rectangle go to infinity. The value of the limit depends on the  
asymptotic behaviour of the ratio of the height of the cylinder over  
the length of its basis. This law of large numbers extends the law of  
large numbers obtained by Grimmett and Kesten (1984) for rectangles of  
particular orientation.

http://arxiv.org/abs/0907.5112


9172. Standard deviation of the longest common subsequence
Author(s): J\"uri Lember and Heinrich Matzinger

Abstract: Let $L_n$ be the length of the longest common subsequence of  
two independent i.i.d. sequences of Bernoulli variables of length $n$.  
We prove that the order of the standard deviation of $L_n$ is $\sqrt{n} 
$, provided the parameter of the Bernoulli variables is small enough.  
This validates Waterman's conjecture in this situation [Philos. Trans.  
R. Soc. Lond. Ser. B 344 (1994) 383--390]. The order conjectured by  
Chvatal and Sankoff [J. Appl. Probab. 12 (1975) 306--315], however, is  
different.

http://arxiv.org/abs/0907.5137


9173. Brunet-Derrida particle systems, free boundary problems and  
Wiener-Hopf equations
Author(s): Rick Durrett and Daniel Remenik

Abstract: We consider a branching-selection system in $\rr$ with $N$  
particles which give birth independently at rate 1 and where after  
each birth the leftmost particle is erased, keeping the number of  
particles constant. We show that, as $N\to\infty$, the empirical  
measure process associated to the system converges in distribution to  
a deterministic measure-valued process whose densities solve a free  
boundary integro-differential equation. We also show that this  
equation has a unique traveling wave solution traveling at speed $c$  
or no such solution depending on whether $c>a$ or $c\leq a$, where $a$  
is the asymptotic speed of the branching random walk obtained by  
ignoring the removal of the leftmost particles in our process. The  
traveling wave solutions correspond to solutions of Wiener-Hopf  
equations.

http://arxiv.org/abs/0907.5180


9174. On ASEP with Step Bernoulli Initial Condition
Author(s): Craig A. Tracy and Harold Widom

Abstract: This paper extends results of earlier work on ASEP to the  
case of step Bernoulli initial condition. The main results are a  
representation in terms of a Fredholm determinant for the probability  
distribution of a fixed particle, and asymptotic results which in  
particular establish KPZ universality for this probability in one  
regime. (And, as a corollary, for the current fluctuations.)

http://arxiv.org/abs/0907.5192


9175. On infinitely cohomologous to zero observables
Author(s): Amanda de Lima and Daniel Smania

Abstract: We show that for a large class of piecewise expanding maps  
T, the bounded p-variation observables u_0 that admits an infinite  
sequence of bounded p-variation observables u_i satisfying u_i(x)= u_{i 
+1}(Tx) -u_{i+1}(x) are constant. The method of the proof consists in  
to find a suitable Hilbert basis for L^2(hm), where hm is the unique  
absolutely continuous invariant probability of T. In terms of this  
basis, the action of the Perron-Frobenious and the Koopan operator on  
L^2(hm) can be easily understood. This result generalizes earlier  
results by Bamon, Kiwi, Rivera-Letelier and Urzua in the case T(x)= n  
x mod 1, n in N-{0,1} and Lipchitizian observables u_0.

http://arxiv.org/abs/0907.5013


9176. A Discussion on Mean Excess Plots
Author(s): Souvik Ghosh and Sidney I Resnick

Abstract: A widely used tool in the study of risk, insurance and  
extreme values is the mean excess plot. One use is for validating a  
Generalized Pareto model for the excess distribution. This paper  
investigates some theoretical and practical aspects of the use of the  
mean excess plot.

http://arxiv.org/abs/0907.5236


9177. A historical law of large numbers for the Marcus Lushnikov process
Author(s): St\'ephanie Jacquot

Abstract: The Marcus-Lushnikov process is a finite stochastic particle  
system, in which each particle is entirely characterized by its mass.  
Each pair of particles with masses $x$ and $y$ merges into a single  
particle at a given rate $K(x,y)$. Under certain assumptions, this  
process converges to the solution to Smoluchowski equation, as the  
number of particles increases to infinity. The Marcus-Lushnikov  
process gives at each time the distribution of masses of the particles  
present in the system, but does not retain the history of formation of  
the particles. In this paper, we set up a historical analogue of the  
Marcus-Lushnikov process (built according the rules of construction of  
the usual Markov-Lushnikov process) each time giving what we call the  
historical tree of a particle. The historical tree of a particle  
present in the Marcus-Lushnikov process at a given time $t$ encodes  
information about the times and masses of the coagulation events that  
have formed that particle. We prove a law of large numbers for the  
empirical distribution of such historical trees. The limit is a  
natural measure on trees which is constructed from a solution to  
Smoluchowski coagulation equation.

http://arxiv.org/abs/0907.5305


9178. A metric analysis of critical Hamilton--Jacobi equations in the  
stationary ergodic setting
Author(s): Andrea Davini and Antonio Siconolfi

Abstract: We adapt the metric approach to the study of stationary  
ergodic Hamilton-Jacobi equations, for which a notion of admissible  
random (sub)solution is defined. For any level of the Hamiltonian  
greater than or equal to a distinguished critical value, we define an  
intrinsic random semidistance and prove that an asymptotic norm does  
exist. Taking as source region a suitable class of closed random sets,  
we show that the Lax formula provides admissible subsolutions. This  
enables us to relate the degeneracies of the critical stable norm to  
the existence/nonexistence of exact or approximate critical admissible  
solutions.

http://arxiv.org/abs/0907.5332


9179. Weak KAM Theory topics in the stationary ergodic setting
Author(s): Andrea Davini and Antonio Siconolfi

Abstract: We perform a qualitative analysis of the critical equation  
associated with a stationary ergodic Hamiltonian through a stochastic  
version of the metric method, where the notion of closed random  
stationary set, issued from stochastic geometry, plays a major role.  
Our purpose is to give an appropriate notion of random Aubry set, to  
single out characterizing conditions for the existence of exact or  
approximate correctors, and write down representation formulae for  
them. For the last task, we make use of a Lax--type formula, adapted  
to the stochastic environment. This material can be regarded as a  
first step of a long--term project to develop a random analog of Weak  
KAM Theory, generalizing what done in the periodic case or, more  
generally, when the underlying space is a compact manifold.

http://arxiv.org/abs/0907.5334


9180. Profiles of permutations
Author(s): Michael Lugo

Abstract: This paper develops an analogy between the cycle structure  
of, on the one hand, random permutations with cycle lengths restricted  
to lie in an infinite set $S$ with asymptotic density $\sigma$ and, on  
the other hand, permutations selected according to the Ewens  
distribution with parameter $\sigma$. In particular we show that the  
asymptotic expected number of cycles of random permutations of $[n]$  
with all cycles even, with all cycles odd, and chosen from the Ewens  
distribution with parameter 1/2 are all ${1 \over 2} \log n + O(1)$,  
and the variance is of the same order. Furthermore, we show that in  
permutations of $[n]$ chosen from the Ewens distribution with  
parameter $\sigma$, the probability of a random element being in a  
cycle longer than $\gamma n$ approaches $(1-\gamma)^\sigma$ for large  
$n$. The same limit law holds for permutations with cycles carrying  
multiplicative weights with average $\sigma$. We draw parallels  
between the Ewens distribution and the asymptotic-density case and  
explain why these parallels should exist using permutations drawn from  
weighted Boltzmann distributions.

http://arxiv.org/abs/0907.5351


9181. Self-interacting diffusions IV: Rate of convergence
Author(s): Michel Benaim (UNINE) and Olivier Raimond (MODAL'X)

Abstract: Self-interacting diffusions are processes living on a  
compact Riemannian manifold defined by a stochastic differential  
equation with a drift term depending on the past empirical measure of  
the process. The asymptotics of this measure is governed by a  
deterministic dynamical system and under certain conditions it  
converges almost surely towards a deterministic measure (see Bena\"im,  
Ledoux, Raimond (2002) and Bena\"im, Raimond (2005)). We are  
interested here in the rate of this convergence. A central limit  
theorem is proved. In particular, this shows that greater is the  
interaction repelling faster is the convergence.

http://arxiv.org/abs/0907.5468


9182. Upper large deviations for the maximal flow through a domain of $ 
\mathbb{R}^d$ in first passage percolation
Author(s): Rapha\"el Cerf and Marie Th\'eret

Abstract: We consider the standard first passage percolation model in  
the rescaled graph $\mathbb{Z}^d/n$ for $d\geq 2$, and a domain $\Omega 
$ of boundary $\Gamma$ in $\mathbb{R}^d$. Let $\Gamma^1$ and $ 
\Gamma^2$ be two disjoint open subsets of $\Gamma$, representing the  
parts of $\Gamma$ through which some water can enter and escape from $ 
\Omega$. We investigate the asymptotic behaviour of the flow $\phi_n$  
through a discrete version $\Omega_n$ of $\Omega$ between the  
corresponding discrete sets $\Gamma^1_n$ and $\Gamma^2_n$. We prove  
that under some conditions on the regularity of the domain and on the  
law of the capacity of the edges, the upper large deviations of $ 
\phi_n/ n^{d-1}$ above a certain constant are of volume order.

http://arxiv.org/abs/0907.5499


9183. Lower large deviations for the maximal flow through a domain of $ 
\mathbb{R}^d$ in first passage percolation
Author(s): Rapha\"el Cerf and Marie Th\'eret

Abstract: We consider the standard first passage percolation model in  
the rescaled graph $\mathbb{Z}^d/n$ for $d\geq 2$, and a domain $\Omega 
$ of boundary $\Gamma$ in $\mathbb{R}^d$. Let $\Gamma^1$ and $ 
\Gamma^2$ be two disjoint open subsets of $\Gamma$, representing the  
parts of $\Gamma$ through which some water can enter and escape from $ 
\Omega$. We investigate the asymptotic behaviour of the flow $\phi_n$  
through a discrete version $\Omega_n$ of $\Omega$ between the  
corresponding discrete sets $\Gamma^1_n$ and $\Gamma^2_n$. We prove  
that under some conditions on the regularity of the domain and on the  
law of the capacity of the edges, the lower large deviations of $ 
\phi_n/ n^{d-1}$ below a certain constant are of surface order.

http://arxiv.org/abs/0907.5501


9184. Law of large numbers for the maximal flow through a domain of $ 
\mathbb{R}^d$ in first passage percolation
Author(s): Rapha\"el Cerf and Marie Th\'eret

Abstract: We consider the standard first passage percolation model in  
the rescaled graph $\mathbb{Z}^d/n$ for $d\geq 2$, and a domain $\Omega 
$ of boundary $\Gamma$ in $\mathbb{R}^d$. Let $\Gamma^1$ and $ 
\Gamma^2$ be two disjoint open subsets of $\Gamma$, representing the  
parts of $\Gamma$ through which some water can enter and escape from $ 
\Omega$. We investigate the asymptotic behaviour of the flow $\phi_n$  
through a discrete version $\Omega_n$ of $\Omega$ between the  
corresponding discrete sets $\Gamma^1_n$ and $\Gamma^2_n$. We prove  
that under some conditions on the regularity of the domain and on the  
law of the capacity of the edges, $\phi_n$ converges almost surely  
towards a constant $\phi_{\Omega}$, which is the solution of a  
continuous non-random min-cut problem. Moreover, we give a necessary  
and sufficient condition on the law of the capacity of the edges to  
ensure that $\phi_{\Omega} >0$.

http://arxiv.org/abs/0907.5504


9185. Monotonicity properties of the asymptotic relative efficiency  
between common correlation statistics in the bivariate normal model
Author(s): Raymond Molzon and Iosif Pinelis

Abstract: Pearson's is the most common correlation statistic, used  
mainly in parametric settings. Most common among nonparametric  
correlation statistics are Spearman's and Kendall's. We show that for  
bivariate normal i.i.d. samples the pairwise asymptotic relative  
efficiency between these three statistics depends monotonically on the  
population correlation coefficient. This monotonicity is a corollary  
to a stronger result. The proofs rely on the use of l'Hospital-type  
rules for monotonicity patterns.

http://arxiv.org/abs/0907.5448


9186. Conditionally monotone independence
Author(s): Takahiro Hasebe

Abstract: We define the notion of conditionally monotone product as a  
part of conditionally free product, which naturally includes monotone  
and Boolean products. Then we define conditionally monotone cumulants  
which are useful to calculate the limit distributions in central limit  
theorem and Poisson's law of small numbers. Moreover, we introduce  
deformed convolutions arising from the conditionally monotone  
convolution of probability measures and compute the limit  
distributions. In order to understand the validity of cumulants, we  
discuss what are cumulants of a given convolution product in general.

http://arxiv.org/abs/0907.5473


9187. Loss of memory of random functions of Markov chains and Lyapunov  
exponents
Author(s): Pierre Collet and Florencia Leonardi

Abstract: In this paper we prove that the asymptotic rate of  
exponential loss of memory of a random function of a Markov chain $(Z_ 
{t})_{t\in\Z}$ is bounded above by the difference of the first two  
Lyapunov exponents of a certain product of matrices. We also show that  
this bound is in fact realized, namely for almost all realization of  
the process $(Z_{t})_{t\in\Z}$, we can find symbols where the  
asymptotic exponential rate of loss of memory attains the difference  
of the first two Lyapunov exponents. This shows that the process has  
infinite memory and leads to a lower bound on the asymptotic  
exponential loss of memory which is saturated (and equal to the upper  
bound for an adequate choice of the symbols) on a set of full measure.

http://arxiv.org/abs/0908.0077


9188. Scaling limits of anisotropic Hastings-Levitov clusters
Author(s): Fredrik Johansson and Alan Sola and Amanda Turner

Abstract: We consider a variation of the standard Hastings-Levitov  
model HL(0), in which growth is anisotropic. Two natural scaling  
limits are established and we give precise descriptions of the effects  
of the anisotropy. We show that the limit shapes can be realised as  
Loewner hulls and that the evolution of harmonic measure on the  
cluster boundary can be described by the solution to a deterministic  
ordinary differential equation related to the Loewner equation. We  
also characterise the stochastic fluctuations around the deterministic  
limit flow.

http://arxiv.org/abs/0908.0086


9189. A stochastic min-driven coalescence process and its  
hydrodynamical limit
Author(s): Anne-Laure Basdevant (IMT) and Philippe Laurencot (IMT) and  
James R. Norris (DPMMS), Clement Rau (IMT)

Abstract: A stochastic system of particles is considered in which the  
sizes of the particles increase by successive binary mergers with the  
constraint that each coagulation event involves a particle with  
minimal size. Convergence of a suitably renormalised version of this  
process to a deterministic hydrodynamical limit is shown and the time  
evolution of the minimal size is studied for both deterministic and  
stochastic models.

http://arxiv.org/abs/0908.0129


9190. Sampling Conditioned Hypoelliptic Diffusions
Author(s): Martin Hairer and Andrew M. Stuart and Jochen Voss

Abstract: A series of recent articles introduced a method to construct  
stochastic partial differential equations (SPDEs) which are invariant  
with respect to the distribution of a given conditioned diffusion.  
These works are restricted to the case of elliptic diffusions where  
the drift has a gradient structure, and the resulting SPDE is of  
second order parabolic type. The present article extends this  
methodology to allow the construction of SPDEs which are invariant  
with respect to the distribution of a class of hypoelliptic diffusion  
processes, subject to a bridge conditioning. This allows the treatment  
of more realistic physical models, for example one can use the  
resulting SPDE to study transitions between meta-stable states in  
mechanical systems with friction and noise. In this situation the  
restriction of the drift being a gradient can also be lifted.

http://arxiv.org/abs/0908.0162


9191. On the Speed of Spread for Fractional Reaction-Diffusion Equations
Author(s): Hans Engler

Abstract: The fractional reaction diffusion equation u_t + Au = g(u)  
is discussed, where A is a fractional differential operator on the  
real line with order \alpha between 0 and 2, the C^1 function g  
vanishes at 0 and 1, and either g is non-negative on (0,1) or g < 0  
near 0. In the case of non-negative g, it is shown that solutions with  
initial support on the positive half axis spread into the left half  
axis with unbounded speed if g satisfies some weak growth condition  
near 0 in the case \alpha > 1, or if g is merely positive on a  
sufficiently large interval near 1 in the case \alpha < 1. On the  
other hand, it shown that solutions spread with finite speed if g'(0)  
< 0. The proofs use comparison arguments and a new family of traveling  
wave solutions for this class of problems.

http://arxiv.org/abs/0908.0024


9192. A strong pair correlation bound implies the CLT for Sinai  
Billiards
Author(s): Mikko Stenlund

Abstract: For Dynamical Systems, a strong bound on multiple  
correlations implies the Central Limit Theorem (CLT) [ChMa]. In  
Chernov's paper [Ch2], such a bound is derived for dynamically Holder  
continuous observables of dispersing Billiards. Here we weaken the  
regularity assumption and subsequently show that the bound on multiple  
correlations follows directly from the bound on pair correlations.  
Thus, a strong bound on pair correlations alone implies the CLT, for a  
wider class of observables. The result is extended to Anosov  
diffeomorphisms in any dimension.

http://arxiv.org/abs/0908.0027


9193. Approximating Eigenvectors by Subsampling
Author(s): Noureddine El Karoui and Alexandre d'Aspremont

Abstract: We show that averaging eigenvectors of randomly sampled  
submatrices efficiently approximates the true eigenvectors of the  
original matrix under certain conditions on the incoherence of the  
spectral decomposition. This incoherence assumption is typically  
milder than those made in matrix completion and allows eigenvectors to  
be sparse. We discuss applications to spectral methods in  
dimensionality reduction and information retrieval.

http://arxiv.org/abs/0908.0137


9194. On the Role of Sparsity in Compressed Sensing and Random Matrix  
Theory
Author(s): Roman Vershynin

Abstract: We discuss applications of some concepts of Compressed  
Sensing in the recent work on invertibility of random matrices due to  
Rudelson and the author. We sketch an argument leading to the optimal  
bound N^{-1/2} on the median of the smallest singular value of an N by  
N matrix with random independent entries. We highlight the parts of  
the argument where sparsity ideas played a key role.

http://arxiv.org/abs/0908.0257


9195. Layering and wetting transitions for an SOS interface
Author(s): Kenneth S. Alexander and Fran\c{c}ois Dunlop and Salvador  
Miracle-Sol\'e

Abstract: We study the solid-on-solid interface model above a  
horizontal wall in three dimensional space, with an attractive  
interaction when the interface is in contact with the wall, at low  
temperatures. There is no bulk external field. The system presents a  
sequence of layering transitions, whose levels increase with the  
temperature, before reaching the wetting transition.

http://arxiv.org/abs/0908.0321


9196. Universal Gaussian fluctuations of non-Hermitian matrix ensembles
Author(s): Ivan Nourdin (PMA) and Giovanni Peccati (MODAL'X)

Abstract: We prove multi-dimensional central limit theorems for the  
spectral moments (of arbitrary degrees) associated with random  
matrices with real-valued i.i.d. entries, satisfying some appropriate  
moment conditions. Our techniques rely on a universality principle for  
the Gaussian Wiener chaos, recently proved by the authors together  
with Gesine Reinert, as well as on some combinatorial estimates.  
Unlike other related results in the probabilistic literature, we do  
not require that the law of the entries has a density with respect to  
the Lebesgue measure. In particular, our results apply to the ensemble  
of Bernoulli random matrices.

http://arxiv.org/abs/0908.0391


9197. Optimal Transport and Tessellation
Author(s): Martin Huesmann

Abstract: Optimal transport from the volume measure to a convex  
combination of Dirac measures yields a tessellation of a Riemannian  
manifold into pieces of arbitrary relative size. This tessellation is  
studied for the cost functions $c_p(z,y)=\frac{1}{p}d^p(z,y)$ and  
$1\leq p<\infty$. Geometric descriptions of the tessellations for all  
$p$ is obtained for compact subsets of the Euclidean space and the  
sphere. For $p=1$ this approach yields Laguerre tessellations for all  
compact Riemannian manifolds.

http://arxiv.org/abs/0908.0442


9198. The Statistical Mechanics of Stretched Polymers
Author(s): Dmitry Ioffe and Yvan Velenik

Abstract: We describe some recent results concerning the statistical  
properties of a self-interacting polymer stretched by an external  
force. We concentrate mainly on the cases of purely attractive or  
purely repulsive self-interactions, but our results are stable under  
suitable small perturbations of these pure cases. We provide in  
particular a precise description of the stretched phase (local limit  
theorems for the end-point and local observables, invariance  
principle, microscopic structure). Our results also characterize  
precisely the (non-trivial, direction-dependent) critical force needed  
to trigger the collapsed/stretched phase transition in the attractive  
case. We also describe some recent progress: first, the determination  
of the order of the phase transition in the attractive case; second, a  
proof that a semi-directed polymer in quenched random environment is  
diffusive in dimensions 4 and higher when the temperature is high  
enough. In addition, we correct an incomplete argument from one of our  
earlier works.

http://arxiv.org/abs/0908.0452


9199. On linear evolution equations with cylindrical L\'evy noise
Author(s): Enrico Priola and Jerzy Zabczyk

Abstract: We study an infinite-dimensional Ornstein-Uhlenbeck process $ 
(X_t)$ in a given Hilbert space $H$. This is driven by a cylindrical  
symmetric L\'evy process without a Gaussian component and taking  
values in a Hilbert space $U$ which usually contains $H$. We give if  
and only if conditions under which $X_t$ takes values in $H$ for some  
$t>0$ or for all $t>0$. Moreover, we prove irreducibility for $(X_t)$.

http://arxiv.org/abs/0908.0356


9200. On the short time asymptotic of the stochastic Allen-Cahn equation
Author(s): Hendrik Weber

Abstract: A description of the short time behavior of solutions of the  
Allen-Cahn equation with a smoothened additive noise is presented. The  
key result is that in the sharp interface limit solutions move  
according to motion by mean curvature with an additional stochastic  
forcing. This extends a similar result of Funaki in spatial dimension  
$n=2$ to arbitrary dimensions.

http://arxiv.org/abs/0908.0580


9201. Upper and Lower Bounds in Exponential Tauberian Theorems
Author(s): Jochen Voss

Abstract: In this text we study, for positive random variables, the  
relation between the behaviour of the Laplace transform near infinity  
and the distribution near zero. A result of de Bruijn shows that $E(e^ 
{-\lambda X}) \sim e^{r\sqrt{\lambda}}$ for $\lambda\to\infty$ and $P(X 
\leq\eps) \sim \e^{s/\eps}$ for $\eps\downarrow0$ are in some sense  
equivalent and gives a relation between the constants $r$ and $s$. We  
illustrate how this result can be used to obtain simple large  
deviation results. For use in more complex situations we also give a  
generalisation of de Bruijn's result to the case when the upper and  
lower limits are different from each other.

http://arxiv.org/abs/0908.0642


9202. Exact solution of a two-type branching process: Clone size  
distribution in cell division kinetics
Author(s): Tibor Antal and P. L. Krapivsky

Abstract: We study a two-type branching process which provides  
excellent description of experimental data on cell dynamics in skin  
tissue (Clayton et al., 2007). The model involves only a single type  
of progenitor cell, and does not require support from a self-renewed  
population of stem cells. The progenitor cells divide and may  
differentiate into post-mitotic cells. We derive an exact solution of  
this model in terms of generating functions for the total number of  
cells, and for the number of cells of different types. We also deduce  
large time asymptotic behaviors drawing on our exact results, and on  
an independent diffusion approximation.

http://arxiv.org/abs/0908.0484


9203. Recurrence and ergodicity of random walks on linear groups and  
on homogeneous spaces
Author(s): Y. Guivarc'h and C. R. E. Raja

Abstract: We discuss recurrence and ergodicity properties of random  
walks and associated skew products for large classes of locally  
compact groups and homogeneous spaces. In particular we show that a  
closed subgroup of a product of finitely many linear groups over local  
fields supports a recurrent random walk if and only if it has at most  
quadratic growth. We give also a detailed analysis of ergodicity  
properties for special classes of random walks on homogeneous spaces.  
The structure of closed subgroups of linear groups over local fields  
and the properties of group actions with respect to stationary  
measures play an important role in the proofs.

http://arxiv.org/abs/0908.0637


9204. A general strong law of large numbers for additive arithmetic  
functions
Author(s): Istvan Berkes and Michel Weber

Abstract: Let $f(n)$ be a strongly additive complex valued arithmetic  
function. Under mild conditions on $f$, we prove the following  
weighted strong law of large numbers: if $ X,X_1,X_2,... $ is any  
sequence of integrable i.i.d. random variables, then $$ \lim_{N\to  
\infty} {\sum_{n=1}^N f(n) X_n \over\sum_{n=1}^N f(n)} \buildrel{a.s.} 
\over{=} \E X . $$

http://arxiv.org/abs/0908.0680


9205. Optimal scalings for local Metropolis--Hastings chains on  
nonproduct targets in high dimensions
Author(s): Alexandros Beskos and Gareth Roberts and Andrew Stuart

Abstract: We investigate local MCMC algorithms, namely the random-walk  
Metropolis and the Langevin algorithms, and identify the optimal  
choice of the local step-size as a function of the dimension $n$ of  
the state space, asymptotically as $n\to\infty$. We consider target  
distributions defined as a change of measure from a product law. Such  
structures arise, for instance, in inverse problems or Bayesian  
contexts when a product prior is combined with the likelihood. We  
state analytical results on the asymptotic behavior of the algorithms  
under general conditions on the change of measure. Our theory is  
motivated by applications on conditioned diffusion processes and  
inverse problems related to the 2D Navier--Stokes equation.

http://arxiv.org/abs/0908.0865


9206. Asymptotic optimality of isoperimetric constants with respect to  
$L^{2}(\pi)$-spectral gaps
Author(s): Achim Wuebker

Abstract: In this paper we investigate the existence of $L^{2}(\pi)$- 
spectral gaps for $\pi$-irreducible, positive recurrent Markov chains  
on general state space. We obtain necessary and sufficient conditions  
for the existence of $L^{2}(\pi)$-spectral gaps in terms of a sequence  
of isoperimetric constants and establish their asymptotic behavior. It  
turns out that in some cases the spectral gap can be understood in  
terms of convergence of an induced probability flow to the uniform  
flow. The obtained theorems can be interpreted as mixing results and  
yield sharp estimates for the spectral gap of some Markov chains.

http://arxiv.org/abs/0908.0867


9207. $L^{2}$-spectral gaps, weak-reversible and very weak-reversible  
Markov chains
Author(s): Achim Wuebker and Zakhar Kabluchko

Abstract: The theory of $L^2$-spectral gaps for reversible Markov  
chains has been studied by many authors. In this paper we consider  
positive recurrent general state space Markov chains with stationary  
transition probabilities. Replacing the assumption of reversibility by  
a less strong one, we still obtain a simple necessary and sufficient  
condition for the spectral gap property of the associated Markov  
operator in terms of isoperimetric constant. Moreover, we define a new  
sequence of isoperimetric constants which provides a necessary and  
sufficient condition for the existence of a spectral gap in a very  
general setting. Finally, these results are used to obtain simple  
sufficient conditions for the existence of a spectral gap in terms of  
the first and second order transition probabilities.

http://arxiv.org/abs/0908.0888


9208. $L^{2}$-spectral gaps for time discrete reversible Markov chains
Author(s): Achim Wuebker

Abstract: In this paper we study the spectral properties of Markov- 
operator on $L^{2}$-spaces. Lawler and Sokal (Trans. Amer. Math. Soc.,  
1988, 309, pp. 557-580) used isoperimetric constants for discrete and  
continuous time Markov chains to obtain a spectral gap at 1. For time  
discrete Markov chains this does not exclude periodic behavior. We  
define a new constant measuring the distance from periodicity and give  
necessary and sufficient conditions for the existence of a global  
spectral gap in terms of this constant.

http://arxiv.org/abs/0908.0897


9209. Robust mean-variance hedging in the single period model
Author(s): R. Tevzadze and T. Uzunashvili

Abstract: We give an explicit solution of robust mean-variance hedging  
problem in the single period model for some type of contingent claims.  
The alternative approach is also considered.

http://arxiv.org/abs/0908.0840


9210. Efficient importance sampling for binary contingency tables
Author(s): Jose H. Blanchet

Abstract: Importance sampling has been reported to produce algorithms  
with excellent empirical performance in counting problems. However,  
the theoretical support for its efficiency in these applications has  
been very limited. In this paper, we propose a methodology that can be  
used to design efficient importance sampling algorithms for counting  
and test their efficiency rigorously. We apply our techniques after  
transforming the problem into a rare-event simulation problem--thereby  
connecting complexity analysis of counting problems with efficiency in  
the context of rare-event simulation. As an illustration of our  
approach, we consider the problem of counting the number of binary  
tables with fixed column and row sums, $c_j$'s and $r_i$'s,  
respectively, and total marginal sums $d=\sum_jc_j$. Assuming that $ 
\max_jc_j=o(d^{1/2})$, $\sum c_j^2=O(d)$ and the $r_j$'s are bounded,  
we show that a suitable importance sampling algorithm, proposed by  
Chen et al. [J. Amer. Statist. Assoc. 100 (2005) 109--120], requires $O 
(d^3\varepsilon^{-2}\delta^{-1})$ operations to produce an estimate  
that has $\varepsilon$-relative error with probability $1-\delta$. In  
addition, if $\max_jc_j=o(d^{1/4-\delta_0})$ for some $\delta_0>0$,  
the same coverage can be guaranteed with $O(d^3\varepsilon^{-2}\log 
(\delta^{-1}))$ operations.

http://arxiv.org/abs/0908.0999


9211. A probabilistic study of neural complexity
Author(s): Jerome Buzzi (LM-Orsay) and Lorenzo Zambotti (PMA)

Abstract: G. Edelman, O. Sporns, and G. Tononi have introduced in  
theoretical biology the neural complexity of a family of random  
variables. They have defined it as a specific average of mutual  
information over subsystems. We show that their choice of weights  
satisfies two natural properties, namely exchangeability and  
additivity. This paper classifies all functionals satisfying these two  
properties (which we call intricacies) in terms of probability laws on  
the unit interval and studies the growth rate of maximal intricacies  
when the size of the system goes to infinity. For systems of a fixed  
size, we show that the maximizers are non-unique and that the maximal  
value is not approached by exchangeable laws.

http://arxiv.org/abs/0908.1006


9212. Selling a stock at the ultimate maximum
Author(s): Jacques du Toit and Goran Peskir

Abstract: Assuming that the stock price $Z=(Z_t)_{0\leq t\leq T}$  
follows a geometric Brownian motion with drift $\mu\in\mathbb{R}$ and  
volatility $\sigma>0$, and letting $M_t=\max_{0\leq s\leq t}Z_s$ for $t 
\in[0,T]$, we consider the optimal prediction problems \[V_1=\inf_ 
{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{M_T}{Z_{\tau}}\biggr)\quadand 
\quad V_2=\sup_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{Z_{\tau}}{M_T} 
\biggr),\] where the infimum and supremum are taken over all stopping  
times $\tau$ of $Z$. We show that the following strategy is optimal in  
the first problem: if $\mu\leq0$ stop immediately; if $\mu\in  
(0,\sigma^2)$ stop as soon as $M_t/Z_t$ hits a specified function of  
time; and if $\mu\geq\sigma^2$ wait until the final time $T$. By  
contrast we show that the following strategy is optimal in the second  
problem: if $\mu\leq\sigma^2/2$ stop immediately, and if $\mu> 
\sigma^2/2$ wait until the final time $T$. Both solutions support and  
reinforce the widely held financial view that ``one should sell bad  
stocks and keep good ones.'' The method of proof makes use of  
parabolic free-boundary problems and local time--space calculus  
techniques. The resulting inequalities are unusual and interesting in  
their own right as they involve the future and as such have a  
predictive element.

http://arxiv.org/abs/0908.1014


9213. An operator approach for Markov chain weak approximations with  
an application to infinite activity L\'{e}vy driven SDEs
Author(s): Hideyuki Tanaka and Arturo Kohatsu-Higa

Abstract: Weak approximations have been developed to calculate the  
expectation value of functionals of stochastic differential equations,  
and various numerical discretization schemes (Euler, Milshtein) have  
been studied by many authors. We present a general framework based on  
semigroup expansions for the construction of higher-order  
discretization schemes and analyze its rate of convergence. We also  
apply it to approximate general L\'{e}vy driven stochastic  
differential equations.

http://arxiv.org/abs/0908.1021


9214. Asymptotic normality of plug-in level set estimates
Author(s): David M. Mason and Wolfgang Polonik

Abstract: We establish the asymptotic normality of the $G$-measure of  
the symmetric difference between the level set and a plug-in-type  
estimator of it formed by replacing the density in the definition of  
the level set by a kernel density estimator. Our proof will highlight  
the efficacy of Poissonization methods in the treatment of large  
sample theory problems of this kind.

http://arxiv.org/abs/0908.1045


9215. Gaussian perturbations of circle maps: A spectral approach
Author(s): John Mayberry

Abstract: In this work, we examine spectral properties of Markov  
transition operators corresponding to Gaussian perturbations of  
discrete time dynamical systems on the circle. We develop a method for  
calculating asymptotic expressions for eigenvalues (in the zero noise  
limit) and show that changes to the number or period of stable orbits  
for the deterministic system correspond to changes in the number of  
limiting modulus 1 eigenvalues of the transition operator for the  
perturbed process. We call this phenomenon a $\lambda$-bifurcation.  
Asymptotic expressions for the corresponding eigenfunctions and  
eigenmeasures are also derived and are related to Hermite functions.

http://arxiv.org/abs/0908.1058


9216. A continuous analogue of the invariance principle and its almost  
sure version
Author(s): E.E. Permyakova

Abstract: We deal with random processes obtained from a homogeneous  
random process with independent increments by replacement of the time  
scale and by multiplication by a norming constant. We prove the  
convergence in distribution of these processes to Wiener process in  
Skorokhod space endowed by the topology of uniform convergence. An  
integral type almost sure version of this theorem is obtained.

http://arxiv.org/abs/0908.1072


9217. Functional limit theorems for Levy processes and their almost- 
sure versions
Author(s): E.E. Permyakova

Abstract: In this paper we prove a criterion of convergence in  
distribution in Skorokhod space. We apply this criterion to some  
special Levy processes and obtain almost-sure versions of limit  
theorems for these processes.

http://arxiv.org/abs/0908.1074


9218. Total progeny in killed branching random walk
Author(s): Louigi Addario-Berry and Nicolas Broutin

Abstract: We consider a branching random walk for which the maximum  
position of a particle in the n'th generation, M_n, has zero speed on  
the linear scale: M_n/n --> 0 as n --> infinity. We further remove  
(``kill'') any particle whose displacement is negative, together with  
its entire descendence. The size $Z$ of the set of un-killed particles  
is almost surely finite. In this paper, we confirm a conjecture of  
Aldous that Exp[Z] < infinity while Exp[Z log Z]=infinity. The proofs  
rely on precise large deviations estimates and ballot theorem-style  
results for the sample paths of random walks.

http://arxiv.org/abs/0908.1083


9219. Asymptotic Behavior of the Finite-Size Magnetization as a  
Function of the Speed of Approach to Criticality
Author(s): Richard S. Ellis and Jonathan Machta and Peter Tak-Hun Otto

Abstract: The main focus of this paper is to determine whether the  
thermodynamic magnetization is a physically relevant estimator of the  
finite-size magnetization. This is done by comparing the asymptotic  
behaviors of these two quantities along parameter sequences converging  
to either a second-order point or the tricritical point in the mean- 
field Blume-Capel model. We show that the thermodynamic magnetization  
and the finite-size magnetization are asymptotic when the parameter  
alpha governing the speed at which the sequence approaches criticality  
is below a certain threshold alpha_0. However, when alpha exceeds  
alpha_0, the thermodynamic magnetization converges to 0 much faster  
than the finite-size magnetization. The asymptotic behavior of the  
finite-size magnetization is proved via a moderate deviation principle  
when 0 < alpha < alpha_0 and via a weak-convergence limit when alpha >  
alpha_0. To the best of our knowledge, our results are the first  
rigorous confirmation of the statistical mechanical theory of finite- 
size scaling for a mean-field model.

http://arxiv.org/abs/0908.1103


9220. On the uniqueness of classical solutions of Cauchy problems
Author(s): Erhan Bayraktar and Hao Xing

Abstract: Given that the terminal condition is of at most linear  
growth, it is well known that a Cauchy problem admits a unique  
classical solution when the coefficient multiplying the second  
derivative (i.e., the volatility) is also a function of at most linear  
growth. In this note, we give a condition on the volatility that is  
necessary and sufficient for a Cauchy problem to admit a unique  
solution.

http://arxiv.org/abs/0908.1086


9221. Cram\'{e}r Type Moderate Deviation for the Maximum of the  
Periodogram with Application to Simultaneous Tests in Gene Expression  
Time Series
Author(s): Weidong Liu and Qi Man Shao

Abstract: In this paper, Cram\'{e}r type moderate deviations for the  
maximum of the periodogram and its studentized version are derived.  
The results are then applied to a simultaneous testing problem in gene  
expression time series. It is shown that the level of the simultaneous  
tests is accurate provided that the number of genes $G$ and the sample  
size $n$ satisfy $G=\exp(o(n^{1/3}))$.

http://arxiv.org/abs/0908.1145


9222. Absorbing-State Phase Transition for Stochastic Sandpiles and  
Activated Random Walks
Author(s): Leonardo T. Rolla and Vladas Sidoravicius

Abstract: We study the long-time behavior of conservative interacting  
particle systems in $\mathbb Z$: The Activated Random Walk Model for  
reaction-diffusion systems and the Stochastic Sandpile. Our main  
result states that both systems locally fixate when the initial  
density of particles is small enough, establishing the existence of a  
non-trivial phase transition in the density parameter. This fact is  
predicted by theoretical physics arguments and supported by numerical  
analysis.

http://arxiv.org/abs/0908.1152


9223. A Ciesielski-Taylor type identity for positive self-similar  
Markov processes
Author(s): A.E. Kyprianou and P. Patie

Abstract: The aim of this note is to give a straightforward proof of a  
general version of the Ciesielski-Taylor identity for positive self- 
similar Markov processes of the spectrally negative type which  
umbrellas all previously known Ciesielski-Taylor identities within the  
latter class. The approach makes use of three fundamental features.  
Firstly a new transformation which maps a subset of the family of  
Laplace exponents of spectrally negative L\'evy processes into itself.  
Secondly some classical features of fluctuation theory for spectrally  
negative L\'evy processes as well as more recent fluctuation  
identities for positive self-similar Markov processes.

http://arxiv.org/abs/0908.1157


9224. A sharp analysis of the mixing time for random walk on rooted  
trees
Author(s): Jason Fulman

Abstract: We define an analog of Plancherel measure for the set of  
rooted unlabeled trees on n vertices, and a Markov chain which has  
this measure as its stationary distribution. Using the combinatorics  
of commutation relations, we show that order n^2 steps are necessary  
and suffice for convergence to the stationary distribution.

http://arxiv.org/abs/0908.1141


9225. Sharp Heat Kernel Estimates for Relativistic Stable Processes in  
Open Sets
Author(s): Zhen-Qing Chen and Panki Kim and Renming Song

Abstract: In this paper, we establish sharp two-sided estimates for  
the transition densities of relativistic stable processes (or  
equivalently, for the heat kernels of the operators $m-(m^{2/\alpha}- 
\Delta)^{\alpha/2}$) in $C^{1, 1}$ open sets. The estimates are  
uniform in $m\in (0, M]$ for each fixed $M>0$. Letting $m\downarrow  
0$, the estimates given in this paper recover the Dirichlet heat  
kernel estimates for $-(-\Delta)^{\alpha/2}$ in $C^{1,1}$-open sets  
obtained in \cite{CKS}. Sharp two-sided estimates are also obtained  
for Green functions of relativistic stable processes in half-space- 
like $C^{1,1}$ open sets and bounded $C^{1,1}$ open sets.

http://arxiv.org/abs/0908.1509


9226. The two-type continuum Richardson model: Non-dependence of the  
survival of both types on the initial configuration
Author(s): Sebastian Carstens and Thomas Richthammer

Abstract: We consider the model of Deijfen et al. for the competing  
growth of two infection types in R^d, based on the Richardson model on  
Z^d. Stochastic ball-shaped infection outbursts transmit the infection  
type of the center of the ball to all points of the ball that are not  
yet infected. Relevant parameters of the model are the initial  
infection configuration, the (type-dependent) growth rates and the  
radius distribution of the infection outbursts. The main question is  
that of coexistence: For what values of the parameters is there a  
positive probability that both types grow unboundedly? It is known  
that for this question the initial configuration basically is  
irrelevant, provided certain technical assumptions on the radius  
distribution are satisfied. Here we show how to get rid of these  
assumptions, introducing a slight generalization of the model, where  
immune regions and delayed initial infection configurations are allowed.

http://arxiv.org/abs/0908.1551


9227. Boundary Harnack principle for $\Delta + \Delta^{\alpha/2}$
Author(s): Zhen-Qing Chen and Panki Kim and Renming Song and Zoran  
Vondra\v{c}ek

Abstract: For $d\geq 1$ and $\alpha \in (0, 2)$, consider the family  
of pseudo differential operators $\{\Delta+ b \Delta^{\alpha/2}; b\in  
[0, 1]\}$ on $\R^d$ that evolves continuously from $\Delta$ to $\Delta  
+ \Delta^{\alpha/2}$. In this paper, we establish a uniform boundary  
Harnack principle (BHP) with explicit boundary decay rate for  
nonnegative functions which are harmonic with respect to $\Delta +b  
\Delta^{\alpha/2}$ (or equivalently, the sum of a Brownian motion and  
an independent symmetric $\alpha$-stable process with constant  
multiple $b^{1/\alpha}$) in $C^{1, 1}$ open sets. Here a "uniform" BHP  
means that the comparing constant in the BHP is independent of $b\in  
[0, 1]$. Along the way, a uniform Carleson type estimate is  
established for nonnegative functions which are harmonic with respect  
to $\Delta + b \Delta^{\alpha/2}$ in Lipschitz open sets. Our method  
employs a combination of probabilistic and analytic techniques.

http://arxiv.org/abs/0908.1559


9228. Conformal loop ensembles and the stress-energy tensor. II.  
Construction of the stress-energy tensor
Author(s): Benjamin Doyon

Abstract: This is the second part of a work aimed at constructing the  
stress-energy tensor of conformal field theory (CFT) as a local  
"object" in conformal loop ensembles (CLE). This work lies in the  
wider context of re-constructing quantum field theory from  
mathematically well-defined ensembles of random objects. In the  
present paper, based on results of the first part, we identify the  
stress-energy tensor in the dilute regime of CLE. This is done by  
deriving both its conformal Ward identities for single insertion in  
CLE probability functions, and its properties under conformal  
transformations involving the Schwarzian derivative. We also give the  
one-point function of the stress-energy tensor in terms of a notion of  
partition function, and we show that this agrees with standard CFT  
arguments. The construction is in the same spirit as that found in the  
context of SLE(8/3) by the author, Riva and Cardy (2006), which had to  
do with the case of zero central charge. The present construction  
generalises this to all central charges between 0 and 1, including all  
minimal models. This generalisation is non-trivial: the application of  
these ideas to the CLE context requires the introduction of a  
renormalised probability, and the derivation of the transformation  
properties and of the one-point function do not have counterparts in  
the SLE context.

http://arxiv.org/abs/0908.1511


9229. The uniqueness of symmetrizing measure and linear diffusions
Author(s): Xing Fang and Jiangang Ying and Minzhi Zhao

Abstract: In this short article, we shall study one-dimensional local  
Dirichlet spaces. One result, which has its independent interest, is  
to prove that irreducibility implies the uniqueness of symmetrizing  
measure for right Markov processes. The other result is to give a  
representation for any 1-dim local, irreducible and regular Dirichlet  
space and a necessary and sufficient condition for a Dirichlet space  
to be regular subspace of another Dirichlet space.

http://arxiv.org/abs/0908.1607


9230. Perfect simulation of Vervaat perpetuities
Author(s): James Allen Fill and Mark Huber

Abstract: We use coupling into and from the past to sample perfectly  
in a simple and provably fast fashion from the Vervaat family of  
perpetuities. The family includes the Dickman distribution, which  
arises both in number theory and in the analysis of the Quickselect  
algorithm, which was the motivation for our work.

http://arxiv.org/abs/0908.1733


9231. Static large deviations of boundary driven exclusion processes
Author(s): Jonathan Farfan

Abstract: We prove that the stationary measure associated to a  
boundary driven exclusion process in any dimension satisfies a large  
deviation principle with rate function given by the quasi potential of  
the Freidlin and Wentzell theory.

http://arxiv.org/abs/0908.1798


9232. Lack of strong completeness for stochastic flows
Author(s): Xue-Mei Li and Michael Scheutzow

Abstract: It is well-known that a stochastic differential equation  
(SDE) on a Euclidean space driven by a Brownian motion with Lipschitz  
coefficients generates a stochastic flow of homeomorphisms. When the  
coefficients are only locally Lipschitz, then a maximal continuous  
flow still exists but explosion in finite time may occur. If -- in  
addition -- the coefficients grow at most linearly, then this flow has  
the property that for each fixed initial condition $x$, the solution  
exists for all times almost surely. If the exceptional set of measure  
zero can be chosen independently $x$, then the maximal flow is called  
{\em strongly complete}. The question, whether an SDE with locally  
Lipschitz continuous coefficients satisfying a linear growth condition  
is strongly complete was open for many years. In this paper, we  
construct a 2-dimensional SDE with coefficients which are even bounded  
(and smooth) and which is {\em not} strongly complete thus answering  
the question in the negative.

http://arxiv.org/abs/0908.1839


9233. Stein's method for dependent random variables occurring in  
Statistical Mechanics
Author(s): Peter Eichelsbacher and Matthias L\"owe

Abstract: We obtain rates of convergence in limit theorems of partial  
sums $S_n$ for certain sequences of dependent, identically distributed  
random variables, which arise naturally in statistical mechanics, in  
particular, in the context of the Curie-Weiss models. Under  
appropriate assumptions there exists a real number $\alpha$, a  
positive real number $\mu$, and a positive integer $k$ such that $ 
(S_n- n \alpha)/n^{1 - 1/2k}$ converges weakly to a random variable  
with density proportional to $\exp(-\mu |x|^{2k} /(2k)!)$. We develop  
Stein's method for exchangeable pairs for a rich class of  
distributional approximations including the Gaussian distributions as  
well as the non-Gaussian limit distributions with density proportional  
to $\exp(-\mu |x|^{2k} /(2k)!)$. Our results include the optimal Berry- 
Esseen rate in the Central Limit Theorem for the total magnetization  
in the classical Curie-Weiss model, for high temperatures as well as  
at the critical temperature $\beta_c=1$, where the Central Limit  
Theorem fails. Moreover, we analyze Berry-Esseen bounds as the  
temperature $1/ \beta_n$ converges to one and obtain a threshold for  
the speed of this convergence. Single spin distributions satisfying  
the Griffiths-Hurst-Sherman (GHS) inequality like models of liquid  
helium or continuous Curie-Weiss models are considered.

http://arxiv.org/abs/0908.1909


9234. Replica Symmetry and Combinatorial Optimization
Author(s): Johan W\"astlund

Abstract: We establish the soundness of the replica symmetric ansatz  
(introduced by M. Mezard and G. Parisi) for minimum matching and the  
traveling salesman problem in the pseudo-dimension d mean field model  
for d\geq 1. The case d=1 of minimum matching corresponds to the  
pi^2/6 limit for the assignment problem established by D. Aldous in  
2001, and the analogous limit for the d=1 case of TSP was recently  
established by the author with a different method. We introduce a game- 
theoretical framework by which we prove the correctness of the replica- 
cavity prediction of the corresponding limits also for d>1.

http://arxiv.org/abs/0908.1920


9235. High order discretization schemes for stochastic volatility models
Author(s): Benjamin Jourdain (CERMICS) and Mohamed Sbai (CERMICS)

Abstract: In usual stochastic volatility models, the process driving  
the volatility of the asset price evolves according to an autonomous  
one-dimensional stochastic differential equation. We assume that the  
coefficients of this equation are smooth. Using It\^o's formula, we  
get rid, in the asset price dynamics, of the stochastic integral with  
respect to the Brownian motion driving this SDE. Taking advantage of  
this structure, we propose - a scheme, based on the Milstein  
discretization of this SDE, with order one of weak trajectorial  
convergence for the asset price, - a scheme, based on the Ninomiya- 
Victoir discretization of this SDE, with order two of weak convergence  
for the asset price. We also propose a specific scheme with improved  
convergence properties when the volatility of the asset price is  
driven by an Orstein-Uhlenbeck process. We confirm the theoretical  
rates of convergence by numerical experiments and show that our  
schemes are well adapted to the multilevel Monte Carlo method  
introduced by Giles [2008a,b].

http://arxiv.org/abs/0908.1926


9236. Filtering equations for partially observable diffusion processes  
with Lipschitz continuous coefficients
Author(s): N.V. Krylov

Abstract: We present several results on smoothness in $L_{p}$ sense of  
filtering densities under the Lipschitz continuity assumption on the  
coefficients of a partially observable diffusion processes. We obtain  
them by rewriting in divergence form filtering equation which are  
usually considered in terms of formally adjoint to operators in  
nondivergence form.

http://arxiv.org/abs/0908.1935


9237. A Characterization Theorem for the Distribution of a Continuous  
Local Martingale and Related Limit Theorems
Author(s): Andriy Yurachkivsky

Abstract: The main result of the article reads: the distribution of a  
continuous starting from zero local martingale whose quadratic  
characteristic is almost surely absolutely continuous with respect to  
some non-random increasing continuous function is determined by the  
distribution of the quadratic characteristic. Functional limit theorem  
based on this characterization are proved.

http://arxiv.org/abs/0908.1939


9238. An application of disc packing to statistical mechanics
Author(s): Matthew Kahle

Abstract: We construct stable configurations of n overlapping discs of  
radius r in a unit square, with r = O(1/n). By a result of Diaconis,  
Lebeau, and Michel, this result is best possible, up to a constant  
factor. A consequence is that the Metropolis algorithm, a well-studied  
Markov chain on the hardcore model, is not irreducible in this range  
of parameters.

http://arxiv.org/abs/0908.1830


9239. A comprehensive connection between the basic results and  
properties derived from two kinds of topologies of a random locally  
convex module
Author(s): Tiexin Guo

Abstract: The purpose of this paper is to make a comprehensive  
connection between the basic results and properties derived from the  
two kinds of topologies (namely the $(\epsilon,\lambda)-$topology  
introduced by the author and locally $L^{0}-$convex topology recently  
introduced by Filipovi$\acute{c}$ et. al) of a random locally convex  
module. First, we give an extremely simple proof of the known Hahn- 
Banach extension theorem of $L^{0}-$linear functions as well as its  
continuous variants. Then we give the essential relations between the  
hyperplane separation theorems in [Filipovi$\acute{c}$ et. al, J.  
Funct. Anal.256(2009)3996--4029] and a basic strict separation theorem  
in [Guo et. al, Nonlinear Anal. 71(2009)3794--3804], and in the  
process obtain a useful and surprising fact that a random locally  
convex module with the countable concatenation property must have the  
same completeness under the two topologies! Based on the relation  
between the two kinds of completeness, we further present the central  
part of this paper: we prove that most of the previously established  
deep results of random conjugate spaces of random normed modules under  
the $(\epsilon,\lambda)-$topology are still valid under the locally $L^ 
{0}-$convex topology, which considerably enriches financial  
applications of random normed modules.

http://arxiv.org/abs/0908.1843


9240. Static large deviations of boundary driven exclusion processes
Author(s): Jonathan Farfan

Abstract: We prove that the stationary measure associated to a  
boundary driven exclusion process in any dimension satisfies a large  
deviation principle with rate function given by the quasi potential of  
the Freidlin and Wentzell theory.

http://arxiv.org/abs/0908.1798


9241. Lack of strong completeness for stochastic flows
Author(s): Xue-Mei Li and Michael Scheutzow

Abstract: It is well-known that a stochastic differential equation  
(SDE) on a Euclidean space driven by a Brownian motion with Lipschitz  
coefficients generates a stochastic flow of homeomorphisms. When the  
coefficients are only locally Lipschitz, then a maximal continuous  
flow still exists but explosion in finite time may occur. If -- in  
addition -- the coefficients grow at most linearly, then this flow has  
the property that for each fixed initial condition $x$, the solution  
exists for all times almost surely. If the exceptional set of measure  
zero can be chosen independently $x$, then the maximal flow is called  
{\em strongly complete}. The question, whether an SDE with locally  
Lipschitz continuous coefficients satisfying a linear growth condition  
is strongly complete was open for many years. In this paper, we  
construct a 2-dimensional SDE with coefficients which are even bounded  
(and smooth) and which is {\em not} strongly complete thus answering  
the question in the negative.

http://arxiv.org/abs/0908.1839


9242. Stein's method for dependent random variables occurring in  
Statistical Mechanics
Author(s): Peter Eichelsbacher and Matthias L\"owe

Abstract: We obtain rates of convergence in limit theorems of partial  
sums $S_n$ for certain sequences of dependent, identically distributed  
random variables, which arise naturally in statistical mechanics, in  
particular, in the context of the Curie-Weiss models. Under  
appropriate assumptions there exists a real number $\alpha$, a  
positive real number $\mu$, and a positive integer $k$ such that $ 
(S_n- n \alpha)/n^{1 - 1/2k}$ converges weakly to a random variable  
with density proportional to $\exp(-\mu |x|^{2k} /(2k)!)$. We develop  
Stein's method for exchangeable pairs for a rich class of  
distributional approximations including the Gaussian distributions as  
well as the non-Gaussian limit distributions with density proportional  
to $\exp(-\mu |x|^{2k} /(2k)!)$. Our results include the optimal Berry- 
Esseen rate in the Central Limit Theorem for the total magnetization  
in the classical Curie-Weiss model, for high temperatures as well as  
at the critical temperature $\beta_c=1$, where the Central Limit  
Theorem fails. Moreover, we analyze Berry-Esseen bounds as the  
temperature $1/ \beta_n$ converges to one and obtain a threshold for  
the speed of this convergence. Single spin distributions satisfying  
the Griffiths-Hurst-Sherman (GHS) inequality like models of liquid  
helium or continuous Curie-Weiss models are considered.

http://arxiv.org/abs/0908.1909


9243. Replica Symmetry and Combinatorial Optimization
Author(s): Johan W\"astlund

Abstract: We establish the soundness of the replica symmetric ansatz  
(introduced by M. Mezard and G. Parisi) for minimum matching and the  
traveling salesman problem in the pseudo-dimension d mean field model  
for d\geq 1. The case d=1 of minimum matching corresponds to the  
pi^2/6 limit for the assignment problem established by D. Aldous in  
2001, and the analogous limit for the d=1 case of TSP was recently  
established by the author with a different method. We introduce a game- 
theoretical framework by which we prove the correctness of the replica- 
cavity prediction of the corresponding limits also for d>1.

http://arxiv.org/abs/0908.1920


9244. High order discretization schemes for stochastic volatility models
Author(s): Benjamin Jourdain (CERMICS) and Mohamed Sbai (CERMICS)

Abstract: In usual stochastic volatility models, the process driving  
the volatility of the asset price evolves according to an autonomous  
one-dimensional stochastic differential equation. We assume that the  
coefficients of this equation are smooth. Using It\^o's formula, we  
get rid, in the asset price dynamics, of the stochastic integral with  
respect to the Brownian motion driving this SDE. Taking advantage of  
this structure, we propose - a scheme, based on the Milstein  
discretization of this SDE, with order one of weak trajectorial  
convergence for the asset price, - a scheme, based on the Ninomiya- 
Victoir discretization of this SDE, with order two of weak convergence  
for the asset price. We also propose a specific scheme with improved  
convergence properties when the volatility of the asset price is  
driven by an Orstein-Uhlenbeck process. We confirm the theoretical  
rates of convergence by numerical experiments and show that our  
schemes are well adapted to the multilevel Monte Carlo method  
introduced by Giles [2008a,b].

http://arxiv.org/abs/0908.1926


9245. Filtering equations for partially observable diffusion processes  
with Lipschitz continuous coefficients
Author(s): N.V. Krylov

Abstract: We present several results on smoothness in $L_{p}$ sense of  
filtering densities under the Lipschitz continuity assumption on the  
coefficients of a partially observable diffusion processes. We obtain  
them by rewriting in divergence form filtering equation which are  
usually considered in terms of formally adjoint to operators in  
nondivergence form.

http://arxiv.org/abs/0908.1935


9246. A Characterization Theorem for the Distribution of a Continuous  
Local Martingale and Related Limit Theorems
Author(s): Andriy Yurachkivsky

Abstract: The main result of the article reads: the distribution of a  
continuous starting from zero local martingale whose quadratic  
characteristic is almost surely absolutely continuous with respect to  
some non-random increasing continuous function is determined by the  
distribution of the quadratic characteristic. Functional limit theorem  
based on this characterization are proved.

http://arxiv.org/abs/0908.1939


9247. An application of disc packing to statistical mechanics
Author(s): Matthew Kahle

Abstract: We construct stable configurations of n overlapping discs of  
radius r in a unit square, with r = O(1/n). By a result of Diaconis,  
Lebeau, and Michel, this result is best possible, up to a constant  
factor. A consequence is that the Metropolis algorithm, a well-studied  
Markov chain on the hardcore model, is not irreducible in this range  
of parameters.

http://arxiv.org/abs/0908.1830


9248. A comprehensive connection between the basic results and  
properties derived from two kinds of topologies of a random locally  
convex module
Author(s): Tiexin Guo

Abstract: The purpose of this paper is to make a comprehensive  
connection between the basic results and properties derived from the  
two kinds of topologies (namely the $(\epsilon,\lambda)-$topology  
introduced by the author and locally $L^{0}-$convex topology recently  
introduced by Filipovi$\acute{c}$ et. al) of a random locally convex  
module. First, we give an extremely simple proof of the known Hahn- 
Banach extension theorem of $L^{0}-$linear functions as well as its  
continuous variants. Then we give the essential relations between the  
hyperplane separation theorems in [Filipovi$\acute{c}$ et. al, J.  
Funct. Anal.256(2009)3996--4029] and a basic strict separation theorem  
in [Guo et. al, Nonlinear Anal. 71(2009)3794--3804], and in the  
process obtain a useful and surprising fact that a random locally  
convex module with the countable concatenation property must have the  
same completeness under the two topologies! Based on the relation  
between the two kinds of completeness, we further present the central  
part of this paper: we prove that most of the previously established  
deep results of random conjugate spaces of random normed modules under  
the $(\epsilon,\lambda)-$topology are still valid under the locally $L^ 
{0}-$convex topology, which considerably enriches financial  
applications of random normed modules.

http://arxiv.org/abs/0908.1843


9249. Random matrices: Universality of local eigenvalue statistics up  
to the edge
Author(s): Terence Tao and Van Vu

Abstract: This is a continuation of our earlier paper on the  
universality of the eigenvalues of Wigner random matrices. The main  
new results of this paper are an extension of the results in that  
paper from the bulk of the spectrum up to the edge. In particular, we  
prove a variant of the universality results of Soshnikov for the  
largest eigenvalues, assuming moment conditions rather than symmetry  
conditions. The main new technical observation is that there is a  
significant bias in the Cauchy interlacing law near the edge of the  
spectrum which allows one to continue ensuring the delocalization of  
eigenvectors.

http://arxiv.org/abs/0908.1982


9250. Optimal co-adapted coupling for a random walk on the hyper- 
complete-grap
Author(s): Stephen B. Connor

Abstract: Let $G_d$ be the complete graph with d vertices, and let X  
and Y be two simple symmetric continuous-time random walks on the  
vertices of $G_d^n$. When d=2, X and Y are random walks on the  
hypercube, for which a stochastically fastest co-adapted coupling is  
described by Connor & Jacka (2008). Here we extend this result to  
random walks on $G_d^n$, once again producing a stochastically optimal  
coupling: as d tends to infinity we show that this optimal co-adapted  
coupling tends to a maximal coupling.

http://arxiv.org/abs/0908.2038


9251. Reconstruction on Trees: Exponential Moment Bounds for Linear  
Estimators
Author(s): Yuval Peres and Sebastien Roch

Abstract: Consider a Markov chain $(\xi_v)_{v \in V} \in [k]^V$ on the  
infinite $b$-ary tree $T = (V,E)$ with irreducible edge transition  
matrix $M$, where $b \geq 2$, $k \geq 2$ and $[k] = \{1,...,k\}$. We  
denote by $L_n$ the level-$n$ vertices of $T$. Assume $M$ has a real  
second-largest (in absolute value) eigenvalue $\lambda$ with  
corresponding real eigenvector $\nu \neq 0$. Letting $\sigma_v = \nu_ 
{\xi_v}$, we consider the following root-state estimator, which was  
introduced by Mossel and Peres (2003) in the context of the  
``recontruction problem'' on trees: \begin{equation*} S_n = (b\lambda)^ 
{-n} \sum_{x\in L_n} \sigma_x. \end{equation*} As noted by Mossel and  
Peres, when $b\lambda^2 > 1$ (the so-called Kesten-Stigum  
reconstruction phase) the quantity $S_n$ has uniformly bounded  
variance. Here, we give bounds on the moment-generating functions of  
$S_n$ and $S_n^2$ when $b\lambda^2 > 1$. Our results have implications  
for the inference of evolutionary trees.

http://arxiv.org/abs/0908.2056


9252. Sequence-Length Requirement of Distance-Based Phylogeny  
Reconstruction: Breaking the Polynomial Barrier
Author(s): Sebastien Roch

Abstract: We introduce a new distance-based phylogeny reconstruction  
technique which provably achieves, at sufficiently short branch  
lengths, a polylogarithmic sequence-length requirement -- improving  
significantly over previous polynomial bounds for distance-based  
methods. The technique is based on an averaging procedure that  
implicitly reconstructs ancestral sequences. In the same token, we  
extend previous results on phase transitions in phylogeny  
reconstruction to general time-reversible models. More precisely, we  
show that in the so-called Kesten-Stigum zone (roughly, a region of  
the parameter space where ancestral sequences are well approximated by  
``linear combinations'' of the observed sequences) sequences of length  
$\poly(\log n)$ suffice for reconstruction when branch lengths are  
discretized. Here $n$ is the number of extant species. Our results  
challenge, to some extent, the conventional wisdom that estimates of  
evolutionary distances alone carry significantly less information  
about phylogenies than full sequence datasets.

http://arxiv.org/abs/0908.2061


9253. Sharp approximation for density dependent Markov chains
Author(s): Kamil Szczegot

Abstract: Consider a sequence (indexed by n) of Markov chains Z^n in  
R^d characterized by transition kernels that approximately (in n)  
depend only on the rescaled state n^{-1} Z^n. Subject to a smoothness  
condition, such a family can be closely coupled on short time  
intervals to a Brownian motion with quadratic drift. This construction  
is used to determine the first two terms in the asymptotic (in n)  
expansion of the probability that the rescaled chain exits a convex  
polytope. The constant term and the first correction of size n^{-1/6}  
admit sharp characterization by solutions to associated differential  
equations and an absolute constant. The error is smaller than O(n^{- 
b}) for any b < 1/4. These results are directly applied to the  
analysis of randomized algorithms at phase transitions. In particular,  
the `peeling' algorithm in large random hypergraphs, or equivalently  
the iterative decoding scheme for low-density parity-check codes over  
the binary erasure channel is studied to determine the finite size  
scaling behavior for irregular hypergraph ensembles.

http://arxiv.org/abs/0908.2088


9254. A Sharp Estimate for Divisors of Bernoulli Sums
Author(s): Michel Weber

Abstract: Let $S_n=\e_1+...+\e_n$, where $ \e_i $ are i.i.d. Bernoulli  
r.v.'s. Let $0\le r_d(n)<2d$ be the least residue of $n$ mod$(2d)$, $ 
\bar r_d(n)= 2d -r_d(n)$ and $\b(n,d)=\max ({1\over d}, {1\over \sqrt  
n})[e^{- {r_d(n)^2/2 n}} +e^{- {\bar r_d(n)^2/2 n}}]$. We show that $$ 
\sup_{2\le d\le n} \big|\P\big\{d|S_n\big\}- E(n,d) \big|= {\cal O}\big 
({\log^{5/2} n \over n^{3/2}}\big), $$ where $E(n,d) $ verifies $c_1\b 
(n,d)\le E(n,d)\le c_2\b(n,d) $ and $c_1,c_2 $ are numerical constants.

http://arxiv.org/abs/0908.2047


9255. Simple Error Scattering Model for improved Information  
Reconciliation
Author(s): Stefan Rass

Abstract: Implementations of quantum key distribution as available  
nowadays suffer from inefficiencies due to post processing of the raw  
key that severely cuts down the final secure key rate. We present a  
simple model for the error scattering across the raw key and derive  
"closed form" expressions for the probability of a parity check  
failure, or experiencing more than some fixed number of errors. Our  
results can serve for improvement for key establishment, as  
information reconciliation via interactive error correction and  
privacy amplification rests on mostly unproven assumptions. We support  
those hypotheses on statistical grounds.

http://arxiv.org/abs/0908.2069


9256. Probabilistic model associated with the pressureless gas dynamics
Author(s): Sergio Albeverio and Anastasia Korshunova and Olga Rozanova

Abstract: Using a method of stochastic perturbation of a Langevin  
system associated with the non-viscous Burgers equation we construct a  
solution to the Riemann problem for the pressureless gas dynamics  
describing sticky particles dynamics. As a bridging step we consider a  
medium consisting of noninteracting particles. We analyze the  
difference in the behavior of discontinuous solutions for these two  
models and the relations between them. In our framework in 1D case we  
obtain a unique entropy solution to the Riemann problem. Moreover, we  
describe how starting from smooth data a $\delta$ - singularity arises  
in one component of the solution.

http://arxiv.org/abs/0908.2084


9257. The Mahonian probability distribution on words is asymptotically  
normal
Author(s): E. Rodney Canfield and Svante Janson and and Doron Zeilberger

Abstract: The Mahonian statistic is the number of inversions in a  
permutation of a multiset with $a_i$ elements of type $i$, $1\le i\le m 
$. The counting function for this statistic is the $q$ analog of the  
multinomial coefficient $\binom{a_1+...+a_m}{a_1,... a_m}$, and the  
probability generating function is the normalization of the latter. We  
give two proofs that the distribution is asymptotically normal. The  
first is {\it computer-assisted}, based on the method of moments. The  
Maple package {\tt MahonianStat}, available from the webpage of this  
article, can be used by the reader to perform experiments and  
calculations. Our second proof uses characteristic functions. We then  
take up the study of a local limit theorem to accompany our central  
limit theorem. Here our result is less general, and we must be content  
with a conjecture about further work. Our local limit theorem permits  
us to conclude that the coeffiecients of the $q$-multinomial are log- 
concave, provided one stays near the center (where the largest  
coefficients reside.)

http://arxiv.org/abs/0908.2089


9258. Random matrices: Universality of local eigenvalue statistics up  
to the edge
Author(s): Terence Tao and Van Vu

Abstract: This is a continuation of our earlier paper on the  
universality of the eigenvalues of Wigner random matrices. The main  
new results of this paper are an extension of the results in that  
paper from the bulk of the spectrum up to the edge. In particular, we  
prove a variant of the universality results of Soshnikov for the  
largest eigenvalues, assuming moment conditions rather than symmetry  
conditions. The main new technical observation is that there is a  
significant bias in the Cauchy interlacing law near the edge of the  
spectrum which allows one to continue ensuring the delocalization of  
eigenvectors.

http://arxiv.org/abs/0908.1982


9259. Optimal co-adapted coupling for a random walk on the hyper- 
complete-grap
Author(s): Stephen B. Connor

Abstract: Let $G_d$ be the complete graph with d vertices, and let X  
and Y be two simple symmetric continuous-time random walks on the  
vertices of $G_d^n$. When d=2, X and Y are random walks on the  
hypercube, for which a stochastically fastest co-adapted coupling is  
described by Connor & Jacka (2008). Here we extend this result to  
random walks on $G_d^n$, once again producing a stochastically optimal  
coupling: as d tends to infinity we show that this optimal co-adapted  
coupling tends to a maximal coupling.

http://arxiv.org/abs/0908.2038


9260. Reconstruction on Trees: Exponential Moment Bounds for Linear  
Estimators
Author(s): Yuval Peres and Sebastien Roch

Abstract: Consider a Markov chain $(\xi_v)_{v \in V} \in [k]^V$ on the  
infinite $b$-ary tree $T = (V,E)$ with irreducible edge transition  
matrix $M$, where $b \geq 2$, $k \geq 2$ and $[k] = \{1,...,k\}$. We  
denote by $L_n$ the level-$n$ vertices of $T$. Assume $M$ has a real  
second-largest (in absolute value) eigenvalue $\lambda$ with  
corresponding real eigenvector $\nu \neq 0$. Letting $\sigma_v = \nu_ 
{\xi_v}$, we consider the following root-state estimator, which was  
introduced by Mossel and Peres (2003) in the context of the  
``recontruction problem'' on trees: \begin{equation*} S_n = (b\lambda)^ 
{-n} \sum_{x\in L_n} \sigma_x. \end{equation*} As noted by Mossel and  
Peres, when $b\lambda^2 > 1$ (the so-called Kesten-Stigum  
reconstruction phase) the quantity $S_n$ has uniformly bounded  
variance. Here, we give bounds on the moment-generating functions of  
$S_n$ and $S_n^2$ when $b\lambda^2 > 1$. Our results have implications  
for the inference of evolutionary trees.

http://arxiv.org/abs/0908.2056


9261. Sequence-Length Requirement of Distance-Based Phylogeny  
Reconstruction: Breaking the Polynomial Barrier
Author(s): Sebastien Roch

Abstract: We introduce a new distance-based phylogeny reconstruction  
technique which provably achieves, at sufficiently short branch  
lengths, a polylogarithmic sequence-length requirement -- improving  
significantly over previous polynomial bounds for distance-based  
methods. The technique is based on an averaging procedure that  
implicitly reconstructs ancestral sequences. In the same token, we  
extend previous results on phase transitions in phylogeny  
reconstruction to general time-reversible models. More precisely, we  
show that in the so-called Kesten-Stigum zone (roughly, a region of  
the parameter space where ancestral sequences are well approximated by  
``linear combinations'' of the observed sequences) sequences of length  
$\poly(\log n)$ suffice for reconstruction when branch lengths are  
discretized. Here $n$ is the number of extant species. Our results  
challenge, to some extent, the conventional wisdom that estimates of  
evolutionary distances alone carry significantly less information  
about phylogenies than full sequence datasets.

http://arxiv.org/abs/0908.2061


9262. Sharp approximation for density dependent Markov chains
Author(s): Kamil Szczegot

Abstract: Consider a sequence (indexed by n) of Markov chains Z^n in  
R^d characterized by transition kernels that approximately (in n)  
depend only on the rescaled state n^{-1} Z^n. Subject to a smoothness  
condition, such a family can be closely coupled on short time  
intervals to a Brownian motion with quadratic drift. This construction  
is used to determine the first two terms in the asymptotic (in n)  
expansion of the probability that the rescaled chain exits a convex  
polytope. The constant term and the first correction of size n^{-1/6}  
admit sharp characterization by solutions to associated differential  
equations and an absolute constant. The error is smaller than O(n^{- 
b}) for any b < 1/4. These results are directly applied to the  
analysis of randomized algorithms at phase transitions. In particular,  
the `peeling' algorithm in large random hypergraphs, or equivalently  
the iterative decoding scheme for low-density parity-check codes over  
the binary erasure channel is studied to determine the finite size  
scaling behavior for irregular hypergraph ensembles.

http://arxiv.org/abs/0908.2088


9263. A Sharp Estimate for Divisors of Bernoulli Sums
Author(s): Michel Weber

Abstract: Let $S_n=\e_1+...+\e_n$, where $ \e_i $ are i.i.d. Bernoulli  
r.v.'s. Let $0\le r_d(n)<2d$ be the least residue of $n$ mod$(2d)$, $ 
\bar r_d(n)= 2d -r_d(n)$ and $\b(n,d)=\max ({1\over d}, {1\over \sqrt  
n})[e^{- {r_d(n)^2/2 n}} +e^{- {\bar r_d(n)^2/2 n}}]$. We show that $$ 
\sup_{2\le d\le n} \big|\P\big\{d|S_n\big\}- E(n,d) \big|= {\cal O}\big 
({\log^{5/2} n \over n^{3/2}}\big), $$ where $E(n,d) $ verifies $c_1\b 
(n,d)\le E(n,d)\le c_2\b(n,d) $ and $c_1,c_2 $ are numerical constants.

http://arxiv.org/abs/0908.2047


9264. Simple Error Scattering Model for improved Information  
Reconciliation
Author(s): Stefan Rass

Abstract: Implementations of quantum key distribution as available  
nowadays suffer from inefficiencies due to post processing of the raw  
key that severely cuts down the final secure key rate. We present a  
simple model for the error scattering across the raw key and derive  
"closed form" expressions for the probability of a parity check  
failure, or experiencing more than some fixed number of errors. Our  
results can serve for improvement for key establishment, as  
information reconciliation via interactive error correction and  
privacy amplification rests on mostly unproven assumptions. We support  
those hypotheses on statistical grounds.

http://arxiv.org/abs/0908.2069


9265. Probabilistic model associated with the pressureless gas dynamics
Author(s): Sergio Albeverio and Anastasia Korshunova and Olga Rozanova

Abstract: Using a method of stochastic perturbation of a Langevin  
system associated with the non-viscous Burgers equation we construct a  
solution to the Riemann problem for the pressureless gas dynamics  
describing sticky particles dynamics. As a bridging step we consider a  
medium consisting of noninteracting particles. We analyze the  
difference in the behavior of discontinuous solutions for these two  
models and the relations between them. In our framework in 1D case we  
obtain a unique entropy solution to the Riemann problem. Moreover, we  
describe how starting from smooth data a $\delta$ - singularity arises  
in one component of the solution.

http://arxiv.org/abs/0908.2084


9266. The Mahonian probability distribution on words is asymptotically  
normal
Author(s): E. Rodney Canfield and Svante Janson and and Doron Zeilberger

Abstract: The Mahonian statistic is the number of inversions in a  
permutation of a multiset with $a_i$ elements of type $i$, $1\le i\le m 
$. The counting function for this statistic is the $q$ analog of the  
multinomial coefficient $\binom{a_1+...+a_m}{a_1,... a_m}$, and the  
probability generating function is the normalization of the latter. We  
give two proofs that the distribution is asymptotically normal. The  
first is {\it computer-assisted}, based on the method of moments. The  
Maple package {\tt MahonianStat}, available from the webpage of this  
article, can be used by the reader to perform experiments and  
calculations. Our second proof uses characteristic functions. We then  
take up the study of a local limit theorem to accompany our central  
limit theorem. Here our result is less general, and we must be content  
with a conjecture about further work. Our local limit theorem permits  
us to conclude that the coeffiecients of the $q$-multinomial are log- 
concave, provided one stays near the center (where the largest  
coefficients reside.)

http://arxiv.org/abs/0908.2089


9267. High order discretization schemes for stochastic volatility models
Author(s): Benjamin Jourdain (CERMICS) and Mohamed Sbai (CERMICS)

Abstract: In usual stochastic volatility models, the process driving  
the volatility of the asset price evolves according to an autonomous  
one-dimensional stochastic differential equation. We assume that the  
coefficients of this equation are smooth. Using It\^o's formula, we  
get rid, in the asset price dynamics, of the stochastic integral with  
respect to the Brownian motion driving this SDE. Taking advantage of  
this structure, we propose - a scheme, based on the Milstein  
discretization of this SDE, with order one of weak trajectorial  
convergence for the asset price, - a scheme, based on the Ninomiya- 
Victoir discretization of this SDE, with order two of weak convergence  
for the asset price. We also propose a specific scheme with improved  
convergence properties when the volatility of the asset price is  
driven by an Orstein-Uhlenbeck process. We confirm the theoretical  
rates of convergence by numerical experiments and show that our  
schemes are well adapted to the multilevel Monte Carlo method  
introduced by Giles [2008a,b].

http://arxiv.org/abs/0908.1926


9268. High order discretization schemes for stochastic volatility models
Author(s): Benjamin Jourdain (CERMICS) and Mohamed Sbai (CERMICS)

Abstract: In usual stochastic volatility models, the process driving  
the volatility of the asset price evolves according to an autonomous  
one-dimensional stochastic differential equation. We assume that the  
coefficients of this equation are smooth. Using It\^o's formula, we  
get rid, in the asset price dynamics, of the stochastic integral with  
respect to the Brownian motion driving this SDE. Taking advantage of  
this structure, we propose - a scheme, based on the Milstein  
discretization of this SDE, with order one of weak trajectorial  
convergence for the asset price, - a scheme, based on the Ninomiya- 
Victoir discretization of this SDE, with order two of weak convergence  
for the asset price. We also propose a specific scheme with improved  
convergence properties when the volatility of the asset price is  
driven by an Orstein-Uhlenbeck process. We confirm the theoretical  
rates of convergence by numerical experiments and show that our  
schemes are well adapted to the multilevel Monte Carlo method  
introduced by Giles [2008a,b].

http://arxiv.org/abs/0908.1926


9269. On the Copula for multivariate Extreme Value distributions
Author(s): Glauco Valle and Marco Aurelio Sanfins

Abstract: We show that all multivariate Extreme Value distributions,  
which are the possible weak limits of the $K$ largest order statistics  
of iid sequences, have the same copula, the so called K-extremal  
copula. This copula is described through exact expressions for its  
density and distribution functions. We also study measures of  
dependence, we obtain a weak convergence result and we propose a  
simulation algorithm for the K-extremal copula.

http://arxiv.org/abs/0908.2144


9270. Simulation reductions for the Ising model
Author(s): Mark L. Huber

Abstract: Polynomial time reductions between problems have long been  
used to delineate problem classes. Simulation reductions also exist,  
where an oracle for simulation from some probability distribution can  
be employed together with an oracle for Bernoulli draws in order to  
obtain a draw from a different distribution. Here linear time  
simulation reductions are given for: the Ising spins world to the  
Ising subgraphs world and the Ising subgraphs world to the Ising spins  
world. This answers a long standing question of whether such a direct  
relationship between these two versions of the Ising model existed.  
Moreover, these reductions result in the first method for perfect  
simulation from the subgraphs world and a new Swendsen-Wang style  
Markov chain for the Ising model. The method used is to write the  
desired distribution with set parameters as a mixture of distributions  
where the parameters are at their extreme values.

http://arxiv.org/abs/0908.2151


9271. Connectivity Bounds for the Vacant Set of Random Interlacements
Author(s): Vladas Sidoravicius and Alain-Sol Sznitman

Abstract: The model of random interlacements on Z^d, d bigger or equal  
to 3, was recently introduced in arXiv:0704.2560. A non-negative  
parameter u parametrizes the density of random interlacements on Z^d.  
In the present note we investigate the connectivity properties of the  
vacant set left by random interlacements at level u, in the non- 
percolative regime, where u is bigger than the non-degenerate critical  
parameter for percolation of the vacant set, see arXiv:0704.2560,  
arXiv:0808.3344. We prove a stretched exponential decay of the  
connectivity function for the vacant set at level u, when u is bigger  
than an other critical parameter. It is presently an open problem  
whether these two critical parameters actually coincide.

http://arxiv.org/abs/0908.2206


9272. Random permutations with cycle weights
Author(s): Volker Betz and Daniel Ueltschi and Yvan Velenik

Abstract: We study the distribution of cycle lengths in models of  
nonuniform random permutations with cycle weights. We identify several  
regimes. Depending on the weights, the length of typical cycles grows  
like the total number n of elements, or a fraction of n, or a  
logarithmic power of n.

http://arxiv.org/abs/0908.2217


9273. The tree length of an evolving coalescent
Author(s): Peter Pfaffelhuber and Anton Wakolbinger and Heinz Weisshaupt

Abstract: A well-established model for the genealogy of a large  
population in equilibrium is Kingman's coalescent. For the population  
together with its genealogy evolving in time, this gives rise to a  
time-stationary tree-valued process. We study the sum of the branch  
lengths, briefly denoted as tree length, and prove that the (suitably  
compensated) sequence of tree length processes converges, as the  
population size tends to infinity, to a limit process with cadlag  
paths, infinite infinitesimal variance, and a Gumbel distribution as  
its equilibrium.

http://arxiv.org/abs/0908.2444


9274. Stochastic integral representation of the $L^{2}$ modulus of  
Brownian local time and a central limit theorem
Author(s): Yaozhong Hu and David Nualart

Abstract: The purpose of this note is to prove a central limit theorem  
for the $L^2$-modulus of continuity of the Brownian local time  
obtained in \cite{CLMR}, using techniques of stochastic analysis. The  
main ingredients of the proof are an asymptotic version of Knight's  
theorem and the Clark-Ocone formula for the $L^2$-modulus of the  
Brownian local time.

http://arxiv.org/abs/0908.2473


9275. Environmental Noise Variability in Population Dynamics Matrix  
Models
Author(s): Michel De Lara (CERMICS)

Abstract: The impact of environmental variability on population size  
growth rate in dynamic models is a recurrent issue in the theoretical  
ecology literature. In the scalar case, R. Lande pointed out that  
results are ambiguous depending on whether the noise is added at  
arithmetic or logarithmic scale, while the matrix case has been  
investigated by S. Tuljapurkar. Our contribution consists first in  
introducing another notion of variability than the widely used  
variance or coefficient of variation, namely the so-called convex  
orders. Second, in population dynamics matrix models, we focus on how  
matrix components depend functionaly on uncertain environmental  
factors. In the log-convex case, we show that, in a sense,  
environmental variability increases both mean population size and mean  
log-population size and makes them more variable. Our main result is  
that specific analytical dependence coupled with appropriate notion of  
variability lead to wide generic results, valid for all times and not  
only asymptotically, and requiring no assumptions of stationarity, of  
normality, of independency, etc. Though the approach is different, our  
conclusions are consistent with previous results in the literature.  
However, they make it clear that the analytical dependence on  
environmental factors cannot be overlooked when trying to tackle the  
influence of variability.

http://arxiv.org/abs/0908.2499


9276. A Backward Particle Interpretation of Feynman-Kac Formulae
Author(s): Pierre Del Moral and Arnaud Doucet and Sumeetpal S. Singh

Abstract: We design a particle interpretation of Feynman-Kac measures  
on path spaces based on a backward Markovian representation combined  
with a traditional mean field particle interpretation of the flow of  
their final time marginals. In contrast to traditional genealogical  
tree based models, these new particle algorithms can be used to  
compute normalized additive functionals "on-the-fly" as well as their  
limiting occupation measures with a given precision degree that does  
not depend on the final time horizon. We provide uniform convergence  
results w.r.t. the time horizon parameter as well as functional  
central limit theorems and exponential concentration estimates. We  
also illustrate these results in the context of computational physics  
and imaginary time Schroedinger type partial differential equations,  
with a special interest in the numerical approximation of the  
invariant measure associated to $h$-processes.

http://arxiv.org/abs/0908.2556


9277. Threshold graph limits and random threshold graphs
Author(s): Persi Diaconis and Susan Holmes and Svante Janson

Abstract: We study the limit theory of large threshold graphs and  
apply this to a variety of models for random threshold graphs. The  
results give a nice set of examples for the emerging theory of graph  
limits.

http://arxiv.org/abs/0908.2448


9278. Phase Transition for the Mixing Time of the Glauber Dynamics for  
Coloring Regular Trees
Author(s): Prasad Tetali and Juan C. Vera and Eric Vigoda and Linji Yang

Abstract: We prove that the mixing time of the Glauber dynamics for  
random $k$-colorings of the complete tree with branching factor $b$  
undergoes a phase transition at $k=b(1+o_b(1))/\ln{b}$. Our main  
result shows nearly sharp bounds on the mixing time of the dynamics on  
the complete tree with $n$ vertices for $k=Cb/\ln{b}$ colors with  
constant $C$. For $C\geq 1$ we prove the mixing time is $O(n^{1+o_b(1)} 
\ln^2{n})$. On the other side, for $C< 1$ the mixing time experiences  
a slowing down, in particular, we prove it is $O(n^{1/C + o_b(1)}\ln^2 
{n})$ and $\Omega(n^{1/C-o_b(1)})$. The critical point C=1 is  
interesting since it coincides (at least up to first order) to the so- 
called reconstruction threshold which was recently established by Sly.  
The reconstruction threshold has been of considerable interest  
recently since it appears to have close connections to the efficiency  
of certain local algorithms, and this work was inspired by our attempt  
to understand these connections in this particular setting.

http://arxiv.org/abs/0908.2665


9279. Stochastic Partial Differential Equations with Unbounded and  
Degenerate Coefficients
Author(s): Xicheng Zhang

Abstract: In this article, using DiPerna-Lions theory \cite{Di-Li}, we  
investigate linear second order stochastic partial differential  
equations with unbounded and degenerate non-smooth coefficients, and  
obtain several conditions for existence and uniqueness. Moreover, we  
also prove the $L^1$-integrability and a general maximal principle for  
generalized solutions of SPDEs. As applications, we study nonlinear  
filtering problem and also obtain the existence and uniqueness of  
generalized solutions for a degenerate nonlinear SPDE.

http://arxiv.org/abs/0908.2695


9280. Probabilistic representation for solutions of an irregular  
porous media type equation: the degenerate case
Author(s): Viorel Barbu and Michael Roeckner (SFB 701) and Francesco  
Russo (LAGA)

Abstract: We consider a possibly degenerate porous media type equation  
over all of $\R^d$ with $d = 1$, with monotone discontinuous  
coefficients with linear growth and prove a probabilistic  
representation of its solution in terms of an associated microscopic  
diffusion. This equation is motivated by some singular behaviour  
arising in complex self-organized critical systems. The main idea  
consists in approximating the equation by equations with monotone non- 
degenerate coefficients and deriving some new analytical properties of  
the solution.

http://arxiv.org/abs/0908.2701


9281. Sharp interface limit for invariant measures of a stochastic  
Allen-Cahn equation
Author(s): Hendrik Weber

Abstract: The invariant measure of a one-dimensional Allen-Cahn  
equation with an additive space-time white noise is studied. This  
measure is absolutely continuous with respect to a Brownian bridge  
with a density which can be interpreted as a potential energy term. We  
consider the sharp interface limit in this setup. In the right scaling  
this corresponds to a Gibbs type measure on a growing interval with  
decreasing temperature. Our main result is that in the limit we still  
see exponential convergence towards a curve of minimizers of the  
energy if the interval does not grow too fast. In the original scaling  
the limit measure is concentrated on configurations with precisely one  
jump. This jump is distributed uniformly.

http://arxiv.org/abs/0908.2717


9282. Hydrodynamic limit of move-to-front rules and search cost  
probabilities
Author(s): Kumiko Hattori and Tetsuya Hattori

Abstract: We study a hydrodynamic limit approach to move-to-front  
rules, namely, a scaling limit as the number of items tends to  
infinity, of the joint distribution of jump rate and position of  
items. As an application of the limit formula, we present asymptotic  
formulas on search cost probability distributions, applicable for  
general jump rate distributions.

http://arxiv.org/abs/0908.3222


9283. Stochastic Evolutions of Point Processes
Author(s): Philippe Robert

Abstract: The asymptotic behavior of birth and death processes of  
particles in a compact space is analyzed. Births: Particles are  
created at rate $\lambda_+$ and their location is independent of the  
current configuration. Deaths are due to negative particles arriving  
at rate $\lambda_-$. The death of a particle occurs when a negative  
particle arrives in its neighborhood and kills it. Several killing  
schemes are considered. The arriving locations of positive and  
negative particles are assumed to have the same distribution. By using  
a combination of monotonicity properties and invariance relations it  
is shown that the configurations of particles converge in distribution  
for several models. The problems of uniqueness of invariant measures  
and of the existence of accumulation points for the limiting  
configurations are also investigated. It is shown for several natural  
models that if $\lambda_+<\lambda_-$ then the asymptotic configuration  
has a finite number of points with probability 1. Examples with $ 
\lambda_+<\lambda_-$ and an infinite number of particles in the limit  
are also presented.

http://arxiv.org/abs/0908.3256


9284. Reflected Brownian motion in Weyl chambers
Author(s): Nizar Demni

Abstract: We supply two different descriptions of the pushing process  
driving the reflected Brownian motion in Weyl chambers, when the  
latter domains are simplexes. The first one shows that a simple root  
lies in one and only one orbit if and only if the pushing process in  
the direction of that simple root increases as the sum of all the  
Brownian local times in the directions of the orbit's positive  
elements. The last one shows that the pushing process may be written  
as the sum of an inward normal vector at the chamber's boundary and an  
inward normal vector at the origin, yielding a kind of a multivoque  
stochastic differential equation for the reflected process. We finally  
give a particles system interpretation of the reflected process and  
construct a multidimensional skew Brownian motion.

http://arxiv.org/abs/0908.3302


9285. A zero-one law for linear transformations of Levy noise
Author(s): Steven N. Evans

Abstract: A L\'evy noise on $\mathbb{R}^d$ assigns a random real  
"mass" $\Pi(B)$ to each Borel subset $B$ of $\mathbb{R}^d$ with finite  
Lebesgue measure. The distribution of $\Pi(B)$ only depends on the  
Lebesgue measure of $B$, and if $B_1, ..., B_n$ is a finite collection  
of pairwise disjoint sets, then the random variables $\Pi(B_1), ...,  
\Pi(B_n)$ are independent with $\Pi(B_1 \cup >... \cup B_n) = \Pi(B_1)  
+ ... + \Pi(B_n)$ almost surely. In particular, the distribution of $ 
\Pi \circ g$ is the same as that of $\Pi$ when $g$ is a bijective  
transformation of $\mathbb{R}^d$ that preserves Lebesgue measure. It  
follows from the Hewitt--Savage zero--one law that any event which is  
almost surely invariant under the mappings $\Pi \mapsto \Pi \circ g$  
for every Lebesgue measure preserving bijection $g$ of $\mathbb{R}^d$  
must have probability 0 or 1. We investigate whether certain smaller  
groups of Lebesgue measure preserving bijections also possess this  
property. We show that if $d \ge 2$, the L\'evy noise is not purely  
deterministic, and the group consists of linear transformations and is  
closed, then the invariant events all have probability 0 or 1 if and  
only if the group is not compact.

http://arxiv.org/abs/0908.3339


9286. Finite-time blowup and existence of global positive solutions of  
a semi-linear SPDE
Author(s): Marco Dozzi (IECN) and Jos\'e Alfredo Lopez

Abstract: We consider stochastic equations of the prototype $du(t,x) = 
(\Delta u(t,x)+u(t,x)^{1+\beta})dt+\kappa u(t,x) dW_{t}$ on a smooth  
domain $D\subset \mathord{\rm I\mkern-3.6mu R\:}^d$, with Dirichlet  
boundary condition, where $\beta$, $\kappa$ are positive constants and  
$\{W_t $, $t\ge0\}$ is a one-dimensional standard Wiener process. We  
estimate the probability of finite time blowup of positive solutions,  
as well as the probability of existence of non-trivial positive global  
solutions.

http://arxiv.org/abs/0908.3364


9287. Limit theorems for random processes with random time substitution
Author(s): Permyakova Elena

Abstract: In this paper the sufficient conditions for convergence in  
Skorokhod space $D[0,1]$ of sequence of random processes with random  
time substitution are obtained.

http://arxiv.org/abs/0908.3395


9288. Poisson Splitting by Factors
Author(s): Alexander E. Holroyd and Russell Lyons and and Terry Soo

Abstract: Given a homogeneous Poisson process on R^d with intensity L,  
we prove that it is possible to partition the points into two sets, as  
a deterministic function of the process, and in an isometry- 
equivariant way, so that each set of points forms a homogeneous  
Poisson process, with any given pair of intensities summing to L. In  
particular, this answers a question of Ball, who proved that in d=1,  
the Poisson points may be similarly partitioned (via a translation- 
equivariant function) so that one set forms a Poisson process of lower  
intensity, and asked whether the same was possible for all d. We do  
not know whether it is possible similarly to add points (again chosen  
as a deterministic function of a Poisson process) to obtain a Poisson  
process of higher intensity, but we prove that this is not possible  
under an additional finitariness condition.

http://arxiv.org/abs/0908.3409


9289. A rule of thumb for riffle shuffling
Author(s): Sami Assaf and Persi Diaconis and K. Soundararajan

Abstract: We study how many riffle shuffles are required to mix n  
cards if only certain features of the deck are of interest, e.g. suits  
disregarded or only the colors of interest. For these features, the  
number of shuffles drops from 3/2 log_2(n) to log_2(n). We derive  
closed formulae and an asymptotic `rule of thumb' formula which is  
remarkably accurate.

http://arxiv.org/abs/0908.3462


9290. Optimal transportation and monotonic quantities on evolving  
manifolds
Author(s): Hong Huang

Abstract: In this note we adapt Topping's $\mathcal{L}$-optimal  
transportation theory for Ricci flow to a more general situation, i.e.  
to a closed manifold $(M,g_{ij}(t)$ evolving by $\partial_tg_{ij}=-2S_ 
{ij}$, where $S_{ij}$ is a symmetric tensor field of (2,0)-type on $M 
$. We extend some of Topping's and Lott's recent results, generalize  
the monotonicity of List's (and hence also of Perelman's) $\mathcal{W} 
$-entropy, and recover the monotonicity of M$\ddot{u}$ller's (and  
hence also of Perelman's) reduced volume.

http://arxiv.org/abs/0908.3293


9291. Rank-based attachment leads to power law graphs
Author(s): Jeannette Janssen and Pawel Pralat

Abstract: We investigate the degree distribution resulting from graph  
generation models based on rank-based attachment. In rank-based  
attachment, all vertices are ranked according to a ranking scheme. The  
link probability of a given vertex is proportional to its rank raised  
to the power -a, for some a in (0,1). Through a rigorous analysis, we  
show that rank-based attachment models lead to graphs with a power law  
degree distribution with exponent 1+1/a whenever vertices are ranked  
according to their degree, their age, or a randomly chosen fitness  
value. We also investigate the case where the ranking is based on the  
initial rank of each vertex; the rank of existing vertices only  
changes to accommodate the new vertex. Here, we obtain a sharp  
threshold for power law behaviour. Only if initial ranks are biased  
towards lower ranks, or chosen uniformly at random, we obtain a power  
law degree distribution with exponent 1+1/a. This indicates that the  
power law degree distribution often observed in nature can be  
explained by a rank-based attachment scheme, based on a ranking scheme  
that can be derived from a number of different factors; the exponent  
of the power law can be seen as a measure of the strength of the  
attachment.

http://arxiv.org/abs/0908.3436


9292. Notes on Feige's gumball machines problem
Author(s): John H. Elton

Abstract: We give a detailed proof, in the identically distributed  
case, of a conjecture of Feige about the maximum probability that the  
sum of n independent non-negative integer valued random variables,  
each of mean 1, exceeds n. The general case is reduced to two-point  
distributions.

http://arxiv.org/abs/0908.3528


9293. Limit theorems for projections of random walk on a hypersphere
Author(s): Max Skipper

Abstract: We show that almost any one-dimensional projection of a  
suitably scaled random walk on a hypercube, inscribed in a  
hypersphere, converges weakly to an Ornstein-Uhlenbeck process as the  
dimension of the sphere tends to infinity. We also observe that the  
same result holds when the random walk is replaced with spherical  
Brownian motion. This latter result can be viewed as a "functional"  
generalisation of Poincar\'e's observation for projections of uniform  
measure on high dimensional spheres; the former result is an analogous  
generalisation of the Bernoulli-Laplace central limit theorem. Given  
the relation of these two classic results to the central limit theorem  
for convex bodies, the modest results provided here would appear to  
motivate a functional generalisation.

http://arxiv.org/abs/0908.3536


9294. Can an infinite product of nonnegative matrices be expressed in  
terms of infinite products of stochastic ones?
Author(s): Alain Thomas (LATP)

Abstract: It is known that if the product $M_n... M_1$ converges to a  
nonnull limit when $n\to\infty$ and if the $M_n$ belong to a finite  
set of complex matrices, then the $M_n$ for $n\ge n_0$ have a common  
right eigenvector $V$ for the eigenvalue 1. In case the $M_n$ are  
nonnegative and $V$ is positive, $\Delta^{-1}M_{n_0}... M_n\Delta$ is  
the product of the stochastic matrices $\Delta^{-1}M_n\Delta$, where  
the diagonal matrix $\Delta$ has on its diagonal the same entries as $V 
$. In the last section we examine what happen when we remove the  
hypothesis that $V$ is positive.

http://arxiv.org/abs/0908.3538


9295. Critical random graphs: limiting constructions and  
distributional properties
Author(s): L. Addario-Berry and N. Broutin and C. Goldschmidt

Abstract: We consider the Erdos--Renyi random graph G(n,p) inside the  
critical window, where p = 1/n + lambda * n^{-4/3} for some lambda in  
R. We proved in a previous paper (arXiv:0903.4730) that considering  
the connected components of G(n,p) as a sequence of metric spaces with  
the graph distance rescaled by n^{-1/3} and letting n go to infinity  
yields a non-trivial sequence of limit metric spaces C = (C_1,  
C_2, ...). These limit metric spaces can be constructed from certain  
random real trees with vertex-identifications. We give here equivalent  
constructions using standard Brownian continuum random trees, their  
recursive construction from inhomogeneous Poisson point processes, and  
Polya's urn scheme. We also characterize the distributions of the  
masses and lengths in the constituant parts of a limit component when  
it is decomposed according to its cycle structure.

http://arxiv.org/abs/0908.3629


9296. Harnack Inequalities and Applications for Multivalued Stochastic  
Evolution Equations
Author(s): Shun-Xiang Ouyang

Abstract: By the method of coupling and Girsanov transformation,  
Harnack inequalities [F.-Y. Wang, 1997] and strong Feller property are  
proved for the transition semigroup associated with the multivalued  
stochastic evolution equation on a Gelfand triple. The concentration  
property of the invariant measure for the semigroup is investigated.  
As applications of Harnack inequalities, explicit upper bounds of the  
$L^p$-norm of the density, contractivity, compactness and entropy-cost  
inequality for the semigroup are also presented.

http://arxiv.org/abs/0908.3630


9297. Applications of Weak Convergence for Hedging of American and  
Game Options
Author(s): Yan Dolinsky

Abstract: This paper studies stability of Dynkin's games value under  
weak convergence. We use these results to approximate game options  
prices with path dependent payoffs in continuous time models by  
sequence of game options prices in discrete time models which can be  
calculated by dynamical programming algorithms. We also show that  
shortfall risks of American options in a sequence of multinomial  
approximations of the multidimensional BS market converge to the  
corresponding quantities for similar American options in the  
multidimensional BS market with path dependent payoffs. In comparison  
to previous papers we work under more general convergence of  
underlying processes, as well, as weaker condition on the payoffs.

http://arxiv.org/abs/0908.3661


9298. On the minimal penalty for Markov order estimation
Author(s): Ramon van Handel

Abstract: We show that large-scale typicality of Markov sample paths  
implies that the likelihood ratio statistic satisfies a law of  
iterated logarithm uniformly to the same scale. As a consequence, the  
penalized likelihood Markov order estimator is strongly consistent for  
penalties growing as slowly as log log n when an upper bound is  
imposed on the order which may grow as rapidly as log n. Our method of  
proof, using techniques from empirical process theory, does not rely  
on the explicit expression for the maximum likelihood estimator in the  
Markov case and could therefore be applicable in other settings.

http://arxiv.org/abs/0908.3666


9299. Zero-one laws for connectivity in random key graphs
Author(s): Osman Yagan and Armand M. Makowski

Abstract: The random key graph is a random graph naturally associated  
with the random key predistribution scheme of Eschenauer and Gligor  
for wireless sensor networks. For this class of random graphs we  
establish a new version of a conjectured zero-one law for graph  
connectivity as the number of nodes becomes unboundedly large. The  
results reported here complement and strengthen recent work on this  
conjecture by Blackburn and Gerke. In particular, the results are  
given under conditions which are more realistic for applications to  
wireless sensor networks.

http://arxiv.org/abs/0908.3644


9300. Randomized Scheduling Algorithm for Queueing Networks
Author(s): Devavrat Shah and Jinwoo Shin

Abstract: There has recently been considerable interest in design of  
low-complexity, myopic, distributed and stable scheduling policies for  
constrained queueing network models that arise in the context of  
emerging communication networks. Here, we consider two representative  
models. One, a model for the collection of wireless nodes  
communicating through a shared medium, that represents randomly  
varying number of packets in the queues at the nodes of networks. Two,  
a buffered circuit switched network model for an optical core of  
future Internet, to capture the randomness in calls or flows present  
in the network. The maximum weight scheduling policy proposed by  
Tassiulas and Ephremide in 1992 leads to a myopic and stable policy  
for the packet-level wireless network model. But computationally it is  
very expensive (NP-hard) and centralized. It is not applicable to the  
buffered circuit switched network due to the requirement of non- 
premption of the calls in the service. As the main contribution of  
this paper, we present a stable scheduling algorithm for both of these  
models. The algorithm is myopic, distributed and performs few logical  
operations at each node per unit time.

http://arxiv.org/abs/0908.3670


9301. Asymptotic regimes for the partition into colonies of a  
branching process with emigration
Author(s): Jean Bertoin (PMA and Dma)

Abstract: We consider a spatial branching process with emigration in  
which children either remain at the same site as their parents or  
migrate to new locations and then found their own colonies. We are  
interested in asymptotics of the partition of the total population  
into colonies for large populations with rare migrations. Under  
appropriate regimes, we establish weak convergence of the rescaled  
partition to some random measure that is constructed from the  
restriction of a Poisson point measure to a certain random region, and  
whose cumulant solves a simple integral equation.

http://arxiv.org/abs/0908.3735


9302. On the absolute continuity of multidimensional Ornstein- 
Uhlenbeck processes
Author(s): Thomas Simon (LPP)

Abstract: Let $X$ be a $n$-dimensional Ornstein-Uhlenbeck process,  
solution of the S.D.E. $$\d X_t = AX_t \d t + \d B_t$$ where $A$ is a  
real $n\times n$ matrix and $B$ a L\'evy process without Gaussian  
part. We show that when $A$ is non-singular, the law of $X_1$ is  
absolutely continuous in $\r^n$ if and only if the jumping measure of  
$B$ fulfils a certain geometric condition with respect to $A,$ which  
we call the exhaustion property. This optimal criterion is much weaker  
than for the background driving L\'evy process $B$, which might be  
very singular and sometimes even have a one-dimensional discrete  
jumping measure. It also solves a difficult problem for a certain  
class of multivariate Non-Gaussian infinitely divisible distributions.

http://arxiv.org/abs/0908.3736


9303. Extremal Subgraphs of Random Graphs: an Extended Version
Author(s): Graham Brightwell and Konstantinos Panagiotou and Angelika  
Steger

Abstract: We prove that there is a constant $c >0$, such that whenever  
$p \ge n^{-c}$, with probability tending to 1 when $n$ goes to  
infinity, every maximum triangle-free subgraph of the random graph $G_ 
{n,p}$ is bipartite. This answers a question of Babai, Simonovits and  
Spencer (Journal of Graph Theory, 1990). The proof is based on a tool  
of independent interest: we show, for instance, that the maximum cut  
of almost all graphs with $M$ edges, where $M >> n$, is ``nearly  
unique''. More precisely, given a maximum cut $C$ of $G_{n,M}$, we can  
obtain all maximum cuts by moving at most $O(\sqrt{n^3/M})$ vertices  
between the parts of $C$.

http://arxiv.org/abs/0908.3778


9304. Mixing time of near-critical random graphs
Author(s): Jian Ding and Eyal Lubetzky and Yuval Peres

Abstract: Let $C_1$ be the largest component of the Erd\H{o}s-R\'enyi  
random graph $G(n,p)$. The mixing time of random walk on $C_1$ in the  
strictly supercritical regime, $p=c/n$ with fixed $c>1$, was shown to  
have order $\log^2 n$ by Fountoulakis and Reed, and independently by  
Benjamini, Kozma and Wormald. In the critical window, $p=(1+\epsilon)/n 
$ where $\lambda=\epsilon^3 n$ is bounded, Nachmias and Peres proved  
that the mixing time on $C_1$ is of order $n$. However, it was unclear  
how to interpolate between these results, and estimate the mixing time  
as the giant component emerges from the critical window. Indeed, even  
the asymptotics of the diameter of $C_1$ in this regime were only  
recently obtained by Riordan and Wormald, as well as the present  
authors and Kim. In this paper we show that for $p=(1+\epsilon)/n$  
with $\lambda=\epsilon^3 n\to\infty$ and $\lambda=o(n)$, the mixing  
time on $C_1$ is with high probability of order $(n/\lambda)\log^2  
\lambda$. In addition, we show that this is the order of the largest  
mixing time over all components, both in the slightly supercritical  
and in the slightly subcritical regime (i.e., $p=(1-\epsilon)/n$ with $ 
\lambda$ as above).

http://arxiv.org/abs/0908.3870


9305. Utility Optimization in Congested Queueing Networks
Author(s): Neil Stuart Walton

Abstract: We consider a multi-class single server queueing network as  
a model of a packet switching network. The rates packets are sent into  
this network are controlled by queues which act as congestion windows.  
By considering a sequence of such congestion windows we allow the  
network to become congested. We show the stationary throughput of  
routes on this sequence of networks converges to an allocation that  
maximizes aggregate utility subject to the network's capacity  
constraints. To perform this analysis we require that our utility  
functions satisfy an exponential concavity condition. This family of  
utilities includes weighted $\alpha$-fair utilities for $\alpha >1$.

http://arxiv.org/abs/0908.3787


9306. Distributed Averaging via Lifted Markov Chains
Author(s): Kyomin Jung and Devavrat Shah and Jinwoo Shin

Abstract: Motivated by applications of distributed linear estimation,  
distributed control and distributed optimization, we consider the  
question of designing linear iterative algorithms for computing the  
average of numbers in a network. Specifically, our interest is in  
designing such an algorithm with the fastest rate of convergence given  
the topological constraints of the network. As the main result of this  
paper, we design an algorithm with the fastest possible rate of  
convergence using a non-reversible Markov chain on the given network  
graph. We construct such a Markov chain by transforming the standard  
Markov chain, which is obtained using the Metropolis-Hastings method.  
We call this novel transformation pseudo-lifting. We apply our method  
to graphs with geometry, or graphs with doubling dimension.  
Specifically, the convergence time of our algorithm (equivalently, the  
mixing time of our Markov chain) is proportional to the diameter of  
the network graph and hence optimal. As a byproduct, our result  
provides the fastest mixing Markov chain given the network topological  
constraints, and should naturally find their applications in the  
context of distributed optimization, estimation and control.

http://arxiv.org/abs/0908.4073


9307. Hydrodynamic limit of the exclusion process in inhomogeneous media
Author(s): Milton Jara

Abstract: We obtain the hydrodynamic limit of a simple exclusion  
process in an inhomogeneous environment of divergence form. Our main  
assumption is a suitable version of Gamma-convergence for the  
environment. In this way we obtain an unified approach to recent works  
on the field.

http://arxiv.org/abs/0908.4120


9308. Contact process in a wedge
Author(s): J. Theodore Cox and Nevena Maric and Rinaldo B. Schinazi

Abstract: We prove that the supercritical one-dimensional contact  
process survives in certain wedge-like space-time regions, and that  
when it survives it couples with the unrestricted contact process  
started from its upper invariant measure. As an application we show  
that a type of weak coexistence is possible in the nearest-neighbor  
``grass-bushes-trees'' successional model introduced in Durrett and  
Swindle (1991).

http://arxiv.org/abs/0908.4125


9309. Kolmogorov equation associated to the stochastic reflection  
problem on a smooth convex set of a Hilbert space
Author(s): Viorel Barbu and Giuseppe Da Prato and Luciano Tubaro

Abstract: We consider the stochastic reflection problem associated  
with a self-adjoint operator $A$ and a cylindrical Wiener process on a  
convex set $K$ with nonempty interior and regular boundary $\Sigma$ in  
a Hilbert space $H$. We prove the existence and uniqueness of a smooth  
solution for the corresponding elliptic infinite-dimensional  
Kolmogorov equation with Neumann boundary condition on $\Sigma$.

http://arxiv.org/abs/0908.4139


9310. The survival of large dimensional threshold contact processes
Author(s): Thomas Mountford and Roberto H. Schonmann

Abstract: We study the threshold $\theta$ contact process on $\mathbb 
{Z}^d$ with infection parameter $\lambda$. We show that the critical  
point $\lambda_{\mathrm{c}}$, defined as the threshold for survival  
starting from every site occupied, vanishes as $d\to\infty$. This  
implies that the threshold $\theta$ voter model on $\mathbb{Z}^d$ has  
a nondegenerate extremal invariant measure, when $d$ is large.

http://arxiv.org/abs/0908.4146


9311. On the extendibility of partially and Markov exchangeable binary  
sequences
Author(s): Davide Di Cecco

Abstract: In [Fortini et al., Stoch. Proc. Appl. 100 (2002), 147--165]  
it is demonstrated that a recurrent Markov exchangeable process in the  
sense of Diaconis and Freedman is essentially a partially exchangeable  
process in the sense of de Finetti. In case of finite sequences there  
is not such an equivalence. We analyze both finite partially  
exchangeable and finite Markov exchangeable binary sequences and  
formulate necessary and sufficient conditions for extendibility in  
both cases.

http://arxiv.org/abs/0908.4158


9312. Asymptotic properties of the columns in the products of  
nonnegative matrices
Author(s): \'Eric Olivier (LATP) and Alain Thomas (LATP)

Abstract: We consider the sequence of column-vectors $R_n=A_1... A_nR$  
associated to a sequence $(A_n)$ of nonnegative $d\times d$ matrices  
and to a positive $d$-dimensional column-vector $R$. The problem to  
know the necessary and sufficient conditions -- on the sequence $(A_n) 
$ -- for $\displaystyle{R_n\over\Vert R_n\Vert}$ to converge is yet  
not solved. Nevertheless we prove this convergence in case the $A_n$  
are -- in a sense -- echeloned and fulfill certain boundness  
conditions. If the $A_n$ do not fulfill the conditions and even if  
they are sparse, it may exist a sequence of integers $(n_k)$ such that  
the matrices $A'_k=A_{n_k+1}... A_{n_{k+1}}$ do: we see in some other  
paper how to proceed in one example, and how to use the obtained  
result to study some continuous singular measure.

http://arxiv.org/abs/0908.4171


9313. On the inverse first-passage-time problem for a Wiener process
Author(s): Cristina Zucca and Laura Sacerdote

Abstract: The inverse first-passage problem for a Wiener process $(W_t) 
_{t\ge0}$ seeks to determine a function $b{}:{}\mathbb{R}_+\to\mathbb 
{R}$ such that \[\tau=\inf\{t>0| W_t\ge b(t)\}\] has a given law. In  
this paper two methods for approximating the unknown function $b$ are  
presented. The errors of the two methods are studied. A set of  
examples illustrates the methods. Possible applications are enlighted.

http://arxiv.org/abs/0908.4213


9314. Extremal shot noises, heavy tails and max-stable random fields
Author(s): Cl\'ement Dombry (LMA)

Abstract: Extremal shot noises naturally appear in extreme value  
theory as a model for spatial extremes and serve as basic models for  
annual maxima of rainfall or for coverage field in telecommunication.  
In this work, we examine their properties such as boundedness,  
regularity, ergodicity ... Connexions with max-stable random fields  
are established: we prove a limit theorem when the distribution of the  
weights is heavy tailed and the intensity of points goes to infinity.  
We use a point process approach strongly connected to the Peak Over  
Threshold method used by hydrologists. Properties of the limit max- 
stable random fields are also investigated.

http://arxiv.org/abs/0908.4221


9315. Stochastic completeness and volume growth
Author(s): Christian Baer and G. Pacelli Bessa

Abstract: It has been suggested in 1999 that a certain volume growth  
condition for geodesically complete Riemannian manifolds might imply  
that the manifold is stochastically complete. This is motivated by a  
large class of examples and by a known analogous criterion for  
recurrence of Brownian motion. We show that the suggested implication  
is not true in general. We also give counter-examples to a converse  
implication.

http://arxiv.org/abs/0908.4222


9316. Matrix factorization identity for almost semi-continuous  
processes on a Markov chain
Author(s): D.V. Gusak and E.V. Karnaukh

Abstract: In this article almost semi-continuous processes with  
stationary independent increments on a finite irreducible Markov chain  
are considered. For these processes the components of matrix  
factorization identity are concretely defined. On the basis of this  
concrete definition the relations for the distributions of extrema and  
distributions of their complements for the almost upper semi- 
continuous processes are established.

http://arxiv.org/abs/0908.4326


9317. Limit laws of transient excited random walks on integers
Author(s): Elena Kosygina and Thomas Mountford

Abstract: We consider excited random walks (ERWs) on integers with a  
bounded number of i.i.d. cookies per site without the non-negativity  
assumption on the drifts induced by the "cookies". E. Kosygina and  
M.P.W. Zerner have shown that when the total expected drift per site,  
delta, is larger than 1 then ERW is transient to the right and,  
moreover, for delta>4 under the averaged measure it obeys the Central  
Limit Theorem. We show that when delta is in (2,4] the limiting  
behavior of an appropriately centered and scaled excited random walk  
is described by a strictly stable law with parameter delta/2. Our  
method also extends the results obtained by A.-L. Basdevant and A.  
Singh for delta in (1,2] under the non-negativity assumption to the  
setting which allows both positive and negative cookies.

http://arxiv.org/abs/0908.4356


9318. Poisson Dirichlet$(\alpha,\theta)$-Bridge Equations and  
Coagulation-Fragmentation Duality
Author(s): Lancelot F. James

Abstract: This paper derives distributional properties of a class of  
exchangeable bridges closely related to the Poisson-Dirichlet $(\alpha, 
\theta)$ family of bridges. We then show that various stochastic  
equations derived for these bridges lead to constructions of a new  
large class of coagulation and fragmentation operators that satisfy a  
duality property, and are otherwise easily manipulated. This class,  
builds on, and includes the duality relations developed in Pitman  
(1999), Bertoin and Goldschmidt (2004), and Dong, Goldschmidt and  
Martin (2006),which we can treat in a unified way. Our exposition also  
suggests an approach to obtain other dualities and related results.

http://arxiv.org/abs/0908.4436


9319. Convergence of Numerical Time-Averaging and Stationary Measures  
via Poisson Equations
Author(s): Jonathan C. Mattingly and Andrew M. Stuart and M.V. Tretyakov

Abstract: Numerical approximation of the long time behavior of a  
stochastic differential equation (SDE) is considered. Error estimates  
for time-averaging estimators are obtained and then used to show that  
the stationary behavior of the numerical method converges to that of  
the SDE. The error analysis is based on using an associated Poisson  
equation for the underlying SDE. The main advantage of this approach  
is its simplicity and universality. It works equally well for a range  
of explicit and implicit schemes including those with simple  
simulation of random variables, and for general hypoelliptic SDEs. An  
analogy between this approach and Stein's method is indicated. Some  
practical implications of the results are discussed.

http://arxiv.org/abs/0908.4450


9320. Time averages, recurrence and transience in the stochastic  
replicator dynamics
Author(s): Josef Hofbauer and Lorens A. Imhof

Abstract: We investigate the long-run behavior of a stochastic  
replicator process, which describes game dynamics for a symmetric two- 
player game under aggregate shocks. We establish an averaging  
principle that relates time averages of the process and Nash  
equilibria of a suitably modified game. Furthermore, a sufficient  
condition for transience is given in terms of mixed equilibria and  
definiteness of the payoff matrix. We also present necessary and  
sufficient conditions for stochastic stability of pure equilibria.

http://arxiv.org/abs/0908.4467


9321. Bubbles, convexity and the Black--Scholes equation
Author(s): Erik Ekstr\"{o}m and Johan Tysk

Abstract: A bubble is characterized by the presence of an underlying  
asset whose discounted price process is a strict local martingale  
under the pricing measure. In such markets, many standard results from  
option pricing theory do not hold, and in this paper we address some  
of these issues. In particular, we derive existence and uniqueness  
results for the Black--Scholes equation, and we provide convexity  
theory for option pricing and derive related ordering results with  
respect to volatility. We show that American options are convexity  
preserving, whereas European options preserve concavity for general  
payoffs and convexity only for bounded contracts.

http://arxiv.org/abs/0908.4468


9322. On convergence to stationarity of fractional Brownian storage
Author(s): Michel Mandjes and Ilkka Norros and Peter Glynn

Abstract: With $M(t):=\sup_{s\in[0,t]}A(s)-s$ denoting the running  
maximum of a fractional Brownian motion $A(\cdot)$ with negative  
drift, this paper studies the rate of convergence of $\mathbb {P}(M(t) 
 >x)$ to $\mathbb{P}(M>x)$. We define two metrics that measure the  
distance between the (complementary) distribution functions $\mathbb{P} 
(M(t)>\cdot)$ and $\mathbb{P}(M>\cdot)$. Our main result states that  
both metrics roughly decay as $\exp(-\vartheta t^{2-2H})$, where $ 
\vartheta$ is the decay rate corresponding to the tail distribution of  
the busy period in an fBm-driven queue, which was computed recently  
[Stochastic Process. Appl. (2006) 116 1269--1293]. The proofs  
extensively rely on application of the well-known large deviations  
theorem for Gaussian processes. We also show that the identified  
relation between the decay of the convergence metrics and busy-period  
asymptotics holds in other settings as well, most notably when G 
\"artner--Ellis-type conditions are fulfilled.

http://arxiv.org/abs/0908.4472


9323. Random recurrence equations and ruin in a Markov-dependent  
stochastic economic environment
Author(s): Jeffrey F. Collamore

Abstract: We develop sharp large deviation asymptotics for the  
probability of ruin in a Markov-dependent stochastic economic  
environment and study the extremes for some related Markovian  
processes which arise in financial and insurance mathematics, related  
to perpetuities and the $\operatorname {ARCH}(1)$ and $\operatorname  
{GARCH}(1,1)$ time series models. Our results build upon work of  
Goldie [Ann. Appl. Probab. 1 (1991) 126--166], who has developed tail  
asymptotics applicable for independent sequences of random variables  
subject to a random recurrence equation. In contrast, we adopt a  
general approach based on the theory of Harris recurrent Markov chains  
and the associated theory of nonnegative operators, and meanwhile  
develop certain recurrence properties for these operators under a  
nonstandard "G\"artner--Ellis" assumption on the driving process.

http://arxiv.org/abs/0908.4479


9324. Non-Markov property of certain eigenvalue processes analogous to  
Dyson's model
Author(s): Ryoki Fukushima and Atsushi Tanida and Kouji Yano

Abstract: It is proven that the eigenvalue process of Dyson's random  
matrix process of size two becomes non-Markov if the common  
coefficient $1/\sqrt{2}$ in the non-diagonal entries is replaced by a  
different positive number.

http://arxiv.org/abs/0908.4481


9325. Optimal reinsurance/investment problems for general insurance  
models
Author(s): Yuping Liu and Jin Ma

Abstract: In this paper the utility optimization problem for a general  
insurance model is studied. The reserve process of the insurance  
company is described by a stochastic differential equation driven by a  
Brownian motion and a Poisson random measure, representing the  
randomness from the financial market and the insurance claims,  
respectively. The random safety loading and stochastic interest rates  
are allowed in the model so that the reserve process is non-Markovian  
in general. The insurance company can manage the reserves through both  
portfolios of the investment and a reinsurance policy to optimize a  
certain utility function, defined in a generic way. The main feature  
of the problem lies in the intrinsic constraint on the part of  
reinsurance policy, which is only proportional to the claim-size  
instead of the current level of reserve, and hence it is quite  
different from the optimal investment/consumption problem with  
constraints in finance. Necessary and sufficient conditions for both  
well posedness and solvability will be given by modifying the  
``duality method'' in finance and with the help of the solvability of  
a special type of backward stochastic differential equations.

http://arxiv.org/abs/0908.4538


9326. Recursive estimation of time-average variance constants
Author(s): Wei Biao Wu

Abstract: For statistical inference of means of stationary processes,  
one needs to estimate their time-average variance constants (TAVC) or  
long-run variances. For a stationary process, its TAVC is the sum of  
all its covariances and it is a multiple of the spectral density at  
zero. The classical TAVC estimate which is based on batched means does  
not allow recursive updates and the required memory complexity is O 
(n). We propose a faster algorithm which recursively computes the  
TAVC, thus having memory complexity of order O(1) and the  
computational complexity scales linearly in $n$. Under short-range  
dependence conditions, we establish moment and almost sure convergence  
of the recursive TAVC estimate. Convergence rates are also obtained.

http://arxiv.org/abs/0908.4540


9327. Asymptotic behavior of unstable INAR(p) processes
Author(s): Matyas Barczy and Marton Ispany and Gyula Pap

Abstract: In this paper the asymptotic behavior of an unstable integer- 
valued autoregressive model of order p (INAR(p)) is described. Under a  
natural assumption it is proved that the sequence of appropriately  
scaled random step functions formed from an unstable INAR(p) process  
converges weakly towards a squared Bessel process. We note that this  
limit behavior is quite different from that of familiar unstable  
autoregressive processes of order p.

http://arxiv.org/abs/0908.4560


9328. Analysis of a Stochastic Predator-Prey Model with Applications  
to Intrahost HIV Genetic Diversity
Author(s): Sivan Leviyang

Abstract: During an infection, HIV experiences strong selection by  
immune system T cells. Recent experimental work has shown that MHC  
escape mutations form an important pathway for HIV to avoid such  
selection. In this paper, we study a model of MHC escape mutation. The  
model is a predator-prey model with two prey, composed of two HIV  
variants, and one predator, the immune system CD8 cells. We assume  
that one HIV variant is visible to CD8 cells and one is not. The model  
takes the form of a system of stochastic differential equations.  
Motivated by well-known results concerning the short life-cycle of HIV  
intrahost, we assume that HIV population dynamics occur on a faster  
time scale then CD8 population dynamics. This separation of time  
scales allows us to analyze our model using an asymptotic approach.  
Using this model we study the impact of an MHC escape mutation on the  
population dynamics and genetic evolution of the intrahost HIV  
population. From the perspective of population dynamics, we show that  
the competition between the visible and invisible HIV variants can  
reach steady states in which either a single variant exists or in  
which coexistence occurs depending on the parameter regime. We show  
that in some parameter regimes the end state of the system is  
stochastic. From a genetics perspective, we study the impact of the  
population dynamics on the lineages of HIV samples taken after an  
escape mutation occurs. We show that the lineages go through severe  
bottlenecks and that the lineage distribution can be characterized by  
a Kingman coalescent.

http://arxiv.org/abs/0908.4569


9329. Stability of a spatial polling system with greedy myopic service
Author(s): Lasse Leskel\"a and Falk Unger

Abstract: This paper studies a spatial queueing system on a circle,  
polled at random locations by a myopic server that can only observe  
customers in a bounded neighborhood. The server operates according to  
a greedy policy, always serving the nearest customer in its  
neighborhood, and leaving the system unchanged at polling instants  
where the neighborhood is empty. This system is modeled as a measure- 
valued random process, which is shown to be positive recurrent under a  
natural stability condition that does not depend on the server's scan  
range. When the interpolling times are light-tailed, the stable system  
is shown to be geometrically ergodic. We also briefly discuss how the  
stationary mean number of customers behaves in light and heavy traffic.

http://arxiv.org/abs/0908.4585


9330. Strict positivity of the density for non-linear spatially  
homogeneous SPDEs
Author(s): Eulalia Nualart

Abstract: In this paper, we consider a system of $k$ second order non- 
linear stochastic differential equations with spatial dimension $d  
\geq 1$, driven by a $k$-dimensional Gaussian noise, which is white in  
time and with some spatially homogeneous covariance. We prove  
existence, smoothness, and strict positivity of the density of the law  
of the solution of this system of equations, on the set where the  
diffusion matrix is invertible, under sufficient conditions on the  
fundamental solution $\Gamma$ of the deterministic equation. For this,  
we apply techniques of Malliavin calculus. We apply this result to the  
case of the stochastic heat equation in any space dimension and the  
the stochastic wave equation in dimension $d\in \{1,2,3\}$, with a  
spatial covariance given by a Riesz kernel. We then study the strict  
positivity of the density for the case of a single equation ($k=1$),  
and apply it to the stochastic heat equation in any space dimension,  
and the stochastic wave equation in dimension $d\in \{1,2,3\}$, with a  
general spatial covariance.

http://arxiv.org/abs/0908.4587


9331. On the spectral dimension of causal triangulations
Author(s): Bergfinnur Durhuus and Thordur Jonsson and John F. Wheater

Abstract: We introduce an ensemble of infinite causal triangulations,  
called the uniform infinite causal triangulation, and show that it is  
equivalent to an ensemble of infinite trees, the uniform infinite  
planar tree. It is proved that in both cases the Hausdorff dimension  
almost surely equals 2. The infinite causal triangulations are shown  
to be almost surely recurrent or, equivalently, their spectral  
dimension is almost surely less than or equal to 2. We also establish  
that for certain reduced versions of the infinite causal  
triangulations the spectral dimension equals 2 both for the ensemble  
average and almost surely. The triangulation ensemble we consider is  
equivalent to the causal dynamical triangulation model of two- 
dimensional quantum gravity and therefore our results apply to that  
model.

http://arxiv.org/abs/0908.3643


9332. Stochastic Cahn-Hilliard equation with double singular  
nonlinearities and two reflections
Author(s): Arnaud Debussche (IRMAR) and Ludovic Gouden\`ege (IRMAR)

Abstract: We consider a stochastic partial differential equation with  
two logarithmic nonlinearities, with two reflections at 1 and -1 and  
with a constraint of conservation of the space average. The equation,  
driven by the derivative in space of a space-time white noise,  
contains a bi-Laplacian in the drift. The lack of the maximum  
principle for the bi-Laplacian generates difficulties for the  
classical penalization method, which uses a crucial monotonicity  
property. Being inspired by the works of Debussche, Gouden\`ege and  
Zambotti, we obtain existence and uniqueness of solution for initial  
conditions in the interval $(-1,1)$. Finally, we prove that the unique  
invariant measure is ergodic, and we give a result of exponential  
mixing.

http://arxiv.org/abs/0908.4295




-----------------------------------
Stefano M. Iacus
Department of Economics,
Business and Statistics
University of Milan
Via Conservatorio, 7
I-20123 Milan - Italy
Ph.: +39 02 50321 461
Fax: +39 02 50321 505
http://www.economia.unimi.it/iacus
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