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


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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:
http://www.gnu.org/philosophy/no-word-attachments.html



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
------------------------------------------------------------------------------------
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  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 valu