[PAS] Probability Abstracts 107
Probability Abstract Service
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Sun Jan 4 01:26:14 CST 2009
Probability Abstracts 107
This document contains abstracts 7696-7953
from November-1-2008 to December-31-2008.
They have been mailed on Jan 4, 2009.
This letter can be also found on line at
http://pas.imstat.org/Letters/letter_107.shtml
Wishing you all a great 2009!
stefano
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7696. LARGE GAPS BETWEEN RANDOM EIGENVALUES
Benedek Valk\'o and B\'alint Vir\'ag
We show that in the point process limit of the bulk eigenvalues of
$\beta$-ensembles of random matrices, the probability of having no
eigenvalue
in a fixed interval of size $\lambda$ is given by $$
(\kappa_\beta+o(1))\lambda^{\gamma_\beta} \exp(-
\frac{\beta}{64}\lambda^2+(\frac{\beta}{4}-\frac18)\lambda) $$ as
$\lambda\to\infty$, where $$ \gamma_\beta={1/4}(\frac\beta{2}+\frac{2}
{\beta}-
3). $$ This is a slightly corrected version of a prediction by Dyson.
Our proof
uses the new Brownian carousel representation of the limit process, as
well as
the Cameron-Martin-Girsanov transformation in stochastic calculus.
http://arxiv.org/abs/0811.0007
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7697. A CRITERION FOR THE VIABILITY OF STOCHASTIC SEMILINEAR CONTROL
SYSTEMS VIA THE QUASI-TANGENCY CONDITION
Dan Goreac
In this paper we study a criterion for the viability of stochastic
semilinear
control systems on a real, separable Hilbert space. The necessary and
sufficient condition is given using the notion of stochastic quasi-
tangency. As
a consequence, we prove that approximate viability and the viability
property
coincide for stochastic linear control systems. The paper generalizes
recent
results from the deterministic framework.
http://arxiv.org/abs/0811.0098
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7698. COMPETITIVE OR WEAK COOPERATIVE STOCHASTIC LOTKA-VOLTERRA
SYSTEMS CONDITIONED TO NON-EXTINCTION
Patrick Cattiaux (IMT) and Sylvie M\'el\'eard (CMAP)
We are interested in the long time behavior of a two-type density-
dependent
biological population conditioned to non-extinction, in both cases of
competition or weak cooperation between the two species. This
population is
described by a stochastic Lotka-Volterra system, obtained as limit of
renormalized interacting birth and death processes. The weak cooperation
assumption allows the system not to blow up. We study the existence and
uniqueness of a quasi-stationary distribution, that is convergence to
equilibrium conditioned to non extinction. To this aim we generalize in
two-dimensions spectral tools developed for one-dimensional
generalized Feller
diffusion processes. The existence proof of a quasi-stationary
distribution is
reduced to the one for a $d$-dimensional Kolmogorov diffusion process
under a
symmetry assumption. The symmetry we need is satisfied under a local
balance
condition relying the ecological rates. A novelty is the outlined
relation
between the uniqueness of the quasi-stationary distribution and the
ultracontractivity of the killed semi-group. By a comparison between the
killing rates for the populations of each type and the one of the global
population, we show that the quasi-stationary distribution can be either
supported by individuals of one (the strongest one) type or supported by
individuals of the two types. We thus highlight two different long time
behaviors depending on the parameters of the model: either the model
exhibits
an intermediary time scale for which only one type (the dominant
trait) is
surviving, or there is a positive probability to have coexistence of
the two
species.
http://arxiv.org/abs/0811.0240
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7699. ASYMPTOTICS FOR THE SURVIVAL PROBABILITY IN A SUPERCRITICAL
BRANCHING RANDOM WALK
Nina Gantert and Yueyun Hu and Zhan Shi
Consider a discrete-time one-dimensional supercritical branching
random walk.
We study the probability that there exists an infinite ray in the
branching
random walk that always lies above the line of slope $\gamma-\epsilon
$, where
$\gamma$ denotes the asymptotic speed of the right-most position in the
branching random walk. Under mild general assumptions upon the
distribution of
the branching random walk, we prove that when $\epsilon\to 0$, the
probability
in question decays like $\exp\{- {\beta + o(1)\over \epsilon^{1/2}}\}
$, where
$\beta$ is a positive constant depending on the distribution of the
branching
random walk. In the special case of i.i.d. Bernoulli$(p)$ random
variables
(with $0<p<{1\over 2}$) assigned on a rooted binary tree, this answers
an open
question of Robin Pemantle.
http://arxiv.org/abs/0811.0262
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7700. DISPERSION OF VOLUME UNDER THE ACTION OF ISOTROPIC BROWNIAN FLOWS
Georgi Dimitroff and Michael Scheutzow
We study transport properties of isotropic Brownian flows. Under a
transience
condition for the two-point motion, we show asymptotic normality of
the image
of a finite measure under the flow and -- under slightly stronger
assumptions
-- asymptotic normality of the distribution of the volume of the image
of a set
under the flow. Finally, we show that for a class of isotropic flows,
the
volume of the image of a nonempty open set (which is a martingale)
converges to
a random variable which is almost surely strictly positive.
http://arxiv.org/abs/0811.0276
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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
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7702. QUENCHED INVARIANCE PRINCIPLE FOR THE KNUDSEN STOCHASTIC
BILLIARD IN A RANDOM TUBE
Francis Comets and Serguei Popov and Gunter M. Sch\"utz and Marina
Vachkovskaia
We consider a stochastic billiard in a random tube which stretches to
infinity in the direction of the first coordinate. This random tube is
stationary and ergodic, and also it is supposed to be in some sense
well-behaved. The stochastic billiard can be described as follows: when
strictly inside the tube, the particle moves straight with constant
speed. Upon
hitting the boundary, it is reflected randomly, according to the
cosine law:
the density of the outgoing direction is proportional to the cosine of
the
angle between this direction and the normal vector. We also consider the
discrete-time random walk formed by the particle's positions at the
moments of
hitting the boundary. Under the condition of existence of the second
moment of
the projected jump length with respect to the stationary measure for the
environment seen from the particle, we prove the quenched invariance
principles
for the projected trajectories of the random walk and the stochastic
billiard.
http://arxiv.org/abs/0811.0366
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7703. BIASED TUG-OF-WAR, THE BIASED INFINITY LAPLACIAN, AND COMPARISON
WITH EXPONENTIAL CONES
Yuval Peres and G\'abor Pete and Stephanie Somersille
We prove that if X\subset \R^n is compact in the path metric, Y\subset
X is
closed, and F is a Lipschitz function on Y, then for each \beta \in \R
there
exists a unique viscosity solution to the \beta-biased infinity
Laplacian
equation
\beta |\nabla u| + \Delta_\infty u=0 on X\setminus Y, where \Delta_
\infty u=
|\nabla u|^{-2} \sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j} . In the proof,
we extend
the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and
define the
\beta-biased \eps-game as follows. The starting position is x_0 \in X
\backslash
Y. At the k-th step the two players toss a suitably biased coin (in
our key
example, player I wins with odds of \exp(\beta\eps) to 1), and the
winner
chooses x_k with d(x_k,x_{k-1}) < \eps. The game ends when x_k \in Y,
and
player II pays the amount F(x_k) to player I. We prove that the value
u^{\eps}(x_0) of this game exists, and that \|u^\eps - u\|_\infty \to
0 as \eps
\to 0, where u is the unique extension of F to X that satisfies
comparison with
\beta-exponential cones. Comparison with exponential cones is a notion
that we
introduce here, and generalizing a theorem of Crandall, Evans and
Gariepy
regarding comparison with linear cones, we show that a continuous
function
satisfies comparison with \beta-exponential cones if and only if it is a
viscosity solution to the \beta-biased infinity Laplacian equation.
http://arxiv.org/abs/0811.0208
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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
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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
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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
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7707. EVENTOLOGICAL THEORY OF DECISION MAKING FOR STOCK MARKETS
Oleg Yu. Vorobyev and Joe J. Goldblatt and Rebecca Finkel
The eventological theory of decision-making, the theory of eventfull
decision-making is a theory of decision-making based on eventological
principles and using results of mathematical eventology; a theoretical
basis of
the practical eventology. The beginnings of this theory which have
arisen from
eventfull representation of the reasonable subject and his decisions
in the
form of eventological distributions (E-distributions) of sets of
events and
which are based on the eventological H-theorem are offered. The
illustrative
example of the eventological decision-making by the reasonable subject
on his
own eventfull behaviour in the financial or share market is considered.
http://arxiv.org/abs/0811.0420
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7708. LARGE DEVIATIONS FOR RANDOM WALKS IN RANDOM ENVIRONMENT ON A
GALTON-WATSON TREE
Elie Aidekon (PMA)
Consider a random walk in random environment on a supercritical
Galton--Watson tree, and let $\tau_n$ be the hitting time of
generation $n$.
The paper presents a large deviation principle for $\tau_n/n$, both in
quenched
and annealed cases. Then we investigate the subexponential situation,
revealing
a polynomial regime similar to the one encountered in one dimension.
The paper
heavily relies on estimates on the tail distribution of the first
regeneration
time.
http://arxiv.org/abs/0811.0438
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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
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7710. GAUSSIAN CORRELATION CONJECTURE FOR SYMMETRIC CONVEX SETS
He-Jing Hong and Ze-Chun Hu
Gaussian correlation conjecture states that the Gaussian measure of the
intersection of two symmetric convex sets is greater or equal to the
product of
the measures. In this paper, firstly we prove that the inequality
holds when
one of the two convex sets is the intersection of finite centered
ellipsoids
and the other one is simply symmetric. Then we prove that any
symmetric convex
set can be approximated by the intersection of finite centered
ellipsoids, and
thus the inequality holds for any two symmetric convex sets in any
dimensional
$\mathbb{R}^n$, i.e. Gaussian correlation conjecture is true.
http://arxiv.org/abs/0811.0488
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7711. FIRST HITTING TIME OF THE BOUNDARY OF THE WEYL CHAMBER BY RADIAL
DUNKL PROCESSES
Nizar Demni
We provide two equivalent approaches for computing the tail
distribution of
the first hitting time of the boundary of the Weyl chamber by a radial
Dunkl
process. The first approach is based on a spectral problem with
initial value.
The second one expresses the tail distribution by means of the $W$-
invariant
Dunkl-Hermite polynomials. Illustrative examples are given by the
irreducible
root systems of types $A$, $B$, $D$. The paper ends with an interest
in the
case of Brownian motions for which our formulae take determinantal
forms.
http://arxiv.org/abs/0811.0504
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7712. GENERALIZED BESSEL FUNCTION OF TYPE D
Nizar Demni
We write down the generalized Bessel function associated with the root
system
of type $D$ by means of multivariate hypergeometric series. Our hint
comes from
the particular case of the Brownian motion in the Weyl chamber of type
$D$.
http://arxiv.org/abs/0811.0507
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7713. DIFFERENCES OF RANDOM CANTOR SETS AND LOWER SPECTRAL RADII
F. Michel Dekking and Bram Kuijvenhoven
We investigate the question under which conditions the algebraic
difference
between two independent random Cantor sets $C_1$ and $C_2$ almost surely
contains an interval, and when not. The natural condition is whether
the sum
$d_1+d_2$ of the Hausdorff dimensions of the sets is smaller (no
interval) or
larger (an interval) than 1. Palis conjectured that \emph{generically}
it
should be true that $d_1+d_2>1$ should imply that $C_1-C_2$ contains an
interval. We prove that for 2-adic random Cantor sets generated by a
vector of
probabilities $(p_0,p_1)$ the interior of the region where the Palis
conjecture
does not hold is given by those $p_0,p_1$ which satisfy $p_0+p_1>
\sqrt{2}$ and
$p_0p_1(1+p_0^2+p_1^2)<1$. We furthermore prove a general result which
characterizes the interval/no interval property in terms of the lower
spectral
radius of a set of $2\times 2$ matrices.
http://arxiv.org/abs/0811.0525
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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
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7715. ON THE DYNAMICS OF SOCIAL BALANCE ON GENERAL NETWORKS (WITH AN
APPLICATION TO XOR-SAT)
Gabriel Istrate
We study nondeterministic and probabilistic versions of a discrete
dynamical
system (due to T. Antal, P. L. Krapivsky, and S. Redner) inspired by
Heider's
social balance theory. We investigate the convergence time of this
dynamics on
several classes of graphs. Our contributions include:
1. We point out the connection between the triad dynamics and a
generalization of annihilating walks to hypergraphs. In particular, this
connection allows us to completely characterize the recurrent states
in graphs
where each edge belongs to at most two triangles.
2. We also solve the case of hypergraphs that do not contain edges
consisting
of one or two vertices.
3. We show that on the so-called "triadic cycle" graph, the
convergence time
is linear.
4. We obtain a cubic upper bound on the convergence time on 2-
regular triadic
simplexes G. This bound can be further improved to a quantity that
depends on
the Cheeger constant of G. In particular this provides some rigorous
counterparts to previous experimental observations.
We also point out an application to the analysis of the random walk
algorithm
on certain instances of the 3-XOR-SAT problem.
http://arxiv.org/abs/0811.0381
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7716. STOCHASTIC CAHN-HILLIARD EQUATION WITH SINGULAR NONLINEARITY
AND REFLECTION
Ludovic Gouden\`ege (IRMAR)
We consider a stochastic partial differential equation with
logarithmic (or
negative power) nonlinearity, with one reflection at 0 and with a
constraint of
conservation of the space average. The equation, driven by the
derivative in
space of a space-time white noise, contains a bi-Laplacian in the
drift. The
lack of the maximum principle for the bi-Laplacian generates
difficulties for
the classical penalization method, which uses a crucial monotonicity
property.
Being inspired by the works of Debussche and Zambotti, we use a method
based on
infinite dimensional equations, approximation by regular equations and
convergence of the approximated semi-group. We obtain existence and
uniqueness
of solution for nonnegative intial conditions, results on the invariant
measures, and on the reflection measures.
http://arxiv.org/abs/0811.0580
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7717. ASYMPTOTIC ANALYSIS AND DIFFUSION LIMIT OF THE PERSISTENT
TURNING WALKER MODEL
Patrick Cattiaux (IMT) and Djalil Chafai (IMT and UPTE) and S
\'ebastien Motsch (IMT)
The Persistent Turning Walker Model (PTWM) was introduced by Gautrais
et al
in Mathematical Biology for the modelling of fish motion. It involves a
nonlinear pathwise functional of a non-elliptic hypo-elliptic
diffusion. This
diffusion solves a kinetic Fokker-Planck equation based on an
Ornstein-Uhlenbeck Gaussian process. The long time "diffusive"
behavior of this
model was recently studied by Degond & Motsch using partial differential
equations techniques. This model is however intrinsically
probabilistic. In the
present paper, we show how the long time diffusive behavior of this
model can
be essentially recovered and extended by using appropriate tools from
stochastic analysis. The approach can be adapted to many other kinetic
"probabilistic" models. Beyond the mathematical results, the aim of
this short
paper is also to contribute to the diffusion of stochastic techniques
in the
domain of partial differential equations. Also, the text aims to be very
accessible for non probabilists.
http://arxiv.org/abs/0811.0600
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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
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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
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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
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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
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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
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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
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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
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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
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