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Probability Abstracts 113
This document contains abstracts 9333-9659
from Nov-1-2009 to Dec-31-2009.
They have been mailed on Jan 1st, 2010.
Author(s): Bruno Jaffuel (PMA)
Abstract: We study a branching random walk on $\r$ with an absorbing barrier. The
position of the barrier depends on the generation. In each generation, only the
individuals born below the barrier survive and reproduce. Given a reproduction
law, Biggins et al. \cite{BLSW91} determined whether a linear barrier allows
the process to survive. In this paper, we refine their result: in the boundary
case in which the speed of the barrier matches the speed of the minimal
position of a particle in a given generation, we add a second order term $a
n^{1/3}$ to the position of the barrier for the $n^\mathrm{th}$ generation and
find an explicit critical value $a_c$ such that the process dies when $aa_c$.
We also obtain the rate of extinction when $a < a_c$ and a lower bound on the
surviving population when $a > a_c$.
http://arxiv.org/abs/0911.2227
Author(s): Paul Lescot (LMRS)
Abstract: We give an exposition, following joint works with J.-C. Zambrini, of the link
between Euclidean Quantum Mechanics, Bernstein processes and isovectors for the
heat equation. A new application to Mathematical Finance is then discussed.
http://arxiv.org/abs/0911.2229
Author(s): Tiefeng Jiang
Abstract: For a beta-Jacobi ensemble determined by parameters a_1, a_2 and n, under the
restriction that the three parameters go to infinity with n and a_1 being of
small orders of a_2, we obtain both the bulk and the edge scaling limits. In
particular, we derive the asymptotic distributions for the largest and the
smallest eigenvalues, the Central Limit Theorems of the eigenvalues, and the
limiting distributions of the empirical distributions of the eigenvalues.
http://arxiv.org/abs/0911.2262
Author(s): Pierre Tarres and Balint Toth and Benedek Valko
Abstract: We study the asymptotic behavior of a self interacting one dimensional
Brownian polymer first introduced by Durrett and Rogers. The polymer describes
a stochastic process with a drift which is a certain average of its local time.
We show that a smeared out version of the local time function as viewed from
the actual position of the process is a Markov process in a suitably chosen
function space, and that this process has a Gaussian stationary measure. As a
first consequence this enables us to partially prove a conjecture about the law
of large numbers for the end-to-end displacement of the polymer formulated by
Durrett and Rogers.
Next we give upper and lower bounds for the variance of the process under the
stationary measure, in terms of the qualitative infrared behavior of the
interaction function. In particular we show that in the locally self-repelling
case (when the process is essentially pushed by the negative gradient of its
own local time) the process is super-diffusive.
http://arxiv.org/abs/0911.2356
Author(s): L. Avena and F. den Hollander and F. Redig
Abstract: In this paper we consider a class of one-dimensional interacting particle
systems in equilibrium, constituting a dynamic random environment, together
with a nearest-neighbor random walk that on occupied/vacant sites has a local
drift to the right/left. We adapt a regeneration-time argument originally
developed by Comets and Zeitouni for static random environments to prove that,
under a space-time mixing property for the dynamic random environment called
cone-mixing, the random walk has an a.s. constant global speed. In addition, we
show that if the dynamic random environment is exponentially mixing in
space-time and the local drifts are small, then the global speed can be written
as a power series in the size of the local drifts. From the first term in this
series the sign of the global speed can be read off. The results can be easily
extended to higher dimensions.
http://arxiv.org/abs/0911.2385
Author(s): Xinghua Zheng
Abstract: We study critical branching random walks (BRWs) $U^{(n)}$ on $\zz{Z}_{+}$
where for each $n$, the displacement of an offspring from its parent has drift
$2\beta/\sqrt{n}$ towards the origin and reflection at the origin. We prove
that conditional on survival to generation $n^{\alpha}$, the maximal
displacement is $O_p(\sqrt{n^\alpha})$ if $\alpha \leq 1$ and is asymptotically
equivalent to $(\alpha-1)/(4\beta)\cdot \sqrt{n}\log n$ if $\alpha>1$. We
further show that for a sequence of critical BRWs with such displacement
distributions, if the initial mass distributions converge, then the
measure-valued processes associated with the BRWs converge to a limiting
measure-valued process which may \emph{not} be a Dawson-Watanabe process.
http://arxiv.org/abs/0911.2401
Author(s): Z. Brzezniak and B. Goldys and P. Imkeller and S. Peszat and E. Priola and J. Zabczyk
Abstract: The paper is concerned with the properties of solutions to linear evolution
equation perturbed by cylindrical L\'evy processes. It turns out that
solutions, under rather weak requirements, do not have c\`adl\`ag modification.
Some natural open questions are also stated.
http://arxiv.org/abs/0911.2418
Author(s): Young Myoung Ko and Natarajan Gautam
Abstract: In this paper, we consider modeling time-dependent multi-server queues that
include abandonments and retrials. For the performance analysis of those, fluid
and diffusion models called "strong approximations" have been widely used in
the literature. Although they are proven to be asymptotically exact, their
effectiveness as approximations in critically loaded regimes needs to be
investigated. To that end, we find that existing fluid and diffusion
approximations might be either inaccurate under simplifying assumptions or
computationally intractable. To address that concern, this paper focuses on
developing a methodology by adjusting the fluid and diffusion models so that
they significantly improve the estimation accuracy. We illustrate the accuracy
of our adjusted models by performing a number of numerical experiments.
http://arxiv.org/abs/0911.2436
Author(s): Timo Sepp\"al\"ainen
Abstract: We study a 1+1-dimensional directed polymer in a random environment on the
integer lattice with log-gamma distributed weights and both endpoints of the
path fixed. Among directed polymers this model is special in the same way as
the last-passage percolation model with exponential or geometric weights is
special among growth models, namely, both permit explicit calculations. With
appropriate boundary conditions the polymer with log-gamma weights satisfies an
analogue of Burke's theorem for queues. Building on this we prove that the
fluctuation exponents for the free energy and the polymer path have their
conjectured values. For the model without boundary conditions we get upper
bounds on the exponents.
http://arxiv.org/abs/0911.2446
Author(s): Sigurdur Orn Stefansson
Abstract: We prove the existence of a limit of the finite volume probability measures
generated by tree growth rules in Ford's alpha model of phylogenetic trees. The
limiting measure is shown to be concentrated on the set of trees consisting of
exactly one infinite spine with finite, identically and independently
distributed outgrowths.
http://arxiv.org/abs/0911.2140
Author(s): Gautam Iyer and Alexei Novikov and Lenya Ryzhik and Andrej Zlatos
Abstract: Let $\Omega\subset\mathbb R^n$ be a bounded domain and for $x\in\Omega$ let
$\tau(x)$ be the expected exit time from $\Omega$ of a diffusing particle
starting at $x$ and advected by an incompressible flow $u$. We are interested
in the question which flows maximize $\|\tau\|_{L^\infty(\Omega)}$, that is,
they are most efficient in the creation of hotspots inside $\Omega$.
Surprisingly, among all simply connected domains in two dimensions, the discs
are the only ones for which the zero flow $u\equiv 0$ maximises
$\|\tau\|_{L^\infty(\Omega)}$. We also show that in any dimension, among all
domains with a fixed volume and all incompressible flows on them,
$\|\tau\|_{L^\infty(\Omega)}$ is maximized by the zero flow on the ball.
http://arxiv.org/abs/0911.2294
Author(s): M.E. Caballero and J.C. Pardo and J.L. Perez
Abstract: In this paper we introduce a new class of L\'evy processes which we call
hypergeometric-stable L\'evy processes, because they are obtained from
symmetric stable processes through several transformations and where the Gauss
hypergeometric function plays an essential role. We characterize the L\'evy
measure of this class and obtain several useful properties such as the Wiener
Hopf factorization, the characteristic exponent and some associated exit
problems.
http://arxiv.org/abs/0911.0712
Author(s): Mikael Falconnet (IF)
Abstract: We show that the Bayesian star paradox, first proved mathematically by Steel
and Matsen for a specific class of prior distribution, occurs in a wider
context.
http://arxiv.org/abs/0911.0733
Author(s): Hui He
Abstract: We consider a new type of lookdown processes where spatial motion of each
individual is influenced by an individual noise and a common noise, which could
be regarded as an environment. Then a class of probability measure-valued
processes on real line $\mbb{R}$ are constructed.
The sample path properties are investigated: the values of this new type
process are either purely atomic measures or absolutely continuous measures
according to the existence of individual noise. When the process is absolutely
continuous with respect to Lebesgue measure, we derive a new stochastic partial
differential equation for the density process. At last we show that such
processes also arise from normalizing a class of measure-valued branching
diffusions in a Brownian medium as the classical result that Dawson-Watanabe
superprocesses, conditioned to have total mass one, are Fleming-Viot
superprocesses.
http://arxiv.org/abs/0911.0773
Author(s): Hui He and Zenghu Li and Xiaowen Zhou
Abstract: This paper concerns the almost sure time dependent local extinction behavior
for super-coalescing Brownian motion $X$ with $(1+\beta)$-stable branching and
Lebesgue initial measure on $\bR$. We first give a representation of $X$ using
excursions of a continuous state branching process and Arratia's coalescing
Brownian flow. For any nonnegative, nondecreasing and right continuous function
$g$, put
\tau:=\sup \{t\geq 0: X_t([-g(t),g(t)])>0 \}. We prove that
$\bP\{\tau=\infty\}=0$ or 1 according as the integral $\int_1^\infty
g(t)t^{-1-1/\beta} dt$ is finite or infinite.
http://arxiv.org/abs/0911.0774
Author(s): Piotr Milos
Abstract: In this paper we consider two related stochastic models. The first one is a
branching system consisting of particles moving according to a Markov family in
R^d and undergoing subcritical branching with a constant rate of V>0. New
particles immigrate to the system according to a homogeneous space time Poisson
random field. The second model is the superprocess corresponding to the
branching particle system. We study rescaled occupation time process and the
process of its fluctuations with very mild assumptions on the Markov family. In
the general setting a functional central limit theorem as well as large and
moderate deviations principles are proved. The subcriticality of the branching
law determines the behaviour in large time scales and in "overwhelms" the
properties of the particles' motion. For this reason the results are the same
for all dimensions and can be obtained for a wide class of Markov processes
(both properties are unusual for systems with critical branching).
http://arxiv.org/abs/0911.0777
Author(s): Gady Kozma and Asaf Nachmias
Abstract: We study the probability that the origin is connected to the sphere of radius
r (an arm event) in critical percolation in high dimensions, namely when the
dimension d is large enough or when d>6 and the lattice is sufficiently spread
out. We prove that this probability decays like 1/r^2. Furthermore, we show
that the probability of having k disjoint arms to distance r emanating from the
vicinity of the origin is 1/r^2k.
http://arxiv.org/abs/0911.0871
Author(s): Elie Aidekon
Abstract: We consider a branching random walk on $\mathbb{R}$ with a killing barrier at
zero. At criticality, the process becomes eventually extinct, and the total
progeny $Z$ is therefore finite. We show that the tail distribution of $Z$
displays a typical behaviour in $(n\ln^2(n))^{-1}$, which confirms the
prediction of Addario-Berry and Broutin.
http://arxiv.org/abs/0911.0877
Author(s): Jason Schweinsberg
Abstract: Suppose $\Pi$ is an exchangeable random partition of the positive integers
and $\Pi_n$ is its restriction to $\{1, ..., n\}$. Let $K_n$ denote the number
of blocks of $\Pi_n$, and let $K_{n,r}$ denote the number of blocks of $\Pi_n$
containing $r$ integers. We show that if $0 < \alpha < 1$ and $K_n/(n^{\alpha}
\ell(n))$ converges in probability to $\Gamma(1-\alpha)$, where $\ell$ is a
slowly varying function, then $K_{n,r}/(n^{\alpha} \ell(n))$ converges in
probability to $\alpha \Gamma(r - \alpha)/r!$. This result was previously known
when the convergence of $K_n/(n^{\alpha} \ell(n))$ holds almost surely, but the
result under the hypothesis of convergence in probability has significant
implications for coalescent theory. We also show that a related conjecture for
the case when $K_n$ grows only slightly slower than $n$ fails to be true.
http://arxiv.org/abs/0911.1793
Author(s): Ivan Matic
Abstract: A deterministic walk in a random environment can be understood as a general
finite-range dependent random walk that starts repeating the loop once it
reaches a site it has visited before. Such process lacks the Markov property.
We study the exponential decay of the probabilities that the walk will reach
sites located far away from the origin.
http://arxiv.org/abs/0911.1809
Author(s): Vincent Bansaye (CMAP) and Jean-Fran\c{c}ois Delmas (CERMICS) and Laurence Marsalle (LPP), Viet Chi Tran (LPP)
Abstract: We study the evolution of a particle system whose genealogy is given by a
supercritical continuous time Galton-Watson tree. The particles move
independently according to a Markov process and when a branching event occurs,
the offspring locations depend on the position of the mother and the number of
offspring. We prove a law of large numbers for the empirical measure of
individuals alive at time $t$. This relies on a probabilistic interpretation of
its intensity by mean of an auxiliary process. This latter has the same
generator as the Markov process along the branches plus additional branching
events, associated with jumps of accelerated rate and biased distribution. This
comes from the fact that choosing an individual uniformly at time $t$ favors
lineages with more branching events and larger offspring number. The central
limit theorem is considered on a special case. Several examples are developed,
including applications to splitting diffusions, cellular aging, branching
L\'evy processes and ancestral lineages.
http://arxiv.org/abs/0911.1973
Author(s): Alexei Borodin and Maurice Duits
Abstract: We study a Markov process on a system of interlacing particles. At large
times the particles fill a domain that depends on a parameter $\eps> 0$. The
domain has two cusps, one pointing up and one pointing down. In the limit
$\varepsilon\downarrow 0$ the cusps touch, thus forming a tacnode. The main
result of the paper is a derivation of the local correlation kernel around the
tacnode in the transition regime $\varepsilon \downarrow 0$. We also prove that
the local process interpolates between the Pearcey process and the GUE minor
process.
http://arxiv.org/abs/0911.1980
Author(s): Francois Baccelli and Bartlomiej Blaszczyszyn and Mir Omid Haji Mirsadeghi
Abstract: We analyze a class of Signal-to-Interference-and-Noise-Ratio (SINR) random
graphs. These random graphs arise in the modeling packet transmissions in
wireless networks. In contrast to previous studies on the SINR graphs, we
consider both a space and a time dimension. The spatial aspect originates from
the random locations of the network nodes in the Euclidean plane. The time
aspect stems from the random transmission policy followed by each network node
and from the time variations of the wireless channel characteristics. The
combination of these random space and time aspects leads to fluctuations of the
SINR experienced by the wireless channels, which in turn determine the
progression of packets in space and time in such a network. This paper studies
optimal paths in such wireless networks in terms of first passage percolation
on this random graph. We establish both "positive" and "negative" results on
the associated time constant. The latter determines the asymptotics of the
minimum delay required by a packet to progress from a source node to a
destination node when the Euclidean distance between the two tends to infinity.
The main negative result states that this time constant is infinite on the
random graph associated with a Poisson point process under natural assumptions
on the wireless channels. The main positive result states that when adding a
periodic node infrastructure of arbitrarily small intensity to the Poisson
point process, the time constant is positive and finite.
http://arxiv.org/abs/0911.3721
Author(s): Jean B\'erard (ICJ) and Jean-Baptiste Gou\'er\'e (MAPMO)
Abstract: We give an alternative proof of a result by N. Gantert, Y. Hu and Z. Shi on
the asymptotic behavior of the survival probability of the branching random
walk killed below a linear boundary.
http://arxiv.org/abs/0911.3755
Author(s): Wouter Kager and Haiyan Liu and Ronald Meester
Abstract: Let \Lambda be a finite subset of Z^d. We study the following sandpile model
on \Lambda. The height at any given vertex x of \Lambda is a positive real
number, and additions are uniformly distributed on some interval [a,b], which
is a subset of [0,1]. The threshold value is 1; when the height at a given
vertex exceeds 1, it topples, that is, its height is reduced by 1, and the
heights of all its neighbours in \Lambda increase by 1/2d. We first establish
that the uniform measure \mu on the so called "allowed configurations" is
invariant under the dynamics. When a < b, we show with coupling ideas that
starting from any initial configuration of heights, the process converges in
distribution to \mu, which therefore is the unique invariant measure for the
process. When a = b, that is, when the addition amount is non-random, and a is
rational, it is still the case that \mu is the unique invariant probability
measure, but in this case we use random ergodic theory to prove this; this
proof proceeds in a very different way. Indeed, the coupling approach cannot
work in this case since we also show the somewhat surprising fact that when a =
b is rational, the process does not converge in distribution at all starting
from any initial configuration.
http://arxiv.org/abs/0911.3782
Author(s): Hubert Lacoin
Abstract: Cette these est consacree a l' etude de differents modeles aleatoires de
polymeres. On s'interesse en particulier a l'influence du desordre sur la
localisation des trajectoires pour les modeles d'accrochage et pour les
polymeres diriges en milieu aleatoire. En plus des modeles classiques dans Zd,
nous abordons l' etude de modeles dit hierarchiques, construits sur une suite
de reseaux auto-similaires, tres present dans la litterature physique. Les
resultats que nous avons obtenus concernent principalement l' energie libre et
le phenomene de surdiffusivite. En particulier, nous prouvons: (1) la
pertinence du desordre a toute temperature dans pour le modele d' accrochage
desordonne en dimension 1 + 1, (2) l' occurence d' un desordre tres fort a
toute temperature en dimension 1 + 2 pour les polymeres diriges en milieu
aleatoire.
This thesis studies models of random directed polymers. We focus on the
influence of disorder on localization of the trajectories for pinning model and
directed polymers in random environment. In addition to the classical Zd
models, we pay a particular attention to so-called hierachical models, buildt
on a sequence of self-similar lattices, that are frequently studied in the
physics literature. The results we obtain concern mainly free energy and
superdiffusivity properties. In particular we present the proof that: (1)
disorder is relevant at arbitrary high temperature for pinning models in
dimension 1 + 1, (2) very strong disorder holds at all temperature in dimension
1 + 2 for directed polymers in random environment.
http://arxiv.org/abs/0911.3824
Author(s): Harry Kesten and Olivier Raimond (MODAL'X) and Bruno Schapira (LM-Orsay)
Abstract: We study recurrence properties and the validity of the (weak) law of large
numbers for (discrete time) processes which, in the simplest case, are obtained
from simple symmetric random walk on $\Z$ by modifying the distribution of a
step from a fresh point. If the process is denoted as $\{S_n\}_{n \ge 0}$, then
the conditional distribution of $S_{n+1} - S_n$ given the past through time $n$
is the distribution of a simple random walk step, provided $S_n$ is at a point
which has been visited already at least once during $[0,n-1]$. Thus in this
case $P\{S_{n+1}-S_n = \pm 1|S_\ell, \ell \le n\} = 1/2$. We denote this
distribution by $P_1$. However, if $S_n$ is at a point which has not been
visited before time $n$, then we take for the conditional distribution of
$S_{n+1}-S_n$, given the past, some other distribution $P_2$. We want to decide
in specific cases whether $S_n$ returns infinitely often to the origin and
whether $(1/n)S_n \to 0$ in probability. Generalizations or variants of the
$P_i$ and the rules for switching between the $P_i$ are also considered.
http://arxiv.org/abs/0911.3886
Author(s): Thilo Meyer-Brandis and Xunyu Zhou and Bernt Oksendal
Abstract: This paper considers a controlled It\^o-L\'evy process where the information
available to the controller is possibly less than the overall information. All
the system coefficients and the objective performance functional are allowed to
be random, possibly non-Markovian. Malliavin calculus is employed to derive a
maximum principle for the optimal control of such a system where the adjoint
process is explicitly expressed.
http://arxiv.org/abs/0911.3720
Author(s): Jan van Neerven
Abstract: We present a survey of the theory of $\gamma$-radonifying operators and their
applications to stochastic integration in Banach spaces.
http://arxiv.org/abs/0911.3788
Author(s): Michael Belfrage and Torsten M\"utze and Reto Sp\"ohel
Abstract: Consider the following probabilistic one-player game: The board is a graph
with $n$ vertices, which initially contains no edges. In each step, a new edge
is drawn uniformly at random from all non-edges and is presented to the player,
henceforth called Painter. Painter must assign one of $r$ available colors to
each edge immediately, where $r \geq 2$ is a fixed integer. The game is over as
soon as a monochromatic copy of some fixed graph $F$ has been created, and
Painter's goal is to 'survive' for as many steps as possible before this
happens.
We present a new technique for deriving upper bounds on the threshold of this
game, i.e., on the typical number of steps Painter will survive with an optimal
strategy. More specifically, we consider a deterministic two-player variant of
the game where the edges are not chosen randomly, but by a second player
Builder. However, Builder has to adhere to the restriction that, for some real
number $d$, the ratio of edges to vertices in all subgraphs of the evolving
board never exceeds $d$. We show that the existence of a winning strategy for
Builder in this deterministic game implies an upper bound of $n^{2-1/d}$ for
the threshold of the original probabilistic game. Moreover, we show that the
best bound that can be derived in this way is indeed the threshold of the game
if $F$ is a forest. We illustrate our technique with several examples, and
derive new explicit bounds for the case when $F$ is a path.
http://arxiv.org/abs/0911.3810
Author(s): Guus Balkema and Natalia Nolde
Abstract: Asymptotic independence of the components of random vectors is a concept used
in many applications. The standard criteria for checking asymptotic
independence are given in terms of distribution functions (dfs). Dfs are rarely
available in an explicit form, especially in the multivariate case. Often we
are given the form of the density or, via the shape of the data clouds, one can
obtain a good geometric image of the asymptotic shape of the level sets of the
density. This paper establishes a simple sufficient condition for asymptotic
independence for light-tailed densities in terms of this asymptotic shape. This
condition extends Sibuya's classic result on asymptotic independence for
Gaussian densities.
http://arxiv.org/abs/0912.4331
Author(s): Pierre Calka (MAP5) and Tomasz Schreiber and J. E. Yukich
Abstract: The paper of Schreiber and Yukich [40] establishes an asymptotic
representation for random convex polytope geometry in the unit ball $\B_d, d
\geq 2,$ in terms of the general theory of stabilizing functionals of Poisson
point processes as well as in terms of the so-called generalized paraboloid
growth process. This paper further exploits this connection, introducing also a
dual object termed the paraboloid hull process. Via these growth processes we
establish local functional and measure-level limit theorems for the properly
scaled radius-vector and support functions as well as for curvature measures
and $k$-face empirical measures of convex polytopes generated by high density
Poisson samples. We use general techniques of stabilization theory to establish
Brownian sheet limits for the defect volume and mean width functionals, and we
provide explicit variance asymptotics and central limit theorems for the
$k$-face and intrinsic volume functionals. We establish extreme value theorems
for radius-vector and support functions of random polytopes and we also
establish versions of the afore-mentioned results for large isotropic cells of
hyperplane tessellations, reducing the study of their asymptotic geometry to
that of convex polytopes via inversion-based duality relations, as in Calka and
Schreiber [14].
http://arxiv.org/abs/0912.4339
Author(s): M. Fraas and D. Krejcirik and Y. Pinchover
Abstract: We study strong ratio limit properties of the quotients of the heat kernels
of subcritical and critical operators which are defined on a noncompact
Riemannian manifold.
http://arxiv.org/abs/0912.4337
Author(s): Daniel D\'iaz
Abstract: Let $\Xi\subset\mathbb R^d$ be a set of centers chosen according to a Poisson
point process in $\mathbb R^d$. Let $\psi$ be an allocation of $\mathbb R^d$ to
$\Xi$ in the sense of the Gale-Shapley marriage problem, with the additional
feature that every center $\xi\in\Xi$ has an appetite given by a nonnegative
random variable $\alpha$. Generalizing some previous results, we study large
deviations for the distance of a typical point $x\in\mathbb R^d$ to its center
$\psi(x)\in\Xi$, subject to some restrictions on the moments of $\alpha$.
http://arxiv.org/abs/0911.1429
Author(s): Xing M. Wang
Abstract: After a brief introduction to Probability Bracket Notation (PBN), indicator
operator and conditional density operator (CDO), we investigate probability
spaces associated with various quantum systems: system with one observable
(discrete or continuous), system with two commutative observables (independent
or dependent) and a system of indistinguishable non-interacting many-particles.
In each case, we derive unified expressions of conditional expectation (CE),
conditional probability (CP), and absolute probability (AP): they have the same
format for discrete or continuous spectrum; they are defined in both Hilbert
space (using Dirac notation) and probability space (using PBN); and they may be
useful to deal with CE of non-commutative observables.
http://arxiv.org/abs/0911.1462
Author(s): Hongzhong Zhang and Olympia Hadjiliadis
Abstract: In this work we study drawdowns and drawups of general diffusion processes.
The drawdown process is defined as the current drop of the process from its
running maximum, while the drawup process is defined as the current increase
over its running minimum. The drawdown and the drawup are the first hitting
times of the drawdown and the drawup processes respectively. In particular, we
derive a closed-form formula for the Laplace transform of the probability
density of the drawdown of a units when it precedes the drawup of b units. We
then separately consider the special case of drifted Brownian motion, for which
we derive a closed form formula for the above-mentioned density by inverting
the Laplace transform. Finally, we apply the results to a problem of interest
in financial risk-management and to the problem of transient signal detection
and identification of two-sided changes in the drift of general diffusion
processes.
http://arxiv.org/abs/0911.1575
Author(s): Nathan Keller and Elchanan Mossel and Arnab Sen
Abstract: We present a new definition of influences in product spaces of continuous
distributions. Our definition is geometric, and for monotone sets it is
identical with the measure of the boundary with respect to uniform enlargement.
We prove analogues of the Kahn-Kalai-Linial (KKL) and Talagrand's influence sum
bounds for the new definition. We further prove an analogue of a result of
Friedgut showing that sets with small "influence sum" are essentially
determined by a small number of coordinates. In particular, we establish the
following tight analogue of the KKL bound: for any set in $\mathbb R^n$ of
Gaussian measure $t$, there exists a coordinate $i$ such that the $i$-th
geometric influence of the set is at least $ct(1-t)\sqrt{\log n}/n$, where $c$
is a universal constant. This result is then used to obtain an isoperimetric
inequality for the Gaussian measure on $\mathbb{R}^n$ and the class of sets
invariant under transitive permutation group of the coordinates.
http://arxiv.org/abs/0911.1601
Author(s): Feng-Yu Wang
Abstract: By constructing a coupling with unbounded time-dependent drift,
dimension-free Harnack inequalities are established for the semigroup
associated to SDEs with non-additive noise.
These inequalities are applied to the study of heat kernel upper bound and
contractivity properties of the semigroup. The main results are extended to
non-constant diffusions on manifolds with (non-convex) boundary where the
dimension-free Harnack inequality has been unknown for a long time.
http://arxiv.org/abs/0911.1644
Author(s): Q. Berger and F. Toninelli (ENS Lyon and CNRS)
Abstract: We consider the Random Walk Pinning Model studied in [3,2]: this is a random
walk X on Z^d, whose law is modified by the exponential of \beta times
L_N(X,Y), the collision local time up to time N with the (quenched) trajectory
Y of another d-dimensional random walk. If \beta exceeds a certain critical
value \beta_c, the two walks stick together for typical Y realizations
(localized phase). A natural question is whether the disorder is relevant or
not, that is whether the quenched and annealed systems have the same critical
behavior. Birkner and Sun proved that \beta_c coincides with the critical point
of the annealed Random Walk Pinning Model if the space dimension is d=1 or d=2,
and that it differs from it in dimension d\ge4 (for d\ge 5, the result was
proven also in [2]). Here, we consider the open case of the marginal dimension
d=3, and we prove non-coincidence of the critical points.
http://arxiv.org/abs/0911.1661
Author(s): Olivier Wintenberger (CEREMADE)
Abstract: In this paper we give new deviation inequalities of Bernstein's type for the
partial sums of weakly dependent time series. The loss from the independent
case is studied carefully. We give non mixing examples such that dynamical
systems and Bernoulli shifts for whom our deviation inequalities hold. The
proofs are based on the blocks technique and different coupling arguments.
http://arxiv.org/abs/0911.1682
Author(s): Luigi Accardi and Hiromichi Ohno and Farrukh Mukhamedov
Abstract: We introduce generalized quantum Markov states and generalized d-Markov
chains which extend the notion quantum Markov chains on spin systems to that on
$C^*$-algebras defined by general graphs. As examples of generalized d-Markov
chains, we construct the entangled Markov fields on tree graphs. The concrete
examples of generalized d-Markov chains on Cayley trees are also investigated.
http://arxiv.org/abs/0911.1667
Author(s): Mindaugas P. Bloznelis
Abstract: We consider the typical distance between vertices of the giant component of a
random intersection graph having a power law (asymptotic) vertex degree
distribution with infinite second moment. Given two vertices from the giant
component we construct O(log log n) upper bound (in probability) for the length
of the shortest path connecting them.
http://arxiv.org/abs/0911.5127
Author(s): B.T. Graham
Abstract: Let $\mu$ be the self-avoiding walk connective constant on $\ZZ^d$. We show
that the asymptotic expansion for $\beta_c=1/\mu$ in powers of $1/(2d)$
satisfies Borel type bounds. This supports the conjecture that the expansion is
Borel summable.
http://arxiv.org/abs/0911.5163
Author(s): J. Berestycki and S.C. Harris and A.E. Kyprianou
Abstract: We formulate the notion of the classical
Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) reaction diffusion equation
associated with a homogeneous conservative fragmentation process and study its
travelling waves. Specifically we establish existence, uniqueness and
asymptotics. In the spirit of classical works such as McKean [31, 32], Neveu
[34] and Chauvin [12] our analysis exposes the relation between travelling
waves certain additive and multiplicative martingales via laws of large numbers
which have been previously studied in the context of Crump- Mode-Jagers (CMJ)
processes by Nerman [33] and in the context of fragmentation processes by
Bertoin and Martinez [9] and Harris et al. [17]. The conclusions and
methodology presented here appeal to a number of concepts coming from the
theory of branching random walks and branching Brownian motion showing their
mathematical robustness even within the context of fragmentation theory.
http://arxiv.org/abs/0911.5179
Author(s): Bj\"orn B\"ottcher and Alexander Schnurr
Abstract: We consider the Euler scheme for stochastic differential equations with
jumps, whose intensity might be infinite and the jump structure may depend on
the position, i.e. the driving Poisson random measure may depend on the current
state. This general type of SDE is explicitly given for Feller processes and a
general convergence condition is presented. In particular the characteristic
functions of the increments of the Euler scheme are calculated in terms of the
symbol of the Feller process in a closed form.
http://arxiv.org/abs/0911.5245
Author(s): Laurent Tournier (ICJ)
Abstract: We give a short proof of Theorem 2.1 from [MR07], stating that the linearly
edge reinforced random walk (ERRW) on a locally finite graph is recurrent if
and only if it returns to its starting point almost surely. This result was
proved in [MR07] by means of the much stronger property that the law of the
ERRW is a mixture of Markov chains. Our proof only uses this latter property on
finite graphs, in which case it is a consequence of De Finetti's theorem on
exchangeability.
http://arxiv.org/abs/0911.5255
Author(s): David J. Aldous
Abstract: In a model of a connected network on random points in the plane, one expects
that the mean length of the shortest route between vertices at distance $r$
apart should grow only as $O(r)$ as $r \to \infty$, but this is not always easy
to verify. We give a general sufficient condition for such linearity, in the
setting of a Poisson point process. In a $L \times L$ square, define a
subnetwork $\GG_L$ to have the edges which are present regardless of the
configuration outside the square; the condition is that the largest component
of $\GG_L$ should contain a proportion $1 - o(1)$ of the vertices, as $L \to
\infty$. The proof is by comparison with oriented percolation. We show that the
general result applies to the relative neighborhood graph, and establishing the
linearity property for this network immediately implies it for a large family
of proximity graphs.
http://arxiv.org/abs/0911.5296
Author(s): David J. Aldous
Abstract: For a connected network on Poisson points in the plane, consider the
route-length $D(r,\theta) $ between a point near the origin and a point near
polar coordinates $(r,\theta)$, and suppose $E D(r,\theta) = O(r)$ as $r \to
\infty$. By analogy with the shape theorem for first-passage percolation, for a
translation-invariant and ergodic network one expects $r^{-1} D(r, \theta)$ to
converge as $r \to \infty$ to a constant $\rho(\theta)$. It turns out there are
some subtleties in making a precise formulation and a proof. We give one
formulation and proof via a variant of the subadditive ergodic theorem wherein
random variables are sometimes infinite.
http://arxiv.org/abs/0911.5301
Author(s): {\L}ukasz D\k{e}bowski
Abstract: We collect several observations that concern variable length coding of
two-sided infinite sequences in a probabilistic setting. Attention is paid to
images and preimages of asymptotically mean stationary measures defined on
subsets of these sequences. We point out sufficient conditions under which the
variable length coding and its inverse preserve asymptotic mean stationarity.
Moreover, conditions for preservation of shift-invariant $\sigma$-fields and
the finite-energy property are discussed and the block entropies for stationary
means of coded processes are related in some cases. Subsequently, we apply
certain of these results to construct a stationary nonergodic process with a
desired linguistic interpretation.
http://arxiv.org/abs/0911.5318
Author(s): Laura Cottatellucci and Ralf R. Mueller and and Merouane Debbah
Abstract: Totally asynchronous code-division multiple-access (CDMA) systems are
addressed. In Part I, the fundamental limits of asynchronous CDMA systems are
analyzed in terms of spectral efficiency and SINR at the output of the optimum
linear detector. The focus of Part II is the design of low-complexity
implementations of linear multiuser detectors in systems with many users that
admit a multistage representation, e.g. reduced rank multistage Wiener filters,
polynomial expansion detectors, weighted linear parallel interference
cancellers. The effects of excess bandwidth, chip-pulse shaping, and time delay
distribution on CDMA with suboptimum linear receiver structures are
investigated. Recursive expressions for universal weight design are given. The
performance in terms of SINR is derived in the large-system limit and the
performance improvement over synchronous systems is quantified. The
considerations distinguish between two ways of forming discrete-time
statistics: chip-matched filtering and oversampling.
http://arxiv.org/abs/0911.5067
Author(s): Benjamin Jourdain (CERMICS) and Michel Vellekoop
Abstract: We analyze the regularity of the optimal exercise boundary for the American
Put option when the underlying asset pays a discrete dividend at a known time
$t_d$ during the lifetime of the option. The ex-dividend asset price process is
assumed to follow Black-Scholes dynamics and the dividend amount is a
deterministic function of the ex-dividend asset price just before the dividend
date. The solution to the associated optimal stopping problem can be
characterised in terms of an optimal exercise boundary which, in contrast to
the case when there are no dividends, is no longer monotone. In this paper we
prove that when the dividend function is positive and concave, then the
boundary tends to 0 as time tends to $t_d^-$ and is non-increasing in a
left-hand neighbourhood of $t_d$. We also show that the exercise boundary is
continuous and a high contact principle holds in such a neighbourhood when the
dividend function is moreover linear in a neighbourhood of 0.
http://arxiv.org/abs/0911.5117
Author(s): J. Berestycki and E. Brunet and J. W. Harris and S. C. Harris
Abstract: In this note we consider a branching Brownian motion (BBM) on $\mathbb{R}$ in
which a particle at spatial position $y$ splits into two at rate $\beta y^2$,
where $\beta>0$ is a constant. This is a critical breeding rate for BBM in the
sense that the expected population size blows up in finite time while the
population size remains finite, almost surely, for all time. We find an
asymptotic for the almost sure rate of growth of the population.
http://arxiv.org/abs/0912.1360
Author(s): Ming Fang and Ofer Zeitouni
Abstract: Let $\mathbb{T}$ denote a rooted $b$-ary tree and let $\{S_v\}_{v\in
\mathbb{T}}$ denote a branching random walk indexed by the vertices of the
tree, where the increments are i.i.d. and possess a logarithmic moment
generating function $\Lambda(\cdot)$. Let $m_n$ denote the minimum of the
variables $S_v$ over all vertices at the $n$th generation, denoted by
$\mathbb{D}_n$. Under mild conditions, $m_n/n$ converges almost surely to a
constant, which for convenience may be taken to be 0. With $\bar
S_v=\max\{S_w:{\rm $w$ is on the geodesic connecting the root to $v$}\}$,
define $L_n=\min_{v\in \mathbb{D}_n} \bar S_v$. We prove that $L_n/n^{1/3}$
converges almost surely to an explicit constant $l_0$. This answers a question
of Hu and Shi.
http://arxiv.org/abs/0912.1392
Author(s): Atilla Yilmaz
Abstract: We consider large deviations for nearest-neighbor random walk in a uniformly
elliptic i.i.d. environment on $Z^d$. There exist variational formulae for the
quenched and the averaged rate functions $I_q$ and $I_a$, obtained by
Rosenbluth and Varadhan, respectively. $I_q$ and $I_a$ are not identically
equal. However, when $d\geq4$ and the walk satisfies the so-called (T)
condition of Sznitman, they have been previously shown to be equal on an open
set $A_{eq}$.
For every $\xi$ in $A_{eq}$, we prove the existence of a positive solution to
a Laplace-like equation involving $\xi$ and the original transition kernel of
the walk. We then use this solution to define a new transition kernel via the
h-transform technique of Doob. This new kernel corresponds to the unique
minimizer of Varadhan's variational formula at $\xi$. It also corresponds to
the unique minimizer of Rosenbluth's variational formula provided that the
latter is slightly modified.
http://arxiv.org/abs/0912.1429
Author(s): Fraser Daly and Claude Lef\`evre and Sergey Utev
Abstract: A stochastic ordering approach is applied with Stein's method for
approximation by the equilibrium distribution of a birth-death process. The
usual stochastic order and the more general s-convex orders are discussed.
Attention is focused on Poisson and translated Poisson approximation of a sum
of dependent Bernoulli random variables, for example k-runs in i.i.d. Bernoulli
trials. Other applications include approximation by polynomial birth--death
distributions.
http://arxiv.org/abs/0912.1448
Author(s): Rene L. Schilling and Alexander Schnurr
Abstract: We consider a stochastic differential equations which is driven by a Levy
process. It turns out that the solution process is a Feller process if the
coefficient of the SDE is bounded. Using a probabilistic formula we calculate
the symbol, which appears in the Fourier representation of the generator,
explicitely. Using the symbol we introduce indices which are generalizations of
the well known Blumenthal-Getoor index. These indices are then used to obtain
some fine properties of the solution process.
http://arxiv.org/abs/0912.1458
Author(s): V. Knopova and R. Schilling
Abstract: We show on- and off-diagonal upper estimates for the transition densities of
symmetric Levy and Levy-type processes. To get the an-diagonal estimates we
prove a Nash type inequality for the related Dirichlet form. For the
off-diagonal estimates we assume that the characteristic function of a Levy
(type) process is analytic, which allows to apply the complex analysis
technique.
http://arxiv.org/abs/0912.1482
Author(s): Qian Lin
Abstract: In this paper, we study the notion of local time and Tanaka formula for the
G-Brownian motion. Moreover, the joint continuity of the local time of the
G-Brownian motion is obtained and its quadratic variation is proven. As an
application, we generalize It^o's formula with respect to the G-Brownian motion
to convex functions.
http://arxiv.org/abs/0912.1515
Author(s): In\'es Armend\'ariz and Michail Loulakis
Abstract: It is known that large deviations of sums of subexponential random variables
are most likely realised by deviations of a single random variable. In this
article we give a detailed picture of how subexponential random variables are
distributed when a large deviation of their sum is observed.
http://arxiv.org/abs/0912.1516
Author(s): Raluca Balan and Sana Louhichi
Abstract: In this article, we consider a stationary array $(X_{j,n})_{1 \leq j \leq n,
n \geq 1}$ of random variables with values in $\bR \verb2\2 \{0\}$ (which
satisfy some asymptotic dependence conditions), and the corresponding sequence
$(N_{n})_{n\geq 1}$ of point processes, where $N_{n}$ has the points $X_{j,n},
1\leq j \leq n$. Our main result identifies some explicit conditions for the
convergence of the sequence $(N_{n})_{n \geq 1}$, in terms of the probabilistic
behavior of the variables in the array.
http://arxiv.org/abs/0912.1561
Author(s): Yilun Shang
Abstract: In this paper, we consider a one-dimensional random geometric graph process
with the inter-nodal gaps evolving according to an exponential AR(1) process,
which may serve as a mobile wireless network model. The transition probability
matrix and stationary distribution are derived for the Markov chains in terms
of network connectivity and the number of components. We characterize an
algorithm for the hitting time regarding disconnectivity. In addition, we also
study topological properties for static snapshots. We obtain the degree
distributions as well as asymptotic precise bounds and strong law of large
numbers for connectivity threshold distance and the largest nearest neighbor
distance amongst others. Both closed form results and limit theorems are
provided.
http://arxiv.org/abs/0912.1412
Author(s): Ben Morris
Abstract: E. Thorp introduced the following card shuffling model. Suppose the number of
cards $n$ is even. Cut the deck into two equal piles. Drop the first card from
the left pile or from the right pile according to the outcome of a fair coin
flip. Then drop from the other pile. Continue this way until both piles are
empty. We show that if $n$ is a power of 2 then the mixing time of the Thorp
shuffle is $O(\log^3 n)$. Previously, the best known bound was $O(\log^4 n)$.
http://arxiv.org/abs/0912.2759
Author(s): Zhen-Qing Chen and Panki Kim and Renming Song and Zoran Vondracek
Abstract: We consider a family of pseudo differential operators $\{\Delta+ a^\alpha
\Delta^{\alpha/2}; a\in [0, 1]\}$ on $\R^d$ that evolves continuously from
$\Delta$ to $\Delta + \Delta^{\alpha/2}$, where $d\geq 1$ and $\alpha \in (0,
2)$. It gives rise to a family of L\'evy processes \{$X^a, a\in [0, 1]\}$,
where $X^a$ is the sum of a Brownian motion and an independent symmetric
$\alpha$-stable process with weight $a$. Using a recently obtained uniform
boundary Harnack principle with explicit decay rate, we establish sharp bounds
for the Green function of the process $X^a$ killed upon exiting a bounded
$C^{1,1}$ open set $D\subset\R^d$. As a consequence, we identify the Martin
boundary of $D$ with respect to $X^a$ with its Euclidean boundary. Finally,
sharp Green function estimates are derived for certain L\'evy processes which
can be obtained as perturbations of $X^a$.
http://arxiv.org/abs/0912.2765
Author(s): Victor Korolev and Irina Shevtsova
Abstract: By a modification of the method that was applied in (Korolev and Shevtsova,
2009), here the inequalities $$ \ud\le\frac{0.335789(\beta^3+0.425)}{\sqrt{n}}
$$ and $$ \ud\le \frac{0.3051(\beta^3+1)}{\sqrt{n}} $$ are proved for the
uniform distance $\rho(F_n,\Phi)$ between the standard normal distribution
function $\Phi$ and the distribution function $F_n$ of the normalized sum of an
arbitrary number $n\ge1$ of independent identically distributed random
variables with zero mean, unit variance and finite third absolute moment
$\beta^3$. The first of these inequalities sharpens the best known version of
the classical Berry--Esseen inequality since
$0.335789(\beta^3+0.425)\le0.335789(1+0.425)\beta^3<0.4785\beta^3$ by virtue of
the condition $\beta^3\ge1$, and $0.4785$ is the best known upper estimate of
the absolute constant in the classical Berry--Esseen inequality. The second
inequality is applied to lowering the upper estimate of the absolute constant
in the analog of the Berry--Esseen inequality for Poisson random sums to
$0.3051$ which is strictly less than the least possible value of the absolute
constant in the classical Berry--Esseen inequality. As a corollary, the
estimates of the rate of convergence in limit theorems for compound mixed
Poisson distributions are refined.
http://arxiv.org/abs/0912.2795
Author(s): Christian Meyer
Abstract: We collect well known and less known facts about the bivariate normal
distribution and translate them into copula language. In addition, we prove a
very general formula for the bivariate normal copula, we compute Gini's gamma,
and we provide improved bounds and approximations on the diagonal.
http://arxiv.org/abs/0912.2816
Author(s): Josh Reed
Abstract: In this paper, we study the $G/\mathit{GI}/N$ queue in the Halfin--Whitt
regime. Our first result is to obtain a deterministic fluid limit for the
properly centered and scaled number of customers in the system which may be
used to provide a first-order approximation to the queue length process. Our
second result is to obtain a second-order stochastic approximation to the
number of customers in the system in the Halfin--Whitt regime. This is
accomplished by first centering the queue length process by its deterministic
fluid limit and then normalizing by an appropriate factor. We then proceed to
obtain an alternative but equivalent characterization of our limiting
approximation which involves the renewal function associated with the service
time distribution. This alternative characterization reduces to the diffusion
process obtained by Halfin and Whitt [Oper. Res. 29 (1981) 567--588] in the
case of exponentially distributed service times.
http://arxiv.org/abs/0912.2837
Author(s): Dalibor Voln\'y
Abstract: Let $(X_i)$ be a stationary and ergodic Markov chain with kernel $Q$, $f$ an
$L^2$ function on its state space. If $Q$ is a normal operator and $f =
(I-Q)^{1/2}g$ (which is equivalent to the convergence of $\sum_{n=1}^\infty
\frac{\sum_{k=0}^{n-1}Q^kf}{n^{3/2}}$ in $L^2$), we have the central limit
theorem (cf\. \cite{D-L 1}, \cite{G-L 2}). Without assuming normality of $Q$,
the CLT is implied by the convergence of $\sum_{n=1}^\infty
\frac{\|\sum_{k=0}^{n-1}Q^kf\|_2}{n^{3/2}}$, in particular by
$\|\sum_{k=0}^{n-1}Q^kf\|_2 = o(\sqrt n/\log^q n)$, $q>1$ by \cite{M-Wu} and
\cite{Wu-Wo} respectively. We shall show that if $Q$ is not normal and $f\in
(I-Q)^{1/2} L^2$, or if the conditions of Maxwell and Woodroofe or of Wu and
Woodroofe are weakened to $\sum_{n=1}^\infty
c_n\frac{\|\sum_{k=0}^{n-1}Q^kf\|_2}{n^{3/2}}<\infty$ for some sequence
$c_n\searrow 0$, or by $\|\sum_{k=0}^{n-1}Q^kf\|_2 = O(\sqrt n/\log n)$, the
CLT need not hold.
http://arxiv.org/abs/0912.2864
Author(s): Serge Cohen (IMT) and Fabien Panloup (IMT)
Abstract: In this paper, we obtain some existence results of stationary solutions to a
class of SDEs driven by continuous Gaussian processes with stationary
increments. We propose a constructive approach based on the study of some
sequences of empirical measures of Euler schemes of these SDEs. In our main
result, we obtain the functional convergence of this sequence to a stationary
solution to the SDE. We also obtain some specific properties of the stationary
solution. In particular, we show that, in contrast to Markovian SDEs, the
initial random value of a stationary solution and the driving Gaussian process
are always dependent. This emphasizes the fact that the concept of invariant
distribution is definitely different to the Markovian case.
http://arxiv.org/abs/0912.2889
Author(s): Holger K\"osters
Abstract: We consider the sample covariance matrices of large data matrices which have
i.i.d. complex matrix entries and which are non-square in the sense that the
difference between the number of rows and the number of columns tends to
infinity. We show that the second-order correlation function of the
characteristic polynomial of the sample covariance matrix is asymptotically
given by the sine kernel in the bulk of the spectrum and by the Airy kernel at
the edge of the spectrum. Similar results are given for real sample covariance
matrices.
http://arxiv.org/abs/0912.2956
Author(s): T. N. Narasimhan
Abstract: Stochastic diffusion equation, which attained prominence with Einstein's work
on Brownian motion at the beginning of the twentieth century, was first
formulated by Laplace a century earlier as part of his work on Central Limit
Theorem. Between 1807 and 1811, Fourier's work on heat diffusion, and
Laplace'swork on probability influenced and inspired each other. This brief
period of interaction between these two illustrious figures must be considered
remarkable for its profound impact on subsequent developments in mathematical
physics, probability theory and pure analysis.
http://arxiv.org/abs/0912.2798
Author(s): S. N. Stelmastchuk
Abstract: Let $P(M,G)$ be a principal fiber bundle and $E(M,N,G,P)$ be an associate
fiber bundle. Our interested is to study harmonic sections of the projection
$\pi_{E}$ of $E$ into $M$. Our first purpose is to give a stochastic
characterization of harmonic section from $M$ into $E$ and a geometric
characterization of harmonic sections with respect to its equivariant lift. The
second purpose is to show a version of Liouville theorem for harmonic sections
and to prove that section $M$ into $E$ is a harmonic section if and only if it
is parallel.
http://arxiv.org/abs/0912.2895
Author(s): Krzysztof Burdzy and Soumik Pal
Abstract: We study Markov processes where the "time" parameter is replaced by paths in
a directed graph from an initial vertex to a terminal one. Along each directed
path the process is Markov and has the same distribution as the one along any
other directed path. If two directed paths do not interact, in a suitable
sense, then the distributions of the processes on the two paths are
conditionally independent, given their values at the common endpoint of the two
paths. Conditions on graphs that support such processes (e.g., hexagonal
lattice) are established. Next we analyze a particularly suitable family of
Markov processes, called harnesses, which includes Brownian motion and other
Levy processes, on such time-like graphs. Finally we investigate continuum
limits of harnesses on a sequence of time-like graphs that admits a limit in a
suitable sense.
http://arxiv.org/abs/0912.0328
Author(s): Louigi Addario-Berry and Simon Griffiths and Ross Kang
Abstract: We study invasion percolation on Aldous' Poisson-weighted infinite tree, and
derive two distinct Markovian representations of the resulting process. One of
these is the sigma to infinity limit of a representation discovered by Angel,
Goodman, den Hollander and Slade (arXiv:math/0608132v2). We also introduce an
exploration process of a randomly weighted Poisson incipient infinite cluster.
The dynamics of the new process are much more straightforward to describe than
those of invasion percolation, but it turns out that the two processes have
extremely similar behavior. Finally, we introduce two new "stationary"
representations of the Poisson incipient infinite cluster as random graphs on Z
which are, in particular, factors of a homogeneous Poisson point process on the
upper half-plane Rx[0,infinity).
http://arxiv.org/abs/0912.0335
Author(s): David Gamarnik and David Goldberg and Theophane Weber
Abstract: We consider a decision network on an undirected graph in which each node
corresponds to a decision variable, and each node and edge of the graph is
associated with a reward function whose value depends only on the variables of
the corresponding nodes. The goal is to construct a decision vector which
maximizes the total reward. This decision problem encompasses a variety of
models, including maximum-likelihood inference in graphical models (Markov
Random Fields), combinatorial optimization on graphs, economic team theory and
statistical physics. The network is endowed with a probabilistic structure in
which costs are sampled from a distribution. Our aim is to identify sufficient
conditions to guarantee average-case polynomiality of the underlying
optimization problem. We construct a new decentralized algorithm called Cavity
Expansion and establish its theoretical performance for a variety of models.
Specifically, for certain classes of models we prove that our algorithm is able
to find near optimal solutions with high probability in a decentralized way.
The success of the algorithm is based on the network exhibiting a correlation
decay (long-range independence) property. Our results have the following
surprising implications in the area of average case complexity of algorithms.
Finding the largest independent (stable) set of a graph is a well known NP-hard
optimization problem for which no polynomial time approximation scheme is
possible even for graphs with largest connectivity equal to three, unless P=NP.
We show that the closely related maximum weighted independent set problem for
the same class of graphs admits a PTAS when the weights are i.i.d. with the
exponential distribution. Namely, randomization of the reward function turns an
NP-hard problem into a tractable one.
http://arxiv.org/abs/0912.0338
Author(s): Yun Xue and Yimin Xiao
Abstract: Spatio-temporal models are widely used for inference in statistics and many
applied areas. In such contexts interests are often in the fractal nature of
the sample surfaces and in the rate of change of the spatial surface at a given
location in a given direction. In this paper we apply the theory of Yaglom
(1957) to construct a large class of space-time Gaussian models with stationary
increments, establish bounds on the prediction errors and determine the
smoothness properties and fractal properties of this class of Gaussian models.
Our results can be applied directly to analyze the stationary space-time models
introduced by Cressie and Huang (1999), Gneiting (2002) and Stein (2005),
respectively.
http://arxiv.org/abs/0912.0285
Author(s): Robert W. Neel
Abstract: After recalling the Dirichlet problem at infinity on a Cartan-Hadamard
manifold, we discuss what is known and the difference between the
two-dimensional and higher-dimensional cases. Turning our attention to the
two-dimensional case, we prove that the Dirichlet problem at infinity on a
two-dimensional Cartan-Hadamard manifold is solvable under the curvature
condition $K\leq (1+\epsilon)/(r^2 \log r)$, outside of a compact set, for some
$\epsilon>0$ in polar coordinates around some pole. This condition on the
curvature is sharp, and improves upon the previously known case of quadratic
curvature decay. Finally, we briefly discuss the issues which arise in trying
to extend this method to higher dimensions.
http://arxiv.org/abs/0912.0330
Author(s): St\'ephane Goutte (LAGA and OPTEA) and Nadia Oudjane (LAGA) and Francesco Russo (LAGA, MathFi, CERMICS)
Abstract: For a large class of vanilla contingent claims, we establish an explicit
F\"ollmer-Schweizer decomposition when the underlying is a process with
independent increments (PII) and an exponential of a PII process. This allows
to provide an efficient algorithm for solving the mean variance hedging
problem. Applications to models derived from the electricity market are
performed.
http://arxiv.org/abs/0912.0372
Author(s): Craig A. Tracy and Harold Widom
Abstract: We review recent progress in limit laws for the one-dimensional asymmetric
simple exclusion process (ASEP) on the integer lattice. The limit laws are
expressed in terms of a certain Painleve II function. Furthermore, we take this
opportunity to give a brief survey of the appearance of Painleve functions in
statistical physics.
http://arxiv.org/abs/0912.2362
Author(s): Bernard Bercu and Ivan Nourdin and Murad Taqqu
Abstract: In this paper, we study almost sure central limit theorems for sequences of
functionals of general Gaussian fields. We apply our result to non-linear
functions of stationary Gaussian sequences. We obtain almost sure central limit
theorems for these non-linear functions when they converge in law to a normal
distribution.
http://arxiv.org/abs/0912.2398
Author(s): Yaozhong Hu and David Nualart
Abstract: The purpose of this note is to prove a central limit theorem for the
$L^3$-modulus of continuity of the Brownian local time using techniques of
stochastic analysis. The main ingredients of the proof are an asymptotic
version of Knight's theorem and the Clark-Ocone formula for the $L^3$-modulus
of the Brownian local time.
http://arxiv.org/abs/0912.2400
Author(s): Dhafer Malouche and Bala Rajaratnam
Abstract: A covariance graph is an undirected graph associated with a multivariate
probability distribution of a given random vector where each vertex represents
each of the different components of the random vector and where the absence of
an edge between any pair of variables implies marginal independence between
these two variables. Covariance graph models have recently received much
attention in the literature and constitute a sub-family of graphical models.
Though they are conceptually simple to understand, they are considerably more
difficult to analyze. Under some suitable assumption on the probability
distribution, covariance graph models can also be used to represent more
complex conditional independence relationships between subsets of variables.
When the covariance graph captures or reflects all the conditional independence
statements present in the probability distribution the latter is said to be
faithful to its covariance graph - though no such prior guarantee exists.
Despite the increasingly widespread use of these two types of graphical models,
to date no deep probabilistic analysis of this class of models, in terms of the
faithfulness assumption, is available. Such an analysis is crucial in
understanding the ability of the graph, a discrete object, to fully capture the
salient features of the probability distribution it aims to describe. In this
paper we demonstrate that multivariate Gaussian distributions that have trees
as covariance graphs are necessarily faithful. The method of proof is original
as it uses an entirely new approach and in the process yields a technique that
is novel to the field of graphical models.
http://arxiv.org/abs/0912.2407
Author(s): Mohsen Bayati and David Gamarnik and Prasad Tetali
Abstract: We establish the existence of free energy limits for several sparse random
hypergraph models corresponding to certain combinatorial models on
Erd{\"o}s-R\'{e}nyi graph $\G(N,c/N)$ and random $r$-regular graph $\G(N,r)$.
For a variety of models, including independent sets, MAX-CUT, Coloring and
K-SAT, we prove that the free energy both at a positive and zero temperature,
appropriately rescaled, converges to a limit as the size of the underlying
graph diverges to infinity. For example, as a special case we prove that the
size of a largest independent set in these graphs, normalized by the number of
nodes converges to a limit w.h.p., thus resolving an open problem, (see
Conjecture 2.20 in \cite{WormaldModelsRandomGraphs}, as well as
\cite{Aldous:FavoriteProblems}, \cite{BollobasRiordanMetrics},
\cite{JansonThomason}, and \cite{AldousSteele:survey}).
Our approach is based on extending and simplifying the interpolation method
developed by Guerra and Toninelli \cite{GuerraTon} and Franz and Leone
\cite{FranzLeone},\cite{FranzLeoneToninelliRegular}. Among other applications,
this method was used to prove the existence of free energy limits for
Viana-Bray and K-SAT models on Erd{\"o}s-R\'{e}nyi graphs. The case of zero
temperature was treated by taking limits of positive temperature models. We
provide instead a simpler combinatorial approach and work with the zero
temperature case (optimization) directly both in the case of
Erd{\"o}s-R\'{e}nyi graph $\G(N,c/N)$ and random regular graph $\G(N,r)$. In
addition we establish the large deviations principle for the satisfiability
property for constraint satisfaction problems such as Coloring, K-SAT and
NAE-K-SAT.
http://arxiv.org/abs/0912.2444
Author(s): Xavier Bardina Carles Rovira
Abstract: We show how from an unique standard Poisson process we can build a family of
processes that converges in law to a $d$-dimensional standard Brownian motion
for any $d \ge 1$.
http://arxiv.org/abs/0912.2457
Author(s): S. P\'ech\'e
Abstract: We consider complex sample covariance matrices $M_N=\frac{1}{N}YY^*$ where
$Y$ is a $N \times p$ random matrix with i.i.d. entries $Y_{ij}, 1\leq i\leq N,
1\leq j \leq p$ with distribution $F$. Under some regularity and decay
assumption on $F$, we prove universality of some local eigenvalue statistics in
the bulk of the spectrum in the limit where $N\to \infty$ and $\lim_{N \to
\infty}p/N =\gamma$ for any real number $\gamma \in (0, \infty)$.
http://arxiv.org/abs/0912.2493
Author(s): Mark M. Meerschaert and Erkan Nane and and Palaniappan Vellaisamy
Abstract: In a fractional Cauchy problem, the usual first order time derivative is
replaced by a fractional derivative. The fractional derivative models time
delays in a diffusion process. The order of the fractional derivative can be
distributed over the unit interval, to model a mixture of delay sources. In
this paper, we provide explicit strong solutions and stochastic analogues for
distributed-order fractional Cauchy problems on bounded domains with Dirichlet
boundary conditions. Stochastic solutions are constructed using a non-Markovian
time change of a killed Markov process generated by a uniformly elliptic second
order space derivative operator.
http://arxiv.org/abs/0912.2521
Author(s): Tianxiao Wang
Abstract: This paper is devoted to the unique solvability of backward stochastic
Volterra integral equations (BSVIEs for short), in terms of both M-solution
introduced in [15] and the adapted solutions in [6], [11]. We prove the
existence and uniqueness of M-solutions of BSVIEs in Lp (11) is also considered, which also generalize the results in [6] and [11].
http://arxiv.org/abs/0912.2567
Author(s): Alexandr Andoni and Constantinos Daskalakis and Avinatan Hassidim and Sebastien Roch
Abstract: Molecular phylogenetic techniques do not generally account for such common
evolutionary events as site insertions and deletions (known as indels). Instead
tree building algorithms and ancestral state inference procedures typically
rely on substitution-only models of sequence evolution. In practice these
methods are extended beyond this simplified setting with the use of heuristics
that produce global alignments of the input sequences--an important problem
which has no rigorous model-based solution. In this paper we consider a new
version of the multiple sequence alignment in the context of stochastic indel
models. More precisely, we introduce the following {\em trace reconstruction
problem on a tree} (TRPT): a binary sequence is broadcast through a tree
channel where we allow substitutions, deletions, and insertions; we seek to
reconstruct the original sequence from the sequences received at the leaves of
the tree. We give a recursive procedure for this problem with strong
reconstruction guarantees at low mutation rates, providing also an alignment of
the sequences at the leaves of the tree. The TRPT problem without indels has
been studied in previous work (Mossel 2004, Daskalakis et al. 2006) as a
bootstrapping step towards obtaining optimal phylogenetic reconstruction
methods. The present work sets up a framework for extending these works to
evolutionary models with indels.
http://arxiv.org/abs/0912.2577
Author(s): Svante Janson
Abstract: We give a survey of a number of simple applications of renewal theory to
problems on random strings and tries: insertion depth, size, insertion mode and
imbalance of tries; variations for b-tries and Patricia tries; Khodak and
Tunstall codes.
http://arxiv.org/abs/0912.2174
Author(s): Xiaopeng Chen and Jinqiao Duan and Xinchu Fu
Abstract: Some properties of random Conley index are obtained and then a sufficient
condition for the existence of abstract bifurcation points for both
discrete-time and continuous-time random dynamical systems is presented. This
stochastic bifurcation phenomenon is demonstrated by a few examples.
http://arxiv.org/abs/0912.2487
Author(s): S.N. Stelmastchuk
Abstract: Let $G$ be a Lie Group with a left invariant connection such that its
connection function is skew-symmetric. Our main goal is to show a version of
Pluzhnikov's Theorem for this kind of connection. To this end, we use the
stochastic logarithm. More exactly, the stochastic logarithm gives
characterizations for Brownian motions and Martingales in $G$, and these
characterzations are used to prove Pluzhnikov's Theorem.
http://arxiv.org/abs/0912.2665
Author(s): Domenico Marinucci (DIPMAT) and Giovanni Peccati (MODAL'X)
Abstract: We investigate the relationship between ergodicity and asymptotic Gaussianity
of isotropic spherical random fields, in the high-resolution (or
high-frequency) limit. In particular, our results suggest that under a wide
variety of circumstances the two conditions are equivalent, i.e. the sample
angular power spectrum may converge to the population value if and only if the
underlying field is asymptotically Gaussian, in the high frequency sense. These
findings may shed some light on the role of Cosmic Variance in Cosmic Microwave
Background (CMB) radiation data analysis.
http://arxiv.org/abs/0911.2502
Author(s): Omer Angel and Alexander E. Holroyd
Abstract: A sorting network is a shortest path from 12...n to n...21 in the Cayley
graph of S_n generated by nearest-neighbor swaps. For m<=n, consider the random
m-particle sorting network obtained by choosing an n-particle sorting network
uniformly at random and then observing only the relative order of m particles
chosen uniformly at random. We prove that the expected number of swaps in
location j in the subnetwork does not depend on n, and we provide a formula for
it. Our proof is probabilistic, and involves a Polya urn with non-integer
numbers of balls. From the case m=4 we obtain a proof of a conjecture of
Warrington. Our result is consistent with a conjectural limiting law of the
subnetwork as n->infinity implied by the great circle conjecture Angel,
Holroyd, Romik and Virag.
http://arxiv.org/abs/0911.2519
Author(s): Joseph Najnudel and Ashkan NIkeghbali
Abstract: In a previous work, we associated with any submartingale $X$ of class
$(\Sigma)$, defined on a filtered probability space $(\Omega, \mathcal{F},
\mathbb{P}, (\mathcal{F}_t)_{t \geq 0})$ satisfying some technical conditions,
a $\sigma$-finite measure $\mathcal{Q}$ on $(\Omega, \mathcal{F})$, such that
for all $t \geq 0$, and for all events $\Lambda_t
\in \mathcal{F}_t$: $$ \mathcal{Q} [\Lambda_t, g\leq t] =
\mathbb{E}_{\mathbb{P}} [\mathds{1}_{\Lambda_t} X_t],$$ where $g$ is the last
hitting time of zero by the process $X$. The measure $\mathcal{Q}$, which was
previously studied in particular cases related with Brownian penalisations and
problems in mathematical finance, enjoys some remarkable properties which are
detailed in this paper. Most of these properties are related to a certain class
of nonnegative martingales, defined as the local densities (with respect to
$\mathbb{P}$) of the finite measures which are absolutely continuous with
respect to $\mathcal{Q}$. From the properties of the measure $\mathcal{Q}$, we
also deduce a universal class of penalisation results of the probability
measure $\mathbb{P}$ with a large class of functionals: the measure
$\mathcal{Q}$ appears to be the unifying object in these problems.
http://arxiv.org/abs/0911.2571
Author(s): D. Alpay and D. Levanony and A. Pinhas
Abstract: We study state space equations within the white noise space setting. A
commutative ring of power series in a countable number of variables plays an
important role. Transfer functions are rational functions with coefficients in
this commutative ring, and are characterized in a number of ways. A major
feature in our approach is the observation that key characteristics of a
linear, time invariant, stochastic system are determined by the corresponding
characteristics associated with the deterministic part of the system, namely
its average behavior.
http://arxiv.org/abs/0911.2574
Author(s): Mikhail V. Menshikov and Andrew R. Wade
Abstract: The study of discrete-time stochastic processes on the half-line with mean
drift at $x$ given by $\mu_1 (x) \to 0$ as $x \to \infty$ is known as
Lamperti's problem. We give sharp almost-sure bounds for processes of this type
in the case where $\mu_1 (x)$ is of order $x^{-\beta}$ for some $\beta \in
(0,1)$. The bounds are of order $t^{1/(1+\beta)}$, so the process is
super-diffusive but sub-ballistic (has zero speed). We make minimal assumptions
on the moments of the increments of the process (finiteness of
$(2+2\beta+\eps)$-moments for our main results, so 4th moments certainly
suffice) and do not assume that the process is time-homogeneous or Markovian.
In the case where $x^\beta \mu_1 (x)$ has a finite positive limit, our results
imply a strong law of large numbers, which strengthens and generalizes earlier
results of Lamperti and Voit. We prove an accompanying central limit theorem,
which appears to be new even in the case of a nearest-neighbour random walk,
although our result is considerably more general. This answers a question of
Lamperti. We also prove transience of the process under weaker conditions than
those that we have previously seen in the literature. Most of our results also
cover the case where $\beta =0$. We illustrate our results with applications to
birth-and-death chains and to multi-dimensional non-homogeneous random walks.
http://arxiv.org/abs/0911.2599
Author(s): Vlad Bally and Nicolas Fournier
Abstract: We consider the 2-dimensional spatially homogeneous Boltzmann equation for
hard potentials. We assume that the initial condition is a probability measure
that has some exponential moments and is not a Dirac mass. We prove some
regularization properties: for a class of very hard potentials, the solution
instantaneously belongs to $H^r$, for some $r\in (-1,2)$ depending on the
parameters of the equation. Our proof relies on the use of a well-suited
Malliavin calculus for jump processes.
http://arxiv.org/abs/0911.2614
Author(s): Vlad Bally and Lucia Caramellino
Abstract: We give a criterion of regularity for a probability measure $\mu$ on $\R^{d}$
based on integration by parts formulas. The standard way to deal with this
problem is to use a Fourier transform argument. Here we give an alternative
approach using the Riesz transform and the machinery of the Sobolev spaces
associated to $\mu$. Finally we apply this criterion in order to improve the
classical regularity criterion for functionals on the Wiener space due to
Malliavin. The basic gain is that we need less regularity for the functionals
at hand.
http://arxiv.org/abs/0911.2631
Author(s): Michael McGettrick
Abstract: We investigate the quantum versions of a one-dimensional random walk, whose
corresponding Markov Chain is of order 2 or 3. This corresponds to the walk
having a memory of up to two previous steps. We derive the amplitudes and
probabilities for these walks, and point out how they differ from both
Classical Random Walks, and Quantum Walks without memory.
http://arxiv.org/abs/0911.1653
Author(s): Svante Janson
Abstract: We study the susceptibility, i.e., the mean cluster size, in random graphs
with given vertex degrees. We show, under weak assumptions, that the
susceptibility converges to the expected cluster size in the corresponding
branching process. In the supercritical case, a corresponding result holds for
the modified susceptibility ignoring the giant component and the expected size
of a finite cluster in the branching process; this is proved using a duality
theorem.
The critical behaviour is studied. Examples are given where the critical
exponents differ on the subcritical and supercritical sides.
http://arxiv.org/abs/0911.2636
Author(s): R. W. R. Darling; E. E. Pyle
Abstract: Fix $\alpha >0$, and sample $N$ integers uniformly at random from
$\{1,2,\ldots ,\lfloor e^{\alpha N}\rfloor \}$. Given $\eta >0$, the
probability that the maximum of the pairwise GCDs lies between $N^{2-\eta }$
and $N^{2+\eta}$ converges to 1 as $N\to \infty $. More precise estimates are
obtained. This is a Birthday Problem: two of the random integers are likely to
share some prime factor of order $N^2/\log [N]$. The proof generalizes to any
arithmetical semigroup where a suitable form of the Prime Number Theorem is
valid.
http://arxiv.org/abs/0911.2660
Author(s): Steven N. Evans and Ben Morris and Arnab Sen
Abstract: A well-known result of Arratia shows that one can make rigorous the notion of
starting an independent Brownian motion at every point of an arbitrary closed
subset of the real line and then building a set-valued process by requiring
particles to coalesce when they collide. Arratia noted that the value of this
process will be almost surely a locally finite set at all positive times, and a
finite set almost surely if the initial value is compact: the key to both of
these facts is the observation that, because of the topology of the real line
and the continuity of Brownian sample paths, at the time when two particles
collide one or the other of them must have already collided with each particle
that was initially between them. We investigate whether such instantaneous
coalescence still occurs for coalescing systems of particles where either the
state space of the individual particles is not locally homeomorphic to an
interval or the sample paths of the individual particles are discontinuous. We
show that Arratia's conclusion is valid for Brownian motions on the Sierpinski
gasket and for stable processes on the real line with stable index greater than
one.
http://arxiv.org/abs/0912.0017
Author(s): Omer Angel and Alexander E. Holroyd and Terry Soo
Abstract: Let Pi and Gamma be homogeneous Poisson point processes on a fixed set of
finite volume. We prove a necessary and sufficient condition on the two
intensities for the existence of a coupling of Pi and Gamma such that Gamma is
a deterministic function of Pi, and all points of Gamma are points of Pi. The
condition exhibits a surprising lack of monotonicity. However, in the limit of
large intensities, the coupling exists if and only if the expected number of
points is at least one greater in Pi than in Gamma.
http://arxiv.org/abs/0912.0047
Author(s): Balint Virag and Benedek Valko
Abstract: It has been conjectured that the eigenvalues of random Schrodinger operators
at the localization transition in dimensions d>=2 behave like the eigenvalues
of the Gaussian Orthogonal Ensemble (GOE). We study a the eigenvalues of long
boxes in dimensions d>=2 for low disorder. We deduce a stochastic differential
equation representation for the limiting process. We show that in dimensions
d>=2 there are sequences of boxes so that the eigenvalues in low disorder
converge to Sine1, the limiting eigenvalue process of the GOE.
http://arxiv.org/abs/0912.0097
Author(s): Jean Bertoin (PMA and DMA) and Mladen Savov
Abstract: The central result of this paper is an analytic duality relation for
real-valued L\'evy processes killed upon exiting a half-line. By Nagasawa's
theorem, this yields a remarkable time-reversal identity involving the L\'evy
process conditioned to stay positive. As examples of applications, we construct
a version of the L\'evy process indexed by the entire real line and started
from $-\infty$ which enjoys a natural spatial-stationarity property, and point
out that the latter leads to a natural Lamperti-type representation for
self-similar Markov processes in $(0,\infty)$ started from the entrance point
0+.
http://arxiv.org/abs/0912.0131
Author(s): Vladislav Kargin
Abstract: In this paper we study the continuous-time quantum walk on Z, Z^d, and
infinite homogeneous trees. We compute the limit of the average probability
distribution for the general isotropic quantum walk on Z, and for the
nearest-neighbor walk on Z^d and on the infinite m-valent tree. In addition, we
compute the asymptotic approximation for the probability of the return to zero
at time t.
http://arxiv.org/abs/0912.0232
Author(s): J. Inglis and V. Kontis and B. Zegarlinski
Abstract: In this paper we study applications of U-bounds to coercive and isoperimetric
problems for probability measures on finite and infinite products of H-type
groups.
http://arxiv.org/abs/0912.0236
Author(s): Wen Lv
Abstract: In this paper, we deal with a class of one-dimensional reflected backward
stochastic differential equations with stochastic Lipschitz coefficient. We
derive the existence and uniqueness of the solutions for those equations via
Snell envelope and the fixed point theorem.
http://arxiv.org/abs/0912.2162
Author(s): Jorge A. Leon and Samy Tindel
Abstract: In this paper we study the existence of a unique solution to a general class
of Young delay differential equations driven by a H\"older continuous function
with parameter greater that 1/2 via the Young integration setting. Then some
estimates of the solution are obtained, which allow to show that the solution
of a delay differential equation driven by a fractional Brownian motion (fBm)
with Hurst parameter H>1/2 has a smooth density. To this purpose, we use
Malliavin calculus based on the Frechet differentiability in the directions of
the reproducing kernel Hilbert space associated with fBm.
http://arxiv.org/abs/0912.2180
Author(s): Tomasz Klimsiak
Abstract: We consider the Cauchy problem for semilinear parabolic equation in
divergence form with obstacle. We show that under natural conditions on the
right-hand side of the eqution and mild conditions on the obstacle a unique
continuous solution of the problem admits a stochastic representation in terms
of reflected backward stochastic differential equations. We derive also some
regularity properties of so- lutions and prove useful approximation results.
http://arxiv.org/abs/0912.2193
Author(s): Peter Major
Abstract: In this paper a result of Latala about the tail behavior of Gaussian
polynomials will be discussed. Latala proved an interesting result about this
problem in paper [2]. But his proof applied an incorrect statement at a crucial
point. Hence the question may arise whether the main result of paper [2] is
valid. The goal of this paper is to settle this problem by presenting such a
proof where the application of the erroneous statement is avoided. I discuss
the proofs in detail even at the price of a longer text and try to give such an
explanation that reveals the ideas behind them better than the original paper.
\
http://arxiv.org/abs/0912.2279
Author(s): S. N. Stelmastchuk
Abstract: Let $\pi:(E,\nabla^{E}) \to (M,g)$ be an affine submersion with horizontal
distribution, where $\nabla^{E}$ is a symmetric connection and $M$ is a
Riemannian manifold. Let $\sigma$ be a section of $\pi$, namely, $\pi \circ
\sigma = Id_{M}$. It is possible to study the harmonic property of section
$\sigma$ in two ways. First, we see $\sigma$ as a harmonic map. Second, we see
$\sigma$ as harmonic section. In the Riemannian context, it means that $\sigma$
is a critical point of the vertical functional energy. Our main goal is to find
conditions to the assertion: $\sigma$ is a harmonic map if and only if $\sigma$
is a harmonic section.
http://arxiv.org/abs/0912.2230
Author(s): Nike Sun (Stanford University)
Abstract: This survey gives an account of the emergence of conformal invariance in the
scaling limit of critical percolation on the triangular lattice, as the lattice
mesh is taken to zero. The main purpose is to provide a mostly self-contained
proof of the result, due to Smirnov and to Camia and Newman, that the
percolation exploration path has a conformally invariant scaling limit. To
motivate this proof, we will review the conformal invariance of planar Brownian
motion, as well as its connection to harmonic functions. We then prove
Smirnov's result on the conformal invariance of crossing probabilities in the
scaling limit. The remainder of the article describes how to pass from this
result to the conformally invariant scaling limit of the exploration path. To
do this we give an introduction to the Schramm-Loewner evolutions SLE(k); it is
known that the exploration path converges to SLE(6). We also discuss how to
make a rigorous definition of the scaling limit of a random curve, and we
present the proof of Aizenman and Burchard which guarantees the existence of
subsequential scaling limits. Finally, we show the conformally invariant
scaling limit for the exploration path, following the work of Smirnov and of
Camia and Newman.
http://arxiv.org/abs/0911.0063
Author(s): Kai Du and Qingxin Meng
Abstract: Backward stochastic partial differential equations of parabolic type with
variable coefficients are considered in the whole Euclidean space. Improved
existence and uniqueness results are given in the Sobolev space $H^n$
($=W^n_2$) under weaker assumptions than those used by X. Zhou [Journal of
Functional Analysis 103, 275--293 (1992)].
http://arxiv.org/abs/0911.0077
Author(s): Roland Speicher
Abstract: Free probability theory was created by Dan Voiculescu around 1985, motivated
by his efforts to understand special classes of von Neumann algebras. His
discovery in 1991 that also random matrices satisfy asymptotically the freeness
relation transformed the theory dramatically. Not only did this yield
spectacular results about the structure of operator algebras, but it also
brought new concepts and tools into the realm of random matrix theory. In the
following we will give, mostly from the random matrix point of view, a survey
on some of the basic ideas and results of free probability theory.
http://arxiv.org/abs/0911.0087
Author(s): Russell Lyons and Fedor Nazarov
Abstract: We prove that in every bipartite Cayley graph of every non-amenable group,
there is a perfect matching that is obtained as a factor of independent uniform
random variables. We also discuss expansion properties of factors and improve
the Hoffman spectral bound on independence number of finite graphs.
http://arxiv.org/abs/0911.0092
Author(s): Pawe{\l} J. Szab{\l}owski
Abstract: We define and study distributions in $\mathbb{R}^{d}$ that we call $q-$%
Normal. For $q=1$ they are really multidimensional Normal, for $q\in (-1,1)$
they have densities, compact support and many properties that resemble
properties of ordinary multidimensional Normal distribution. We also consider
some generalizations of these distributions.
http://arxiv.org/abs/0911.0109
Author(s): Gideon Amir
Abstract: In this paper we study the structure of the limit aggregate (the union of all
finite-time aggregates) of the one-dimensional long range diffusion limited
aggregation process defined in [arXiv:0910.4416] . We show (under some
regularity conditions) that for walks with finite third moment the limit
aggregate has renewal structure and positive density, while for walks with
finite variance the renewal structure no longer exists and the limit aggregate
has 0 density. We define a tree structure on the aggregates and show some
results on the degrees and number of ends of these random trees.
http://arxiv.org/abs/0911.0122
Author(s): Mark Adler and Jonathan Delepine and Pierre van Moerbeke and Pol Vanhaecke
Abstract: Consider non-intersecting Brownian motions on the real line, starting from
the origin at t=0, with a number of particles forced to reach p distinct target
points at time t=1. This work shows that the transition probability, that is
the probability for the particles to pass through windows E_k at times t_k,
satisfies, in a new set of variables, a non-linear PDE which can be expressed
as a near-Wronskian; that is a determinant of a matrix of size p+1, with each
row being a derivative of the previous, except for the last column. It is an
interesting open question to understand those equations from a more
probabilistic point of view.
As an application of these equations, let the number of particles forced to
the extreme target points (the first and the last one) tend to infinity; keep
the number of particles forced to intermediate target points fixed (inliers),
but let the target points themselves go to infinity according to a proper
scale. A new critical process appears at the point of bifurcation, where the
bulk of the particles forced to the first target point depart from those going
to the last target point. These statistical fluctuations near that point of
bifurcation are specified by a kernel, which is a rational perturbation of the
Pearcey kernel. Finally, the paper contains a conjecture.
http://arxiv.org/abs/0911.0152
Author(s): S. Albeverio and V.S. Koroliuk and I.V.Samoilenko
Abstract: Regular and singular parts of asymptotic expansions of semi-Markov random
evolutions are given. Regularity of boundary conditions is shown. An algorithm
for calculation of initial conditions is proposed.
http://arxiv.org/abs/0911.0162
Author(s): I.V.Samoilenko
Abstract: Is studied asymptotic expansion for solution of singularly perturbed equation
for Markov random evolution in Rd. The views of regular and singular parts of
solution are found.
http://arxiv.org/abs/0911.0163
Author(s): I.V.Samoilenko
Abstract: Weak convergence of the stochastic evolutionary system to the average
evolutionary system is proved. The method proposed by R.Liptser in for
semimartingales is used. But we apply a solution of singular perturbation
problem instead of ergodic theorem.
http://arxiv.org/abs/0911.0164
Author(s): I.V.Samoilenko
Abstract: Obvious view of distribution function of Markovian random evolution is found
in terms of Bessel functions of n+1-th order.
http://arxiv.org/abs/0911.0165
Author(s): V.S. Koroliuk and N. Limnios and I.V. Samoilenko
Abstract: In this paper, the weak convergence of impulsive recurrent process with
Markov switching in the scheme of Levy approximation is proved. For the
relative compactness, a method proposed by R. Liptser for semimartingales is
used with a modification, where we apply a solution of a singular perturbation
problem instead of an ergodic theorem.
http://arxiv.org/abs/0911.0168
Author(s): S. Albeverio and V. Koshmanenko and and I. Samoilenko
Abstract: We construct and study a discrete time model describing the conflict
interaction between two complex systems with non-trivial internal structures.
The external conflict interaction is based on the model of alternative
interaction between a pair of non-annihilating opponents. The internal conflict
dynamics is similar to the one of a predator-prey model. We show that the
typical trajectory of the complex system converges to an asymptotic attractive
cycle. We propose an interpretation of our model in terms of migration
processes.
http://arxiv.org/abs/0911.0170
Author(s): Zenghu Li and Li Wang
Abstract: We prove a fluctuating limit theorem of a sequence of super-Brownian motions
over $\mbb{R}$ with a single point catalyst. The weak convergence of the
processes on the space of Schwarz distributions is established. The limiting
process is an Ornstein-Uhlenbeck type process solving a Langevin type equation
driven by a one-dimensional Brownian motion.
http://arxiv.org/abs/0911.0219
Author(s): Yves F. Atchade and Gersende Fort
Abstract: We prove a central limit theorem for a general class of adaptive Markov Chain
Monte Carlo algorithms driven by sub-geometrically ergodic Markov kernels. We
discuss in detail the special case of stochastic approximation. We use the
result to analyze the asymptotic behavior of an adaptive version of the
Metropolis Adjusted Langevin algorithm with a heavy tailed target density.
http://arxiv.org/abs/0911.0221
Author(s): James R. Lee and Yuval Peres
Abstract: We prove that on any infinite, connected, locally finite, transitive graph G,
the probability of the random walk being within $\eps \sqrt{t}$ of the origin
after t steps is at most $O(\eps)$. A similar statement holds for finite
graphs, up to the relaxation time of the walk. Our approach uses non-constant
equivariant harmonic mappings taking values in a Hilbert space. For the special
case of discrete, amenable groups, we present a more explicit proof of the
Mok-Korevaar-Schoen theorem on existence of such harmonic maps by constructing
them from the heat flow on a Folner set.
http://arxiv.org/abs/0911.0274
Author(s): Thomas Cass and Zhongmin Qian and Jan Tudor
Abstract: We prove existence and uniqueness results for (mild) solutions to some
non-linear parabolic evolution equations with a rough forcing term. Our method
of proof relies on a careful exploitation of the interplay between the spatial
and time regularity of the solution by capitialising some of Kato's ideas in
semigroup theory. Classical Young integration theory is then shown to provide a
means of interpreting the equation. As an application we consider the three
dimensional Navier-Stokes system with a stochastic forcing term arising from a
fractional Brownian motion with h > 1/2.
http://arxiv.org/abs/0911.0281
Author(s): Micahel R\"ockner and Feng-Yu Wang
Abstract: A logarithmic type Harnack inequality is established for the semigroup of
solutions to a stochastic differential equation in Hilbert spaces with
non-additive noise. As applications, the strong Feller property as well as the
entropy-cost inequality for the semigroup are derived with respect to the
corresponding distance (cost function).
http://arxiv.org/abs/0911.0290
Author(s): Andrea Collevecchio and Tom Schmitz
Abstract: We develop a technique that provides a lower bound on the speed of transient
random walk in a random environment on regular trees. A refinement of this
technique yields upper bounds on the first regeneration level and regeneration
time. In particular, a lower and upper bound on the covariance in the annealed
invariance principle follows. We emphasize the fact that our methods are
general and also apply in the case of once-reinforced random walk. Durrett,
Kesten and Limic (2002) prove an upper bound of the form $b/(b+\delta)$ for the
speed on the $b$-ary tree, where $\delta$ is the reinforcement parameter. For
$\delta>1$ we provide a lower bound of the form $\gamma^2 b/(b+\delta)$, where
$\gamma$ is the survival probability of an associated branching process.
http://arxiv.org/abs/0911.0305
Author(s): Edward Crane and Nicholas Georgiou and Stanislav Volkov and Andrew Wade and Robert Waters
Abstract: We study a generalized Polya urn model with two types of ball. If the drawn
ball is red it is replaced together with a black ball, but if the drawn ball is
black it is replaced and a red ball is thrown out of the urn. When only black
balls remain, the roles of the colours are swapped and the process restarts. We
prove that the resulting Markov chain is transient but that if we throw out a
ball every time the colours swap, the process is positive-recurrent. We show
that the embedded process obtained by observing the number of balls in the urn
at the swapping times has a scaling limit that is essentially the square of a
Bessel diffusion. We consider an oriented percolation model naturally
associated with the urn process, and obtain detailed information about its
structure, showing that the open subgraph is an infinite tree with a single
end. We also study a natural continuous-time embedding of the urn process that
demonstrates the relation to the simple harmonic oscillator; in this setting
our transience result addresses an open problem in the recurrence theory of
two-dimensional linear birth and death processes due to Kesten and Hutton.
We obtain results on the area swept out by the process. We make use of
connections between the urn process and birth-death processes, a uniform
renewal process, the Eulerian numbers, and Lamperti's problem on processes with
asymptotically small drifts; we prove some new results on some of these
classical objects that may be of independent interest. For instance, we give
sharp new asymptotics for the first two moments of the counting function of the
uniform renewal process. Finally we discuss some related models of independent
interest, including a "Poisson earthquakes" Markov chain on the homeomorphisms
of the plane.
http://arxiv.org/abs/0911.0321
Author(s): Mathew D. Penrose and J. E. Yukich
Abstract: We consider the sum of power weighted nearest neighbor distances in a sample
of size n from a multivariate density f of possibly unbounded support. We give
various criteria guaranteeing that this sum satisfies a law of large numbers
for large n, correcting some inaccuracies in the literature on the way.
Motivation comes partly from the problem of consistent estimation of certain
entropies of f.
http://arxiv.org/abs/0911.0331
Author(s): Erkan Nane and Dongsheng Wu and Yimin Xiao
Abstract: For $0<\alpha \leq 2$ and $0
http://arxiv.org/abs/0911.0357
Author(s): Ernst Eberlein and Kathrin Glau and Antonis Papapantoleon
Abstract: This paper considers the valuation of exotic path-dependent options in L\'evy
models, in particular options on the supremum and the infimum of the asset
price process. Using the Wiener-Hopf factorization, we derive expressions for
the analytically extended characteristic function of the supremum and the
infimum of a L\'evy process. Combined with general results on Fourier methods
for option pricing, we provide formulas for the valuation of one-touch options,
lookback options and equity default swaps in L\'evy models.
http://arxiv.org/abs/0911.0373
Author(s): Roman Vershynin
Abstract: For probability distributions on R^n, we study the optimal sample size
N=N(n,p) that suffices to uniformly approximate the p-th moments of all
one-dimensional marginals. Under the assumption that the support of the
distribution lies in the Euclidean ball of radius \sqrt{n} and the marginals
have bounded 4p moments, we obtain the optimal bound N = O(n^{p/2}) for p > 2.
This bound goes in the direction of bridging the two recent results: a theorem
of Guedon and Rudelson which has an extra logarithmic factor in the sample
size, and a recent result of Adamczak, Litvak, Pajor and Tomczak-Jaegermann
which requires stronger subexponential moment assumptions.
http://arxiv.org/abs/0911.0391
Author(s): Wolfgang Woess
Abstract: This is a continuation of the study, begun by Ceccherini-Silberstein and
Woess, of context-free pairs of groups and the related context-free graphs in
the sense of Muller and Schupp. Instead of the cones (connected components with
respect to deletion of finite balls with respect to the graph metric), a more
general approach to context-free graphs is proposed via tree sets consisting of
cuts of the graph, and associated structure trees. The existence of tree sets
with certain "good" properties is studied. With a tree set, a natural
context-free grammar is associated. These investigations of the structure of
context free pairs, resp. graphs are then applied to study random walk
asymptotics via complex analysis.
http://arxiv.org/abs/0911.0134
Author(s): Wilfried Huss and Ecaterina Sava and Wolfgang Woess
Abstract: A language L over a finite alphabet is growth-sensitive (or entropy
sensitive) if forbidding any set of subwords F yields a sub-language L^F whose
exponential growth rate (entropy) is smaller than that of L. Let (X, E, l) be
an infinite, oriented, labelled graph. Considering the graph as an (infinite)
automaton, we associate with any pair of vertices x,y in X the language
consisting of all words that can be read as the labels along some path from x
to y. Under suitable, general assumptions we prove that these languages are
growth-sensitive. This is based on using Markov chains with forbidden
transitions.
http://arxiv.org/abs/0911.0142
Author(s): Cl\'ement Dombry (LMA) and Christian Mazza and Vincent Bansaye (CMAP)
Abstract: Organisms adapt to fluctuating environments by regulating their dynamics, and
by adjusting their phenotypes to environmental changes. We model population
growth using multitype branching processes in random environments, where the
offspring distribution of some organism having trait $t\in\cT$ in environment
$e\in\cE$ is given by some (fixed) distribution $\Upsilon_{t,e}$ on $\bbN$.
Then, the phenotypes are attributed using a distribution (strategy) $\pi_{t,e}$
on the trait space $\cT$. We look for the optimal strategy $\pi_{t,e}$,
$t\in\cT$, $e\in\cE$ maximizing the net growth rate or Lyapounov exponent, and
characterize the set of optimal strategies. This is considered for various
models of interest in biology: hereditary versus non-hereditary strategies and
strategies involving or not involving a sensing mechanism. Our main results are
obtained in the setting of non-hereditary strategies: thanks to a reduction to
simple branching processes in random environment, we derive an exact expression
for the net growth rate and a characterisation of optimal strategies. We also
focus on typical genealogies, that is, we consider the problem of finding the
typical lineage of a randomly chosen organism.
http://arxiv.org/abs/0912.1194
Author(s): James Parkinson and Bruno Schapira
Abstract: In this paper we outline an approach for analysing random walks on the
chambers of buildings. The types of walks that we consider are those which are
well adapted to the structure of the building: Namely walks with transition
probabilities $p(c,d)$ depending only on the Weyl distance $\delta(c,d)$. We
carry through the computations for thick locally finite affine buildings of
type $\tilde{A}_2$ to prove a local limit theorem for these buildings. The
technique centres around the representation theory of the associated Hecke
algebra. This representation theory is particularly well developed for affine
Hecke algebras, with elegant harmonic analysis developed by Opdam. We give an
introductory account of this theory in the second half of this paper.
http://arxiv.org/abs/0912.1301
Author(s): Yuri G. Kondratiev and Tobias Kuna and Maria Jo\~ao Oliveira and Jos\'e Lu\'is da Silva and Ludwig Streit
Abstract: An infinite particle system of independent jumping particles is considered.
Their constructions is recalled, further properties are derived, the relation
with hierarchical equations, Poissonian analysis, and second quantization are
discussed. The hydrodynamic limit for a general initial distribution satisfying
a mixing condition is derived. The large time asymptotic is computed under an
extra assumption.
http://arxiv.org/abs/0912.1312
Author(s): Terence Tao and Van Vu
Abstract: We study the eigenvalues values of the covariance matrix $\frac{1}{n} M^\ast
M$ of a large rectangular matrix $M = M_{n,p} = (\zeta_{ij})_{1 \leq i \leq p;
1 \leq j \leq n}$ whose entries are iid random variables of mean zero, variance
one, and having finite $C_0^{th}$ moment for some sufficiently large $C_0$.
The main result of this paper is a Four Moment Theorem for iid covariance
matrices (analogous to the Four Moment Theorem for Wigner matrices established
by the authors). Indeed, our arguments here draw heavily from those in our
previous work. As in that paper, we can use this theorem together with existing
results to establish universality of local statistics of eigenvalues under mild
conditions.
As a byproduct of our arguments, we also extend our previous results on
Wigner matrices to the case in which the entries have finite $C_0^{th}$ moment
rather than exponential decay.
http://arxiv.org/abs/0912.0966
Author(s): Cameron E. Freer and Daniel M. Roy
Abstract: We prove a uniformly computable version of de Finetti's theorem on
exchangeable sequences of real random variables. As a consequence, exchangeable
stochastic processes in probabilistic functional programming languages can be
automatically rewritten as procedures that do not modify non-local state. Along
the way, we prove that a distribution on the unit interval is computable if and
only if its moments are uniformly computable.
http://arxiv.org/abs/0912.1072
Author(s): Matthieu Josuat-Verg\`es
Abstract: We consider a partially asymmetric exclusion process (PASEP) on a finite
number of sites with open and directed boundary conditions. Its partition
function was calculated by Blythe, Evans, Colaiori, and Essler. It is known to
be a generating function of permutation tableaux by the combinatorial
interpretation of Corteel and Williams.
We prove bijectively two new combinatorial interpretations. The first one is
in terms of weighted Motzkin paths called Laguerre histories and is obtained by
refining a bijection of Foata and Zeilberger. Secondly we show that this
partition function is the generating function of permutations with respect to
right-to-left minima, right-to-left maxima, ascents, and 31-2 patterns, by
refining a bijection of Francon and Viennot.
Then we give a new formula for the partition function which generalizes the
one of Blythe & al. It is proved in two combinatorial ways. The first proof is
an enumeration of lattice paths which are known to be a solution of the Matrix
Ansatz of Derrida & al. The second proof relies on a previous enumeration of
rook placements, which appear in the combinatorial interpretation of a related
normal ordering problem. We also obtain a closed formula for the moments of
Al-Salam-Chihara polynomials.
http://arxiv.org/abs/0912.1279
Author(s): Francesco Russo (LAGA and CERMICS and MathFi) and Frederi G. Viens
Abstract: This paper considers the class of stochastic processes $X$ which are Volterra
convolutions of a martingale $M$. When $M$ is Brownian motion, $X$ is Gaussian,
and the class includes fractional Brownian motion and other Gaussian processes
with or without homogeneous increments. Let $m$ be an odd integer. Under some
technical conditions on the quadratic variation of $M$, it is shown that the
$m$-power variation exists and is zero when a quantity $\delta^{2}(r) $ related
to the variance of an increment of $M$ over a small interval of length $r$
satisfies $\delta(r) = o(r^{1/(2m)}) $. In the case of a Gaussian process with
homogeneous increments, $\delta$ is $X$'s canonical metric and the condition on
$\delta$ is proved to be necessary, and the zero variation result is extended
to non-integer symmetric powers. In the non-homogeneous Gaussian case, when
$m=3$, the symmetric (generalized Stratonovich) integral is defined, proved to
exist, and its It\^o's formula is proved to hold for all functions of class
$C^{6}$.
http://arxiv.org/abs/0912.0782
Author(s): Allan Gut (Uppsala University) and Ulrich Stadtmueller (University of Ulm)
Abstract: In some earlier work we have considered extensions of Lai's (1974) law of the
single logarithm for delayed sums to a multiindex setting with the same as well
as different expansion rates in the various dimensions. A further
generalization concerns window sizes that are regularly varying with index 1
(on the line). In the present paper we establish multiindex versions of the
latter as well as for some mixtures of expansion rates. In order to keep things
within reasonable size we confine ourselves to some special cases for the index
set $\mathbb{Z}_+^2$.
http://arxiv.org/abs/0912.0871
Author(s): Tilmann Gneiting
Abstract: Typically, point forecasting methods are compared and assessed by means of an
error measure or scoring function, such as the absolute error or the squared
error. The individual scores are then averaged over forecast cases, to result
in a summary measure of the predictive performance, such as the mean absolute
error or the (root) mean squared error. I demonstrate that this common practice
can lead to grossly misguided inferences, unless the scoring function and the
forecasting task are carefully matched.
Effective point forecasting requires that the scoring function be specified
ex ante, or that the forecaster receives a directive in the form of a
statistical functional, such as the mean or a quantile of the predictive
distribution. If the scoring function is specified ex ante, the forecaster can
issue the optimal point forecast, namely, the Bayes rule. If the forecaster
receives a directive in the form of a functional, it is critical that the
scoring function be consistent for it, in the sense that the expected score is
minimized when following the directive.
A functional is elicitable if there exists a scoring function that is
strictly consistent for it. Expectations, ratios of expectations and quantiles
are elicitable. For example, a scoring function is consistent for the mean
functional if and only if it is a Bregman function. It is consistent for a
quantile if and only if it is generalized piecewise linear. Similar
characterizations apply to ratios of expectations and to expectiles. Weighted
scoring functions are consistent for functionals that adapt to the weighting in
peculiar ways. Not all functionals are elicitable; for instance, conditional
value-at-risk is not, despite its popularity in quantitative finance.
http://arxiv.org/abs/0912.0902
Author(s): Satya N. Majumdar and Alain Comtet and Julien Randon-Furling
Abstract: In this paper we study the statistical properties of convex hulls of $N$
random points in a plane chosen according to a given distribution. The points
may be chosen independently or they may be correlated. After a non-exhaustive
survey of the somewhat sporadic literature and diverse methods used in the
random convex hull problem, we present a unifying approach, based on the notion
of support function of a closed curve and the associated Cauchy's formulae,
that allows us to compute exactly the mean perimeter and the mean area enclosed
by the convex polygon both in case of independent as well as correlated points.
Our method demonstrates a beautiful link between the random convex hull problem
and the subject of extreme value statistics. As an example of correlated
points, we study here in detail the case when the points represent the vertices
of $n$ independent random walks. In the continuum time limit this reduces to
$n$ independent planar Brownian trajectories for which we compute exactly, for
all $n$, the mean perimeter and the mean area of their global convex hull. Our
results have relevant applications in ecology in estimating the home range of a
herd of animals. Some of these results were announced recently in a short
communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].
http://arxiv.org/abs/0912.0631
Author(s): Shirshendu Chatterjee and Rick Durrett
Abstract: We consider a model for gene regulatory networks that is a modification of
Kauffmann's (1969) random Boolean networks. There are three parameters: $n =$
the number of nodes, $r =$ the number of inputs to each node, and $p =$ the
expected fraction of 1's in the Boolean functions at each node. Following a
standard practice in the physics literature, we use a threshold contact process
on a random graph on $n$ nodes, in which each node has in degree $r$, to
approximate its dynamics. We show that if $r\ge 3$ and $r \cdot 2p(1-p)>1$,
then the threshold contact process persists for a long time, which correspond
to chaotic behavior of the Boolean network. Unfortunately, we are only able to
prove the persistence time is $\ge \exp(cn^{b(p)})$ with $b(p)>0$ when $r\cdot
2p(1-p)> 1$, and $b(p)=1$ when $(r-1)\cdot 2p(1-p)>1$.
http://arxiv.org/abs/0911.5339
Author(s): Louis H.Y. Chen and Xiao Fang and Qi-Man Shao
Abstract: Stein's method is applied to obtain a Cram\'er type moderate deviation result
for dependent random variables whose dependence is defined in terms of a Stein
identity. The result is optimal when applied to the combinatorial central limit
theorem, the binary expansion of a random integer, the anti-voter model on a
complete graph, and the Curie-Weiss model.
http://arxiv.org/abs/0911.5373
Author(s): Ori Gurel-Gurevich and Ron Peled
Abstract: Let X be a Poisson point process of intensity lambda on the real line. A
thickening of it is a (deterministic) measurable function f such that the union
of X and f(X) is a Poisson point process of intensity lambda' where
lambda'>lambda. An equivariant thickening is a thickening which commutes with
all shifts of the line. We show that a thickening exists but an equivariant
thickening does not. We prove similar results for thickenings which commute
only with integer shifts and in the discrete and multi-dimensional settings.
This answers 3 questions of Holroyd, Lyons and Soo.
http://arxiv.org/abs/0911.5377
Author(s): Saul Jacka and Jon Warren and Peter Windridge
Abstract: Suppose we have three independent copies of a regular diffusion on [0,1] with
absorbing boundaries. Of these diffusions, either at least two are absorbed at
the upper boundary or at least two at the lower boundary. In this way, the
three diffusions determine a majority decision between 0 and 1. We show that
the strategy that always runs the process whose value is currently between the
other two reveals the majority decision whilst minimising the total time spent
running the diffusions.
http://arxiv.org/abs/0911.5413
Author(s): Yuri Bakhtin and Carl Mueller
Abstract: We introduce a probabilistic representation for solutions of quasilinear wave
equation with analytic nonlinearities. We use stochastic cascades to prove
existence and uniqueness of the solution.
http://arxiv.org/abs/0911.5450
Author(s): Vladislav Vysotsky
Abstract: Let $S_n$ be a centered random walk with a finite variance, and define the
new sequence $A_n:=\sum_{i=1}^n S_i$, which we call an integrated random walk.
We are interested in the asymptotics of $$p_N:=P(\min_{1 \le k \le N} A_k \ge
0)$$ as $N \to \infty$. Sinai (1992) proved that $p_N \asymp N^{-1/4}$ if $S_n$
is a simple random walk. We show that $p_N \asymp N^{-1/4}$ for some other
types of random walks that include double-sided exponential and double-sided
geometric walks, both not necessarily symmetric. We also prove that $p_N \le c
N^{-1/4}$ for lattice walks and for upper exponential walks, that are the walks
such that $Law (S_1 | S_1>0)$ is an exponential distribution.
http://arxiv.org/abs/0911.5456
Author(s): Raluca Balan and Sana Louhichi
Abstract: In this article, we examine the connection between the limit representation
of an infinitely divisible point process $N$ on $\bR \verb2\2 \{0\}$, and its
cluster representation. More precisely, if $(N_{i,n})_{i \leq k_n}$ are i.i.d.
point processes and $N_n=\sum_{i=1}^{k_n} N_{i,n}$ converges in distribution to
an ID point process $N$, which admits the cluster representation $N=\sum_{i
\geq 1}N_i$, then the pairs $(Y_i,N_i)$ of cluster centers and cluster members
can be obtained as the limit (in the point process sense) of the pairs
$(Y_{i,n},N_{i,n})_{i \leq k_n}$, where $Y_i$ and $Y_{i,n}$ are the "maximal"
points of $N_i$, respectively $N_{i,n}$ (in a certain sense).
http://arxiv.org/abs/0911.5471
Author(s): Alexey M. Kulik
Abstract: New relations between ergodic rate, L_p convergence rates, and asymptotic
behavior of tail probabilities for hitting times of a time homogeneous Markov
process are established. For L_p convergence rates and related spectral and
functional properties (spectral gap and Poincare inequality) sufficient
conditions are given in the terms of an exponential \phi-coupling. This
provides sufficient conditions for L_p convergence rates in the terms of
appropriate combination of `local mixing' and `recurrence' conditions on the
initial process, typical in the ergodic theory of Markov processes. The range
of application of the approach includes time-irreversible processes. In
particular, sufficient conditions for spectral gap property for Levy driven
Ornstein-Uhlenbeck process are established.
http://arxiv.org/abs/0911.5473
Author(s): V. Cammarota and A. Lachal and E. Orsingher
Abstract: We derive in detail four important results on integrals of Bessel functions
from which three combinatorial identities are extracted. We present the
probabilistic interpretation of these identities in terms of different types of
random walks, including asymmetric ones. This work extends the results of a
previous paper concerning the Darling-Siegert interpretation of similar
formulas emerging in the analysis of random flights.
http://arxiv.org/abs/0911.5519
Author(s): L. Avena and F. den Hollander and F. Redig
Abstract: Consider a one-dimensional shift-invariant attractive spin-flip system in
equilibrium, constituting a dynamic random environment, together with a
nearest-neighbor random walk that on occupied sites has a local drift to the
right but on vacant sites has a local drift to the left. In previous work we
proved a law of large numbers for dynamic random environments satisfying a
space-time mixing property called cone-mixing. If an attractive spin-flip
system has a finite average coupling time at the origin for two copies starting
from the all-occupied and the all-vacant configuration, respectively, then it
is cone-mixing.
In the present paper we prove a large deviation principle for the empirical
speed of the random walk, both quenched and annealed, and exhibit some
properties of the associated rate functions. Under an exponential space-time
mixing condition for the spin-flip system, which is stronger than cone-mixing,
the two rate functions have a unique zero, i.e., the slow-down phenomenon known
to be possible in a static random environment does not survive in a fast mixing
dynamic random environment. In contrast, we show that for the simple symmetric
exclusion dynamics, which is not cone-mixing (and which is not a spin-flip
system either), slow-down does occur.
http://arxiv.org/abs/0911.5629
Author(s): Christophe Gomez
Abstract: In shallow-water waveguides a propagating field can be decomposed over three
kinds of modes: the propagating modes, the radiating modes and the evanescent
modes. In this paper we consider the propagation of a wave in a randomly
perturbed waveguide and we analyze the coupling between these three kinds of
modes using an asymptotic analysis based on a separation of scales technique.
Then, we derive the asymptotic form of the distribution of the mode amplitudes
and the coupled power equation for propagating modes. From this equation, we
show that the total energy carried by the propagating modes decreases
exponentially with the size of the random section and we give an expression of
the decay rate. Moreover, we show that the mean propagating mode powers
converge to the solution of a diffusion equation in the high-frequency regime.
http://arxiv.org/abs/0911.5646
Author(s): Bo Chen and Matthias Winkel
Abstract: We introduce the notion of a restricted exchangeable partition of
$\mathbb{N}$ and study natural classes of such partitions. We obtain integral
representations, study associated coalescents and fragmentations, embeddings
into continuum random trees and convergence to such limit trees. As an
application, we deduce from the general theory developed here a particular
limit result conjectured previously for Ford's alpha model and its non-binary
extension, the alpha-gamma model, where restricted exchangeability arises
naturally.
http://arxiv.org/abs/0911.5647
Author(s): Nicholas Crawford and Allan Sly
Abstract: We study limit laws for simple random walks on supercritical long range
percolation clusters on $\Z^d, d \geq 1$. For the long range percolation model,
the probability that two vertices $x, y$ are connected behaves asymptotically
as $\|x-y\|_2^{-s}$. {When} $s\in(d, d+1)$, we prove that the scaling limit of
simple random walk on the infinite component converges to an $\alpha$-stable
L\'evy process with $\alpha = s-d$ establishing a conjecture of Berger and
Biskup \cite{Berger-Biskup}. The convergence holds in both the quenched and
annealed senses. In the case where $d=1$ and $s>2$ we show that the simple
random walk converges to a Brownian motion.
http://arxiv.org/abs/0911.5668
Author(s): Vassili N. Kolokoltsov
Abstract: Ito's construction of Markovian solutions to stochastic equations driven by a
L\'evy noise is extended to nonlinear distribution dependent integrands aiming
at the effective construction of linear and nonlinear Markov semigroups and the
corresponding processes with a given pseudo-differential generator. It is shown
that a conditionally positive integro-differential operator (of the
L\'evy-Khintchine type) with variable coefficients (diffusion, drift and L\'evy
measure) depending Lipschitz continuously on its parameters (position and/or
its distribution) generates a linear or nonlinear Markov semigroup, where the
measures are metricized by the Wasserstein-Kantorovich metrics. This is a
nontrivial but natural extension to general Markov processes of a long known
fact for ordinary diffusions.
http://arxiv.org/abs/0911.5688
Author(s): Sourav Chatterjee and Partha S. Dey
Abstract: We prove that first-passage percolation times across thin cylinders of the
form $[0,n]\times [-h_n,h_n]^{d-1}$ obey Gaussian central limit theorems as
long as $h_n$ grows slower than $n^{1/(d+1)}$. It is an open question as to
what is the fastest that $h_n$ can grow so that a Gaussian CLT still holds. We
conjecture that $n^{1/(d+1)}$ is the right answer for $d\ge 2$.
http://arxiv.org/abs/0911.5702
Author(s): Laura Cottatellucci and Ralf R. Mueller and and Merouane Debbah
Abstract: Spectral efficiency for asynchronous code division multiple access (CDMA)
with random spreading is calculated in the large system limit allowing for
arbitrary chip waveforms and frequency-flat fading. Signal to interference and
noise ratios (SINRs) for suboptimal receivers, such as the linear minimum mean
square error (MMSE) detectors, are derived. The approach is general and
optionally allows even for statistics obtained by under-sampling the received
signal.
All performance measures are given as a function of the chip waveform and the
delay distribution of the users in the large system limit. It turns out that
synchronizing users on a chip level impairs performance for all chip waveforms
with bandwidth greater than the Nyquist bandwidth, e.g., positive roll-off
factors. For example, with the pulse shaping demanded in the UMTS standard,
user synchronization reduces spectral efficiency up to 12% at 10 dB normalized
signal-to-noise ratio. The benefits of asynchronism stem from the finding that
the excess bandwidth of chip waveforms actually spans additional dimensions in
signal space, if the users are de-synchronized on the chip-level. The analysis
of linear MMSE detectors shows that the limiting interference effects can be
decoupled both in the user domain and in the frequency domain such that the
concept of the effective interference spectral density arises. This generalizes
and refines Tse and Hanly's concept of effective interference.
In Part II, the analysis is extended to any linear detector that admits a
representation as multistage detector and guidelines for the design of low
complexity multistage detectors with universal weights are provided.
http://arxiv.org/abs/0911.5385
Author(s): Benoit Collins and Piotr Sniady
Abstract: We study asymptotics of representations of the unitary groups U(n) in the
limit n\to\infty and we show that in many aspects they behave like large random
matrices. In particular, we show that the highest weight of a random
irreducible component in the Kronecker tensor product of two irreducible
representations behaves asymptotically in the same way as the spectrum of the
sum of two large random matrices with prescribed eigenvalues. This agreement
happens not only on the level of the mean values (and thus can be described
within Voiculescu's free probability theory) but also on the level of
fluctuations (and thus can be described within the framework of higher order
free probability).
http://arxiv.org/abs/0911.5546
Author(s): Matthias R\"oger and Hendrik Weber
Abstract: We study an Allen-Cahn equation perturbed by a multiplicative stochastic
noise which is white in time and correlated in space. Formally this equation
approximates a stochastically forced mean curvature flow. We derive uniform
energy bounds and prove tightness of the approximating sequence and convergence
to phase-indicator functions.
http://arxiv.org/abs/0911.5706
Author(s): Florian Voss and Catherine Gloaguen and Volker Schmidt
Abstract: We consider spatial stochastic models, which can be applied e.g. to
telecommunication networks with two hierarchy levels. In particular, we
consider two Cox processes concentrated on the edge set of a random
tessellation, where the points can describe the locations of low-level and
high-level network components, respectively, and the edge set the underlying
infrastructure of the network, like road systems, railways, etc. Furthermore,
each low-level component is marked with the shortest path along the edge set to
the nearest high-level component. We investigate the typical shortest path
length of the resulting marked point process, which is an important
characteristic e.g. in performance analysis and planning of telecommunication
networks. In particular, we show that its distribution converges to simple
parametric limit distributions if a certain scaling factor converges to zero
and infinity, respectively. This can be used to approximate the density of the
typical shortest path length by analytical formulae.
http://arxiv.org/abs/0912.4516
Author(s): Mirko D'Ovidio
Abstract: In this paper we deal with the generalized Gamma processes and their
compositions. For the compositions of two or more than two generalized Gamma
processes we give, when possible, the explicit law whereas, in the other cases
the representations in terms of Fox's H-functions are given. We also study the
connections between iteration and product of random processes by exploiting the
properties of the generalized Gamma processes, such a study allows us to obtain
some striking result about the compositions of the Cauchy processes or
fractional Brownian motions. Furthermore, we find out the partial differential
equations governing the generalized Gamma processes and their compositions
http://arxiv.org/abs/0912.4522
Author(s): Rafa{\l} {\L}ochowski
Abstract: In the paper "On Truncated Variation of Brownian Motion with Drift" (Bull.
Pol. Acad. Sci. Math. 56 (2008), no.4, 267 - 281) we defined truncated
variation of Brownian motion with drift, $W_t = B_t + \mu t, t\geq 0,$ where
$(B_t)$ is a standard Brownian motion. For positive $c$ we define two related
quantities - upward truncated variation UTV^c_{\mu}[a,b]=\sup_n \sup_{a \leq
t_1 < s_1 < ... < t_n < s_n \leq b} \sum_{i=1}^{n} \max {W_{s_i} - W_{t_i} - c,
0} and, analogously, downward truncated variation DTV^c_{\mu}[a,b]=\sup_n
\sup_{a \leq t_1 < s_1 < ... < t_n < s_n \leq b} \sum_{i=1}^{n} \max {W_{t_i} -
W_{s_i} - c, 0} We prove that exponential moments of the above quantities are
finite (in opposite to the regular variation, corresponding to $TV^0$, which is
infinite almost surely). We present estimates of the expected value of $% UTV^c
$ up to universal constants.
As an application we give some estimates of the maximal possible gain from
trading a financial asset in the presence of flat commission (proportional to
the value of the transaction) when the dynamics of the prices of the asset
follows a geometric Browniam motion process. In the presented estimates upward
truncated variation appears naturally.
http://arxiv.org/abs/0912.4533
Author(s): Federico Dalmao and Ernesto Mordecki
Abstract: Consider a flock of birds that fly interacting between them. The interactions
are modelled through a hierarchical system in which each bird, at each time
step, adjusts its own velocity according to his past velocity and a weighted
mean of the relative velocities of its superiors in the hierarchy. We consider
the additional fact, that each of the birds can fail to see any of its
superiors with certain probability, that can depend on the distances between
them. For this model with random interactions we prove that the flocking
phenomena, obtained for similar deterministic models, holds true.
http://arxiv.org/abs/0912.4535
Author(s): Kenneth S. Alexander
Abstract: We consider reflecting random walks on the nonnegative integers with drift of
order 1/x at height x. We establish explicit asymptotics for various
probabilities associated to such walks, including the distribution of the
hitting time of 0 and first return time to 0, and the probability of being at a
given height k at time n (uniformly in a large range of k.) In particular, for
drift of form -\delta/2x + o(1/x) with \delta > -1, we show that the
probability of a first return to 0 at time n is asymptotically n^{-c}\phi(n),
where c = (3+\delta)/2 and \phi is a slowly varying function given explicitly
in terms of the o(1/x) terms.
http://arxiv.org/abs/0912.4550
Author(s): P. Caputo and A. Faggionato and T. Prescott
Abstract: We consider a random walk on a homogeneous Poisson point process with energy
marks. The jump rates decay exponentially in the A-power of the jump length and
depend on the energy marks via a Boltzmann--like factor. The case A=1
corresponds to the phonon-induced Mott variable range hopping in disordered
solids in the regime of strong Anderson localization. We prove that for almost
every realization of the marked process, the diffusively rescaled random walk,
with arbitrary start point, converges to a Brownian motion whose diffusion
matrix is positive definite, and independent of the environment. Finally, we
extend the above result to other point processes including diluted lattices.
http://arxiv.org/abs/0912.4591
Author(s): Sergio Albeverio and Astrid Hilbert and Vassily Kolokoltsov
Abstract: We analyze the asymptotic behaviour of the heat kernel defined by a
stochastically perturbed geodesic flow on the cotangent bundle of a Riemannian
manifold for small time and small diffusion parameter. This extends WKB-type
methods to a particular case of a degenerate Hamiltonian. We derive uniform
bounds for the solution of the degenerate Hamiltonian boundary value problem
for small time. From this equivalence of solutions of the Hamiltonian equations
and the corresponding Hamilton Jacobi equation follows. The results are
exploited to derive two sided estimates and multiplicative asymptotics for the
heat kernel and the trace.
http://arxiv.org/abs/0912.4683
Author(s): Yueyun Hu and Davar Khoshnevisan
Abstract: We study the large-time behavior of the charged-polymer Hamiltonian $H_n$ of
Kantor and Kardar [Bernoulli case] and Derrida, Griffiths, and Higgs [Gaussian
case], using strong approximations to Brownian motion. Our results imply, among
other things, that in one dimension the process $\{H_{[nt]}\}_{0\le t\le 1}$
behaves like a Brownian motion, time-changed by the intersection local-time
process of an independent Brownian motion. Chung-type LILs are also discussed.
http://arxiv.org/abs/0911.3895
Author(s): Masayuki Kumon and Akimichi Takemura and Kei Takeuchi
Abstract: We propose a sequential optimizing betting strategy in the multi-dimensional
bounded forecasting game in the framework of game-theoretic probability of
Shafer and Vovk (2001). By studying the asymptotic behavior of its capital
process, we prove a generalization of the strong law of large numbers, where
the convergence rate of the sample mean vector depends on the growth rate of
the quadratic variation process. The growth rate of the quadratic variation
process may be slower than the number of rounds or may even be zero. We also
introduce an information criterion for selecting efficient betting items. These
results are then applied to multiple asset trading strategies in discrete-time
and continuous-time games. In the case of continuous-time game we present a
measure of the jaggedness of a vector-valued continuous process. Our results
are examined by several numerical examples.
http://arxiv.org/abs/0911.3933
Author(s): Fredrik Johansson and Gregory F. Lawler
Abstract: The tip multifractal spectrum of a two-dimensional curve is one way to
describe the behavior of the uniformizing map of the complement near the tip.
We give the tip multifractal spectrum for a Schramm-Loewner evolution (SLE)
curve, we prove that the spectrum is valid with probability one, and we give
applications to the scaling of harmonic measure at the tip.
http://arxiv.org/abs/0911.3983
Author(s): Christian Benes and Fredrik Johansson and Michael J. Kozdron
Abstract: We derive a rate of convergence of the Loewner driving function for
loop-erased random walk to Brownian motion with speed 2 on the unit circle, the
Loewner driving function for radial SLE(2).
http://arxiv.org/abs/0911.3988
Author(s): Paul-Olivier Dehaye and Dirk Zeindler
Abstract: The second author had previously obtained explicit generating functions for
moments of characteristic polynomials of permutation matrices (n points). In
this paper, we generalize many aspects of this situation. We introduce random
shifts of the eigenvalues of the permutation matrices, in two different ways:
independently or not for each subset of eigenvalues associated to the same
cycle. We also consider vastly more general functions than the characteristic
polynomial of a permutation matrix, by first finding an equivalent definition
in terms of cycle-type of the permutation. We consider other groups than the
symmetric group, for instance the alternating group and other Weyl groups.
Finally, we compute some asymptotics results when n tends to infinity. This
last result requires additional ideas: it exploits properties of the Feller
coupling, which gives asymptotics for the lengths of cycles in permutations of
many points.
http://arxiv.org/abs/0911.4038
Author(s): Stefano Bonaccorsi and Ciprian Tudor (LPP)
Abstract: We study a class of stochastic evolution equations with a dissipative forcing
nonlinearity and additive noise. The noise is assumed to satisfy rather general
assumptions about the form of the covariance function; our framework covers
examples of Gaussian processes, like fractional and bifractional Brownian
motion and also non Gaussian examples like the Hermite process. We give an
application of our results to the study of the stochastic version of a common
model of potential spread in a dendritic tree. Our investigation is specially
motivated by possibility to introduce long-range dependence in time of the
stochastic perturbation.
http://arxiv.org/abs/0911.4092
Author(s): I. Arhosalo and E. J\"arvenp\"a\"a and M. J\"arvenp\"a\"a and M. Rams and P. Shmerkin
Abstract: We study dimensional properties of visible parts of fractal percolation in
the plane. Provided that the dimension of the fractal percolation is at least
1, we show that, conditioned on non-extinction, almost surely all visible parts
from lines are 1-dimensional. Furthermore, almost all of them have positive and
finite Hausdorff measure. We also verify analogous results for visible parts
from points. These results are motivated by an open problem on the dimensions
of visible parts.
http://arxiv.org/abs/0911.3931
Author(s): Sourav Chatterjee
Abstract: The Ghirlanda-Guerra identities are one of the most mysterious features of
spin glasses. We prove the GG identities in a large class of models that
includes the Edwards-Anderson model, the random field Ising model, and the
Sherrington-Kirkpatrick model in the presence of a random external field.
Previously, the GG identities were rigorously proved only `on average' over a
range of temperatures or under small perturbations.
http://arxiv.org/abs/0911.4520
Author(s): Alexandre Gaudilliere (LATP) and Julien Reygner (CMAP)
Abstract: Through a Metropolis-like algorithm with single step computational cost of
order one, we build a Markov chain that relaxes to the canonical Fermi
statistics for k non-interacting particles among m energy levels. Uniformly
over the temperature as well as the energy values and degeneracies of the
energy levels we give an explicit upper bound with leading term km(ln k) for
the mixing time of the dynamics. We obtain such construction and upper bound as
a special case of a general result on (non-homogeneous) products of ultra
log-concave measures (like binomial or Poisson laws) with a global constraint.
As a consequence of this general result we also obtain a disorder-independent
upper bound on the mixing time of a simple exclusion process on the complete
graph with site disorder. This general result is based on an elementary
coupling argument and extended to (non-homogeneous) products of log-concave
measures.
http://arxiv.org/abs/0911.4565
Author(s): Eva Loecherbach
Abstract: We consider an interacting particle system on $\Z^d$ with finite state space
and interactions of infinite range in a high-noise regime. Assuming that the
rate of change is continuous and that a Dobrushin-like condition holds, we show
that the process is recurrent in the sense of Harris and construct explicit
regeneration times for the process in restriction to finite cylinder sets. We
show that the length of a regeneration period admits exponential moments. The
proof that regeneration times are almost surely finite relies on a coupled
construction of generalized house-of-cards chains.
http://arxiv.org/abs/0911.4572
Author(s): Isabelle Camilier (LTCI) and Laurent Decreusefond (LTCI)
Abstract: Determinantal and permanental processes are point processes with a
correlation function given by a determinant or a permanent. Their atoms exhibit
mutual attraction of repulsion, thus these processes are very far from the
uncorrelated situation encountered in Poisson models. We establish a
quasi-invariance result : we show that if atoms locations are perturbed along a
vector field, the resulting process is still a determinantal (respectively
permanental) process, the law of which is absolutely continuous with respect to
the original distribution. Based on this formula, following Bismut approach of
Malliavin calculus, we then give an integration by parts formula.
http://arxiv.org/abs/0911.4638
Author(s): Felix Lindner and Ren\'e L. Schilling
Abstract: We study the approximation of the distribution of $X_T$, where
$(X_t)_{t\in[0,T]}$ is a Hilbert space valued stochastic process that solves a
linear parabolic stochastic partial differential equation driven by an
impulsive space time noise, \[dX_t+AX_t dt= Q^{1/2} dZ_t,\quad X_0=x_0\in
H,\quad t\in [0,T].\] Here $(Z_t)_{t\in[0,T]}$ is an impulsive cylindrical
process and $Q$ is the covariance operator of the noise; we assume that
$A^{-\alpha}$ has finite trace for some $\alpha>0$ and that $A^\beta Q$ is
bounded for some $\beta\in (\alpha-1,\alpha]$. A discretized solution
$(X_h^n)_{n\in\{0,1,...,N\}}$ is defined via the finite element method in space
(parameter $h>0$) and a $\theta$-method in time (parameter $\Delta t=T/N$). For
$\varphi \in C^2_b(H;\R)$ we show an integral representation for the error
$|\E\varphi(X^N_h)-\E\varphi(X_T)|$ and prove that
\[|\E\varphi(X^N_h)-\E\varphi(X_T)|=O(h^{2\gamma}+(\Delta t)^{\gamma})\] where
$\gamma<1-\alpha+\beta$. This is the same order of convergence as in the case
of a Gaussian space time noise, which has been obtained in a paper by A.
Debussche and J. Printems \cite{DebPrin}. Our result also holds for a
combination of impulsive and Gaussian space time noise.
http://arxiv.org/abs/0911.4681
Author(s): Victoria P. Knopova and Alexey M. Kulik
Abstract: We develop a version of the saddle point method which allows us to give exact
symptotic behavior of (a) the transition probability density of a real-valued
Levy process; (b) the transition probability density of a Levy driven
Ornstein-Uhlenbeck process; (c) the density of the invariant distribution of a
Levy driven Ornstein-Uhlenbeck process. Using this method we give explicit
asymptotic expressions of transition probability densities and describe the
asymptotic behavior of the ratio of invariant distribution densities.
http://arxiv.org/abs/0911.4683
Author(s): Makoto Maejima and Yohei Ueda
Abstract: In this paper, three topics on semi-selfdecomposable distributions are
studied. The first one is to characterize semi-selfdecomposable distributions
by stochastic integrals with respect to Levy processes. This characterization
defines a mapping from an infinitely divisible distribution with finite
log-moment to a semi-selfdecomposable distribution. The second one is to
introduce and study a Langevin type equation and the corresponding
Ornstein-Uhlenbecktype process whose limiting distribution is
semi-selfdecomposable. Also, semi-stationary Ornstein-Uhlenbeck type processes
with semi-selfdecomposable distributions are constructed. The third one is to
study the iteration of the mapping above. The iterated mapping is expressed as
a single mapping with a different integrand. Also, nested subclasses of the
class of semi-selfdecomposable distributions are considered, andit is shown
that the limit of these nested subclasses is the closure of the class of
semi-stable distributions.
http://arxiv.org/abs/0911.3449
Author(s): Mingshang Hu and Shige Peng
Abstract: We introduce G-L\'{e}vy processes which develop the theory of processes with
independent and stationary increments under the framework of sublinear
expectations. We then obtain the L\'{e}vy-Khintchine formula and the existence
for G-L\'{e}vy processes. We also introduce G-Poisson processes.
http://arxiv.org/abs/0911.3533
Author(s): Mihai Gradinaru (IRMAR) and Yoann Offret (IRMAR)
Abstract: Let $X$ a solution of the time-inhomogeneous stochastic differential equation
driven by a Brownian motion with drift coefficient $b(t,x)=\rho\,{\rm
sgn}(x)\frac{|x|^\alpha}{t^\beta}$. This process can be viewed as a distorted
Brownian motion in a potential possibly singular, depending on time. After
obtaining results on the existence and the uniqueness of solution, we study its
asymptotic behaviour and made a precise description, in terms of parameters
$\rho,\alpha$ and $\beta$, of the recurrence, transience and convergence. More
precisely, asymptotic distributions, iterated logarithm type laws and rates of
transience are proved for such processes.
http://arxiv.org/abs/0911.3534
Author(s): Le Yang (LMA)
Abstract: In this paper, we define the geometric median of a probability measure on a
Riemannian manifold, give its characterization and a natural condition to
ensure its uniqueness. In order to calculate the median in practical cases, we
also propose a subgradient algorithm and prove its convergence as well as
estimating the error of approximation and the rate of convergence. The
convergence property of this subgradient algorithm, which is a generalization
of the classical Weiszfeld algorithm in Euclidean spaces to the context of
Riemannian manifolds, also answers a recent question in P. T. Fletcher et al.
[13]
http://arxiv.org/abs/0911.3474
Author(s): Michael Lugo
Abstract: In this article we consider the cycle structure of compositions of pairs of
involutions in the symmetric group S_n chosen uniformly at random. These can be
modeled as modified 2-regular graphs, giving rise to exponential generating
functions. A composition of two random involutions in S_n typically has about
n^(1/2) cycles, and the cycles are characteristically of length n^(1/2).
Compositions of two random fixed-point-free involutions, on the other hand,
typically have about log n cycles and are closely related to permutations with
all cycle lengths even. The number of factorizations of a random permutation
into two involutions appears to be asymptotically lognormally distributed,
which we prove for a closely related probabilistic model. This study is
motivated by the observation that the number of involutions in [n] is
(n!)^(1/2) times a subexponential factor; more generally the number of
permutations with all cycle lengths in a finite set S is n!^(1-1/m) times a
subexponential factor, and the typical number of k-cycles is nearly n^(k/m)/k.
Connections to pattern avoidance in involutions are also considered.
http://arxiv.org/abs/0911.3604
Author(s): Zhen-Qing Chen and Kazuhiro Kuwae
Abstract: We establish the equivalence of the analytic and probabilistic notions of
subharmonicity in the framework of general symmetric Hunt processes on locally
compact separable metric spaces, extending an earlier work of the first named
author on the equivalence of the analytic and probabilistic notions of
harmonicity. As a corollary, we prove a strong maximum principle for locally
bounded finely continuous subharmonic functions in the space of functions
locally in the domain of the Dirichlet form under some natural conditions.
http://arxiv.org/abs/0912.3290
Author(s): Nikolai Leonenko and Ludmila Sakhno
Abstract: We study the representations of tensor random fields on the sphere basing on
the theory of representations of the rotation group. Introducing specific
components of a tensor field and imposing the conditions of weak isotropy and
mean square continuity, we derive their spectral decompositions in terms of
generalized spherical functions. The properties of random coefficients of the
decompositions are characterized, including such an important question as
conditions of Gaussianity.
http://arxiv.org/abs/0912.3389
Author(s): Gesine Reinert and Adrian R\"ollin
Abstract: In a recent paper by the authors, a new approach--called the "embedding
method"--was introduced, which allows to make use of exchangeable pairs for
normal and multivariate normal approximation with Stein's method in cases where
the corresponding couplings do not satisfy a certain linearity condition. The
key idea is to embed the problem into a higher dimensional space in such a way
that the linearity condition is then satisfied. Here we apply the embedding to
U-statistics as well as to subgraph counts in random graphs.
http://arxiv.org/abs/0912.3425
Author(s): Ying Jiao (PMA)
Abstract: We study multiple defaults where the global market information is modelled as
progressive enlargement of filtrations. We shall provide a general pricing
formula by establishing a relationship between the enlarged filtration and the
reference default-free filtration in the random measure framework. On each
default scenario, the formula can be interpreted as a Radon-Nikodym derivative
of random measures. The contagion risks are studied in the multi-defaults
setting where we consider the optimal investment problem in a contagion risk
model and show that the optimization can be effectuated in a recursive manner
with respect to the default-free filtration.
http://arxiv.org/abs/0912.3132
Author(s): S. N. Stelmastchuk
Abstract: Let $(M,g)$ be any Riemannian manifold. Our goal is to show that if $g$ and
Ricci tensor $r_{g}$ are no locally constant, if, locally, their product is
non-negative (respectively, non-positive), and if its scalar curvature $s_{g}$
is non-negative (respectively, non-positive), then $(M,g)$ is an Einstein
manifolds. This result is a generalization of the characterization for compacts
Einstein manifolds given by Hilbert.
http://arxiv.org/abs/0912.3436
Author(s): Ruoting Gong and Christian Houdr\'e
Abstract: We develop a stochastic control system from a continuous-time
Principal-Agent model in which both the principal and the agent have
imperfect information and different beliefs about the project. We consider the
agent's problem in this stochastic control system, i.e., we attempt to optimize
the agent's utility function under the agent's belief. Via the corresponding
Hamilton-Jacobi-Bellman equation the value function is shown to be jointly
continuous and to satisfy the Dynamic Programming Principle. These properties
directly lead to the conclusion that the value function is a viscosity solution
of the HJB equation. Uniqueness is then also established.
http://arxiv.org/abs/0911.0956
Author(s): Michal Baran and Jerzy Zabczyk
Abstract: The problem of existence of solution for the Heath-Jarrow-Morton equation
with linear volatility and purely jump random factor is studied. Sufficient
conditions for existence and non-existence of the solution in the class of
bounded fields are formulated. It is shown that if the first derivative of the
Levy-Khinchin exponent grows slower then logarithmic function then the answer
is positive and if it is bounded from below by a fractional power function of
any positive order then the answer is negative. Numerous examples including
models with Levy measures of stable type are presented.
http://arxiv.org/abs/0911.1119
Author(s): C. Houdr\'e and H. Matzinger
Abstract: We investigate the nature of the alignment with gaps corresponding to a
Longest Common Subsequence (LCS) of two random sequences. We show that such an
alignment, which we call optimal, typically matches pieces of similar length.
This is of importance in order to understand the structure of optimal
alignments. We also establish a method for showing that a certain class of
properties typically holds in most parts of the optimal alignment. The
assumption being that the property considered has high probability to hold for
strings of similar short length. The present result is part of our general
effort to obtain the order of the variance of the LCS of random strings.
http://arxiv.org/abs/0911.2031
Author(s): Matus Telgarsky
Abstract: An alternate form for the binomial tail is presented, which leads to a
variety of bounds for the central tail. A few can be weakened into the
corresponding Chernoff and Slud bounds, which not only demonstrates the quality
of the presented bounds, but also provides alternate proofs for the classical
bounds.
http://arxiv.org/abs/0911.2077
Author(s): P. G. L. Porta Mana
Abstract: What is the relationship between plausibility logic and the principle of
maximum entropy? When does the principle give unreasonable or wrong results?
When is it appropriate to use the rule `expectation = average'? Can
plausibility logic give the same answers as the principle, and better answers
if those of the principle are unreasonable? To try to answer these questions,
this study offers a numerical collection of plausibility distributions given by
the maximum-entropy principle and by plausibility logic for a set of fifteen
simple problems: throwing dice.
http://arxiv.org/abs/0911.2197
Author(s): Pierre Br\'emaud and Serguei Foss
Abstract: We prove ergodicity of a point process earthquake model combining the
classical stress release model for primary shocks with the Hawkes model for
aftershocks.
http://arxiv.org/abs/0912.0551
Author(s): Pieter C. Allaart
Abstract: Let X_t, 0<=t<=T be a one-dimensional stochastic process with independent and
stationary increments. This paper considers the problem of stopping the process
X_t "as close as possible" to its eventual supremum M_T:=sup{X_t: 0<=t<=T},
when the reward for stopping with a stopping time tau<=T is a nonincreasing
convex function of M_T-X_tau. Under fairly general conditions on the process
X_t, it is shown that the optimal stopping time tau is of "bang-bang" form: it
is either optimal to stop at time 0 or at time T. For the case of random walk,
the rule tau=T is optimal if the steps of the walk stochastically dominate
their opposites, and the rule tau=0 is optimal if the reverse relationship
holds. For Le'vy processes X_t with finite Le'vy measure, an analogous result
is proved assuming that the jumps of X_t satisfy the above condition, and the
drift of X_t has the same sign as the mean jump. Finally, conditions are given
under which the result can be extended to the case of nonfinite Le'vy measure.
http://arxiv.org/abs/0912.0615
Author(s): Patricia Goncalves and Milton Jara
Abstract: We consider the one-dimensional totally asymmetric zero-range starting from a
step decreasing profile leading in the hydrodynamic limit to the rarefaction
fan of the associated hydrodynamic equation. We show that the sum of joint
probabilities for second class particles sharing the same site, is convergent
and we compute its limit. We derive the Law of Large Numbers for the position
of a second class particle initially at the origin under the initial state in
which all positive sites are occupied and all negative sites are empty and also
for a slight perturbation of the invariant state.
http://arxiv.org/abs/0912.0640
Author(s): Andrea Montanari and Elchanan Mossel and Allan Sly
Abstract: We consider the Ising model with inverse temperature beta and without
external field on sequences of graphs G_n which converge locally to the
k-regular tree. We show that for such graphs the Ising measure locally weak
converges to the symmetric mixture of the Ising model with + boundary
conditions and the - boundary conditions on the k-regular tree with inverse
temperature \beta. In the case where the graphs G_n are expanders we derive a
more detailed understanding by showing convergence of the Ising measure
condition on positive magnetization (sum of spins) to the + measure on the
tree.
http://arxiv.org/abs/0912.0719
Author(s): Ilya Tyurin
Abstract: We investigate the convergence rate in the Lyapunov theorem when the third
absolute moments exist. By means of convex analysis we obtain the sharp
estimate for the distance in the mean metric between a probability distribution
and its zero bias transformation. This bound allows to derive new estimates of
the convergence rate in terms of Kolmogorov's metric as well as the metrics
$\zeta_r$ (r=1,2,3) introduced by Zolotarev. The estimate for $\zeta_3$ is
optimal. Moreover, we show that the constant in the classical Berry-Esseen
theorem can be taken as 0.4785. In addition, the non-i.i.d. analogue of this
theorem with the constant 0.5606 is provided.
http://arxiv.org/abs/0912.0726
Author(s): Oliver Johnson and Ioannis Kontoyiannis and Mokshay Madiman
Abstract: Sufficient conditions are developed, under which the compound Poisson
distribution has maximal entropy within a natural class of probability measures
on the nonnegative integers. Recently, one of the authors [O. Johnson, {\em
Stoch. Proc. Appl.}, 2007] used a semigroup approach to show that the Poisson
has maximal entropy among all ultra-log-concave distributions with fixed mean.
We show via a non-trivial extension of this semigroup approach that the natural
analog of the Poisson maximum entropy property remains valid if the compound
Poisson distributions under consideration are log-concave, but that it fails in
general. A parallel maximum entropy result is established for the family of
compound binomial measures. Sufficient conditions for compound distributions to
be log-concave are discussed and applications to combinatorics are examined;
new bounds are derived on the entropy of the cardinality of a random
independent set in a claw-free graph, and a connection is drawn to Mason's
conjecture for matroids. The present results are primarily motivated by the
desire to provide an information-theoretic foundation for compound Poisson
approximation and associated limit theorems, analogous to the corresponding
developments for the central limit theorem and for Poisson approximation. Our
results also demonstrate new links between some probabilistic methods and the
combinatorial notions of log-concavity and ultra-log-concavity, and they add to
the growing body of work exploring the applications of maximum entropy
characterizations to problems in discrete mathematics.
http://arxiv.org/abs/0912.0581
Author(s): Rapha\"el Rossignol and Marie Th\'eret
Abstract: Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random capacities.
A law of large numbers is known for the maximal flow crossing a rectangle in
$\mathbb{R}^2$ when the side lengths of the rectangle go to infinity. We prove
that the lower large deviations are of surface order, and we prove the
corresponding large deviation principle from below. This extends and improves
previous large deviations results of Grimmett and Kesten (1984) obtained for
boxes of particular orientation.
http://arxiv.org/abs/0912.3601
Author(s): David Nualart and Lluis Quer-Sardanyons
Abstract: In this note, we establish optimal lower and upper Gaussian bounds for the
density of the solution to a class of stochastic integral equations driven by
an additive spatially homogeneous Gaussian random field. The proof is based on
the techniques of the Malliavin calculus and a density formula obtained by
Nourdin and Viens. Then, the main result is applied to the mild solution of a
general class of SPDEs driven by a Gaussian noise which is white in time and
has a spatially homogeneous correlation. In particular, this covers the case of
the stochastic heat and wave equations in $\mathbb{R}^d$ with $d\geq 1$ and
$d\leq 3$, respectively. The upper and lower Gaussian bounds have the same form
and are given in terms of the variance of the stochastic integral term in the
mild form of the equation.
http://arxiv.org/abs/0912.3707
Author(s): Hayato Saigo
Abstract: In the case of monotone independence, the transparent understanding of the
mechanism to validate the central limit theorem (CLT) has been lacking, in
sharp contrast to commutative, free and Boolean cases. We have succeeded in
clarifying it by making use of simple combinatorial structure of peakless pair
partitions.
http://arxiv.org/abs/0912.3728
Author(s): Hubert Lacoin
Abstract: In this paper we study a model of Brownian polymer in $\R^+\times \R^d$,
introduced by Rovira and Tindel. Our investigation focuses mainly on the effect
of strong spatial correlation in the environment in that model in terms of
free-energy, fluctuation exponent and volume exponent. In particular we prove
that under some assumption, very-strong disorder and superdiffusivity hold at
all temperature when d>2 and provide a novel approach to Petermanns
superdiffusivity result in dimension one. We also derive results for a Brownian
model of pinning in a non-random potential with power-law decay at infinity.
http://arxiv.org/abs/0912.3732
Author(s): Pierre Nolin
Abstract: We study the diffusion front for a natural two-dimensional model where many
particles starting at the origin diffuse independently. It turns out that this
model can be described using properties of near-critical percolation, and
provides a natural example where critical fractal geometries spontaneously
arise.
http://arxiv.org/abs/0912.3770
Author(s): Sim\'on Lunag\'omez and Sayan Mukherjee and Robert L. Wolpert
Abstract: A parametrization of hypergraphs based on the geometry of points in
$\rr^\Dim$ is developed. Informative prior distributions on hypergraphs are
induced through this parametrization by priors on point configurations via
spatial processes. This prior specification is used to infer conditional
independence models or Markov structure of multivariate distributions.
Specifically, we can recover both the junction tree factorization as well as
the hyper Markov law. This approach offers greater control on the distribution
of graph features than Erd\"os-R\'enyi random graphs, supports inference of
factorizations that cannot be retrieved by a graph alone, and leads to new
Metropolis\slash Hastings Markov chain Monte Carlo algorithms with both local
and global moves in graph space. We illustrate the utility of this
parametrization and prior specification using simulations.
http://arxiv.org/abs/0912.3648
Author(s): Erwan Hillion
Abstract: This work is devoted to the geometric analysis of metric-measure spaces
satisfying a Prekopa-Leindler or a more general Borell-Brascamp-Lieb
inequality.
Completing the early investigations by Cordero-Erausquin, McCann and
Schmuckenschlager, we show that these functional inequalities characterize
lower bounds on the Ricci curvature on a Riemannian manifold, providing thus an
alternate version of Ricci curvature lower bounds in measured length spaces to
the recent developments by Lott, Villani and Sturm. We also investigate
stability properties and geometric and functional inequalities, such as
logarithmic Sobolev inequality and Bishop-Gromov diameter estimate, in measured
length spaces satisfying a Prekopa-Leindler or a Borell-Brascamp-Lieb
inequality.
http://arxiv.org/abs/0912.3593
Author(s): Gert de Cooman and Erik Quaeghebeur
Abstract: Sets of desirable gambles constitute a quite general type of uncertainty
model with an interesting geometrical interpretation. We give a general
discussion of such models and their rationality criteria. We study
exchangeability assessments for them, and prove counterparts of de Finetti's
finite and infinite representation theorems. We show that the finite
representation in terms of count vectors has a very nice geometrical
interpretation, and that the representation in terms of frequency vectors is
tied up with multivariate Bernstein (basis) polynomials. We also lay bare the
relationships between the representations of updated exchangeable models, and
discuss conservative inference (natural extension) under exchangeability and
the extension of exchangeable sequences.
http://arxiv.org/abs/0911.4727
Author(s): Rodolphe Garbit (LAREMA)
Abstract: We prove that a random walk in the plane with bounded increments and mean
zero conditioned to stay in a cone converges weakly to the corresponding
Brownian meander if and only if the tail distribution of the exit time from the
cone is regularly varying. This condition is satisfied in many natural
situations.
http://arxiv.org/abs/0911.4774
Author(s): C. Houdr\'e and Z. Talata
Abstract: The rate of convergence of the distribution of the length of the longest
increasing subsequence, towards the maximum eigenvalue of certain matrix
ensemble, is investigated. For finite-alphabet uniform and non-uniform iid
sources, a rate of $\log n /\sqrt{n}$ is obtained. The uniform binary case is
further explored, and an improved $1/\sqrt{n}$ rate obtained.
http://arxiv.org/abs/0911.4917
Author(s): Zhiyi Chi
Abstract: In a recent work (arXiv:0910.2517), for nonlinear models with sparse
underlying linear structures, we studied the error bounds of
$\ell_0$-regularized estimation. In this note, we show that
$\ell_1$-regularized estimation in some important cases can achieve the same
order of error bounds as those in the aforementioned work.
http://arxiv.org/abs/0911.4899
Author(s): Sho Matsumoto and Jonathan Novak
Abstract: We show that many important properties of unitary matrix integrals, such as
$1/N$ expansion, character expansion, and in some cases even explicit formulas,
are rooted in properties of the Jucys-Murphy elements. The class of integrals
to which our results apply are the correlation functions of elements of
Haar-distributed random unitary matrices. In the course of our study we obtain
various results on the conjugacy class expansion of symmetric functions in
Jucys-Murphy elements, a topic of interest in algebraic combinatorics.
http://arxiv.org/abs/0905.1992
Author(s): Roberto Imbuzeiro Oliveira
Abstract: We study random k-lifts of large, but otherwise arbitrary graphs G. We prove
that, with high probability, all eigenvalues of the adjacency matrix of the
lift that are not eigenvalues of G are of the order (D ln (kn))^{1/2}, where D
is the maximum degree of G. Similarly, and also with high probability, the
"new" eigenvalues of the Laplacian of the lift are all in an interval of length
(ln (nk)/d)^{1/2} around 1, where d is the minimum degree of G. We also prove
that, from the point of view of Spectral Graph Theory, there is very little
difference between a random k_1k_2 ... k_r-lift of a graph and a random
k_1-lift of a random k_2-lift of ... of a random k_r-lift of the same graph.
The main proof tool is a concentration inequality for sums of random matrices
that was recently introduced by the author.
http://arxiv.org/abs/0911.4741
Author(s): Ole E. Barndorff-Nielsen and Andreas Basse-O'Connor
Abstract: The question of existence and properties of stationary solutions to Langevin
equations driven by noise processes with stationary increments is discussed,
with particular focus on noise processes of pseudo moving average type. On
account of the Wold-Karhunen decomposition theorem such solutions are in
principle representable as a moving average (plus a drift like term) but the
kernel in the moving average is generally not available in explicit form. A
class of cases is determined where an explicit expression of the kernel can be
given, and this is used to obtain information on the asymptotic behavior of the
associated autocorrelation functions, both for small and large lags.
Applications to Gaussian and Levy driven fractional Ornstein-Uhlenbeck
processes are presented. As an element in the derivations a Fubini theorem for
Levy bases is established.
http://arxiv.org/abs/0912.3091
Author(s): Krzysztof Debicki and Agata Tomanek
Abstract: Let $B_H(\cdot)$ be a fractional Brownian motion with Hurst parameter
$H\in(0,1]$. Motivated by applications to maximal inequalities for fractional
Brownian motion, in this note we derive bounds for
K_T(H,\gamma):=E[\sup_{t\in[0,T]}|B_H(t)|]^\gamma, with $\gamma, T>0$.
http://arxiv.org/abs/0912.3117
Author(s): Alexandra Chronopoulou and Ciprian Tudor (LPP) and Frederi Viens
Abstract: The Rosenblatt process is a self-similar non-Gaussian process which lives in
second Wiener chaos, and occurs as the limit of correlated random sequences in
so-called \textquotedblleft non-central limit theorems\textquotedblright. It
shares the same covariance as fractional Brownian motion. We study the
asymptotic distribution of the quadratic variations of the Rosenblatt process
based on long filters, including filters based on high-order finite-difference
and wavelet-based schemes. We find exact formulas for the limiting
distributions, which we then use to devise strongly consistent estimators of
the self-similarity parameter $H$. Unlike the case of fractional Brownian
motion, no matter now high the filter orders are, the estimators are never
asymptotically normal, converging instead in the mean square to the observed
value of the Rosenblatt process at time 1.
http://arxiv.org/abs/0912.3148
Author(s): Jean Picard
Abstract: We discuss the relationships between some classical representations of the
fractional Brownian motion, as a stochastic integral with respect to a standard
Brownian motion, or as a series of functions with independent Gaussian
coefficients. The basic notions of fractional calculus which are needed for the
study are introduced. As an application, we also prove some properties of the
Cameron-Martin space of the fractional Brownian motion, and compare its law
with the law of some of its variants. Several of the results which are given
here are not new; our aim is to provide a unified treatment of some previous
literature, and to give alternative proofs and additional results; we also try
to be as self-contained as possible.
http://arxiv.org/abs/0912.3168
Author(s): Jean-Baptiste Bardet (IRMAR and LMRS) and H\'el\`ene Guerin (IRMAR) and Florent Malrieu (IRMAR)
Abstract: Let $Y$ be an Ornstein-Uhlenbeck diffusion governed by an ergodic finite
state Markov process $X$: $dY_t=-\lambda(X_t)Y_tdt+\sigma(X_t)dB_t$, $Y_0$
given. Under ergodicity condition, we get quantitative estimates for the long
time behavior of $Y$. We also establish a trichotomy for the tail of the
stationary distribution of $Y$: it can be heavy (only some moments are finite),
exponential-like (only some exponential moments are finite) or Gaussian-like
(its Laplace transform is bounded below and above by Gaussian ones). The
critical moments are characterized by the parameters of the model.
http://arxiv.org/abs/0912.3231
Author(s): Jean-Baptiste Bardet (IRMAR and LMRS) and H\'el\`ene Guerin (IRMAR) and Florent Malrieu (IRMAR)
Abstract: We consider the random variables $R$ which are solutions of the
distributional equation $R\overset{\cL}{=}MR+Q$, where $(Q,M)$ is independent
of $R$ and $\ABS{M}\leq 1$. Goldie and Gr\"ubel showed that the tails of $R$
are no heavier than exponential. In this note we provide the exact lower and
upper bounds of the domain of the Laplace transform of $R$.
http://arxiv.org/abs/0912.3232
Author(s): E. Kowalski and A. Nikeghbali
Abstract: In the context of mod-Gaussian convergence, as defined previously in our work
with J. Jacod, we obtain lower bounds for local probabilities for a sequence of
random vectors which are approximately Gaussian with increasing covariance.
This is motivated by the conjecture concerning the density of the set of values
of the Riemann zeta function on the critical line. We obtain evidence for this
fact, and derive unconditional results for random matrices in compact classical
groups, as well as for certain families of L-functions over finite fields.
http://arxiv.org/abs/0912.3237
Author(s): A.D. Barbour and E. Kowalski and A. Nikeghbali
Abstract: In this paper, we consider approximating expansions for the distribution of
integer valued random variables, in circumstances in which convergence in law
cannot be expected. The setting is one in which the simplest approximation to
the $n$'th random variable $X_n$ is by a particular member $R_n$ of a given
family of distributions, whose variance increases with $n$. The basic
assumption is that the ratio of the characteristic function of $X_n$ and that
of R_n$ converges to a limit in a prescribed fashion. Our results cover a
number of classical examples in probability theory, combinatorics and number
theory.
http://arxiv.org/abs/0912.1886
Author(s): Irmingard Eder and Claudia Kl\"uppelberg
Abstract: For the sum process $X=X^1+X^2$ of a bivariate L\'evy process $(X^1,X^2)$
with possibly dependent components, we derive a quintuple law describing the
first upwards passage event of $X$ over a fixed barrier, caused by a jump, by
the joint distribution of five quantities: the time relative to the time of the
previous maximum, the time of the previous maximum, the overshoot, the
undershoot and the undershoot of the previous maximum. The dependence between
the jumps of $X^1$ and $X^2$ is modeled by a L\'evy copula. We calculate these
quantities for some examples, where we pay particular attention to the
influence of the dependence structure. We apply our findings to the ruin event
of an insurance risk process.
http://arxiv.org/abs/0912.1925
Author(s): Hernan Awad and Peter Glynn
Abstract: We consider a stationary fluid queue with fractional Brownian motion input.
Conditional on the workload at time zero being greater than a large value $b$,
we provide the limiting distribution for the amount of time that the workload
process spends above level $b$ over the busy cycle straddling the origin, as
$b\to\infty$. Our results can be interpreted as showing that long delays occur
in large clumps of size of order $b^{2-1/H}$. The conditional limit result
involves a finer scaling of the queueing process than fluid analysis, thereby
departing from previous related literature.
http://arxiv.org/abs/0912.1928
Author(s): Noureddine El Karoui
Abstract: We place ourselves in the setting of high-dimensional statistical inference,
where the number of variables $p$ in a data set of interest is of the same
order of magnitude as the number of observations $n$. More formally, we study
the asymptotic properties of correlation and covariance matrices, in the
setting where $p/n\to\rho\in(0,\infty),$ for general population covariance. We
show that, for a large class of models studied in random matrix theory,
spectral properties of large-dimensional correlation matrices are similar to
those of large-dimensional covarance matrices. We also derive a
Mar\u{c}enko--Pastur-type system of equations for the limiting spectral
distribution of covariance matrices computed from data with elliptical
distributions and generalizations of this family. The motivation for this study
comes partly from the possible relevance of such distributional assumptions to
problems in econometrics and portfolio optimization, as well as robustness
questions for certain classical random matrix results. A mathematical theme of
the paper is the important use we make of concentration inequalities.
http://arxiv.org/abs/0912.1950
Author(s): Ross G. Pinsky
Abstract: Consider a diffusion process corresponding to the operator
$L=\frac12a\frac{d^2}{dx^2}+b\frac d{dx}$ and which is transient to $+\infty$.
For $c>0$, we give an explicit criterion in terms of the coefficients $a$ and
$b$ which determines whether or not the diffusion almost surely eventually
stops making down-crossings of length $c$. As a particular case, we show that
if $a=1$, then the diffusion almost surely stops making down-crossings of
length $c$ if $b(x)\ge\frac1{2c}\log x+\frac\gamma c\log\log x$, for some
$\gamma>1$ and for large $x$, but makes down-crossings of length $c$ at
arbitrarily large times if $b(x)\le\frac1{2c}\log x+\frac1c\log\log x$, for
large $x$.
http://arxiv.org/abs/0912.1973
Author(s): Elizabeth Meckes
Abstract: Let $X$ be a $d$-dimensional random vector and $X_\theta$ its projection onto
the span of a set of orthonormal vectors $\{\theta_1,...,\theta_k\}$.
Conditions on the distribution of $X$ are given such that if $\theta$ is chosen
according to Haar measure on the Stiefel manifold, the bounded-Lipschitz
distance from $X_\theta$ to a Gaussian distribution is concentrated at its
expectation; furthermore, a bound is given for the expected distance, in terms
of $d$, $k$, and the distribution of $X$. The results are applied in the
setting of projection pursuit, showing that most $k$-dimensional projections of
$n$ data points in $\R^d$ are close to Gaussian, when $n$ and $d$ are large and
$k=c\log(d)$ for a small constant $c$.
http://arxiv.org/abs/0912.2044
Author(s): Evelyn Buckwar and Thorsten Sickenberger
Abstract: In this article we compare the mean-square stability properties of the
Theta-Maruyama and Theta-Milstein method that are used to solve stochastic
differential equations. As a simple extension of the standard geometric
Brownian motion as a test equation for the linear stability analysis, we
consider a scalar linear test equation with several multiplicative noise terms.
This test equation allows to begin investigating the influence of
multi-dimensional noise on the stability behaviour of the methods while the
analysis is still tractable. Our findings include: (i) the stability condition
for the Theta-Milstein method and thus, for some choices of Theta, the
conditions on the step-size, are much more restrictive than those for the
Theta-Maruyama method; (ii) the precise stability region of the Theta-Milstein
method explicitly depends on the noise terms. Further, we investigate the
effect of introducing (partially) implicitness in the diffusion approximation
terms of Milstein-type methods, thus obtaining the possibility to control the
stability properties of these methods with a further method parameter Sigma.
Numerical examples illustrate the results and provide a comparison of the
stability behaviour of the different methods.
http://arxiv.org/abs/0912.1968
Author(s): Matthias Birkner and Rongfeng Sun
Abstract: We study the continuous time version of the random walk pinning model, where
conditioned on a continuous time random walk Y on Z^d with jump rate \rho>0,
which plays the role of disorder, the law up to time t of a second independent
random walk X with jump rate 1 is Gibbs transformed with weight e^{\beta
L_t(X,Y)}, where L_t(X,Y) is the collision local time between X and Y up to
time t. As the inverse temperature \beta varies, the model undergoes a
localization-delocalization transition at some critical \beta_c>=0. A natural
question is whether or not there is disorder relevance, namely whether or not
\beta_c differs from the critical point \beta_c^{ann} for the annealed model.
In Birkner and Sun [BS09], it was shown that there is disorder irrelevance in
dimensions d=1 and 2, and disorder relevance in d>=4. For d>=5, disorder
relevance was first proved by Birkner, Greven and den Hollander [BGdH08]. In
this paper, we prove that if X and Y have the same jump probability kernel,
which is irreducible and symmetric with finite second moments, then there is
also disorder relevance in the critical dimension d=3. Our proof employs coarse
graining and fractional moment techniques, which have recently been
successfully applied by Giacomin, Lacoin and Toninelli [GLT09] to establish
disorder relevance for the random pinning model in the critical dimension, and
by Lacoin [L09] to the directed polymer model in random environment. Along the
way, we also prove a continuous time version of Doney's local limit theorem
[D97] for renewal processes with infinite mean.
http://arxiv.org/abs/0912.1663
Author(s): Joseph Najnudel and Ashkan Nikeghbali
Abstract: In this paper, we give a global view of the results we have obtained in
relation with a remarkable class of submartingales, called $(\Sigma)$, and its
links with a universal sigma-finite measure and penalization problems on the
space of continuous and cadlag paths.
http://arxiv.org/abs/0912.1693
Author(s): Pawe{\l} Hitczenko and Jacek Weso{\l}owski
Abstract: We consider the tail behavior of random variables $R$ which are solutions of
the distributional equation $R\stackrel{d}{=}Q+MR$, where $(Q,M)$ is
independent of $R$ and $|M|\le 1$. Goldie and Gr\"{u}bel showed that the tails
of $R$ are no heavier than exponential and that if $Q$ is bounded and $M$
resembles near 1 the uniform distribution, then the tails of $R$ are
Poissonian. In this paper, we further investigate the connection between the
tails of $R$ and the behavior of $M$ near 1. We focus on the special case when
$Q$ is constant and $M$ is nonnegative.
http://arxiv.org/abs/0912.1694
Author(s): Shirshendu Chatterjee and Rick Durrett
Abstract: If we consider the contact process with infection rate $\lambda$ on a random
graph on $n$ vertices with power law degree distributions, mean field
calculations suggest that the critical value $\lambda_c$ of the infection rate
is positive if the power $\alpha>3$. Physicists seem to regard this as an
established fact, since the result has recently been generalized to bipartite
graphs by G\'{o}mez-Garde\~{n}es et al. [Proc. Natl. Acad. Sci. USA 105 (2008)
1399--1404]. Here, we show that the critical value $\lambda_c$ is zero for any
value of $\alpha>3$, and the contact process starting from all vertices
infected, with a probability tending to 1 as $n\to\infty$, maintains a positive
density of infected sites for time at least $\exp(n^{1-\delta})$ for any
$\delta>0$. Using the last result, together with the contact process duality,
we can establish the existence of a quasi-stationary distribution in which a
randomly chosen vertex is occupied with probability $\rho(\lambda)$. It is
expected that $\rho(\lambda)\sim C\lambda^{\beta}$ as $\lambda \to0$. Here we
show that $\alpha-1\le\beta\le2\alpha-3$, and so $\beta>2$ for $\alpha>3$. Thus
even though the graph is locally tree-like, $\beta$ does not take the mean
field critical value $\beta=1$.
http://arxiv.org/abs/0912.1699
Author(s): Alexander M. G. Cox and David G. Hobson and Jan K. Ob{\l}\'oj
Abstract: We solve explicitly the following problem: for a given probability measure
mu, we specify a generalised martingale diffusion X which, stopped at an
independent exponential time T, is distributed according to mu. The process X
is specified via its speed measure m. We present three proofs. First we show
how the result can be derived from the solution of Bertoin and Le Jan (1992) to
the Skorokhod embedding problem. Secondly, we give a proof exploiting
applications of Krein's spectral theory of strings to the study of linear
diffusions. Finally, we present a novel direct probabilistic proof based on a
coupling argument.
http://arxiv.org/abs/0912.1719
Author(s): Pao-Liu Chow
Abstract: The paper is concerned with the problem of explosive solutions for a class of
nonlinear stochastic wave equations in a domain
$\mathcal{D}\subset\mathbb{R}^d$ for $d\leq3$. Under appropriate conditions on
the initial data, the nonlinear term and the noise intensity is proved in
Theorem 3.1 that the $L^2$-norm of the solution will blow up at a finite time
in the mean-square sense. An example is given to show an application of the
theorem.
http://arxiv.org/abs/0912.1735
Author(s): In\'es Armend\'ariz and Stefan Grosskinsky and Michail Loulakis
Abstract: Zero-range processes with decreasing jump rates exhibit a continuous
condensation transition, where a finite fraction of all particles condenses on
a single lattice site when the total density exceeds a critical value. We study
the onset of condensation, i.e. the behaviour of the maximum occupation number
after adding a subextensive excess mass of particles at the critical density.
We establish a law of large numbers for the excess mass fraction in the
maximum, which turns out to jump from 0 to a positive value at a critical
scale. Our results also include distributional limits for the fluctuations of
the maximum in both regimes, which change from standard extreme value
statistics to Gaussian. We identify the detailed behaviour at the critical
scale including sub-leading terms, providing a full understanding of the
crossover between the two regimes.
http://arxiv.org/abs/0912.1793
Author(s): Brice Franke and Michael Stolz
Abstract: We propose an approach to the aggregation of risks which is based on
estimation of simple quantities (such as covariances) associated to a vector of
dependent random variables, and which avoids the use of parametric families of
copulae. Our main result demonstrates that the method leads to bounds on the
worst case Value at Risk for a sum of dependent random variables. Its proof
applies duality theory for infinite dimensional linear programs.
http://arxiv.org/abs/0912.1841
Author(s): Christos A. Athanasiadis and Persi Diaconis
Abstract: Many seemingly disparate Markov chains are unified when viewed as random
walks on the set of chambers of a hyperplane arrangement. These include the
Tsetlin library of theoretical computer science and various shuffling schemes.
If only selected features of the chains are of interest, then the mixing times
may change. We study the behavior of hyperplane walks, viewed on a
subarrangement of a hyperplane arrangement. These include many new examples,
for instance a random walk on the set of acyclic orientations of a graph. All
such walks can be treated in a uniform fashion, yielding diagonalizable
matrices with known eigenvalues, stationary distribution and good rates of
convergence to stationarity.
http://arxiv.org/abs/0912.1686
Author(s): Alexei Borodin
Abstract: We present a list of algebraic, combinatorial, and analytic mechanisms that
give rise to determinantal point processes.
http://arxiv.org/abs/0911.1153
Author(s): Yves F. Atchade
Abstract: In this paper we study kernel methods for the estimation of asymptotic
variances (or long run variances) for a class of adaptive Markov chains. We
prove that these estimators are $L^p$-consistent and strongly consistent.
Although the motivation comes from Markov Chain Monte Carlo, these results
apply more generally. In the special case of Markov chains, the results improve
on the existing literature by imposing weaker moments conditions. We illustrate
the results with applications to the GARCH$(1,1)$ Markov model and to adaptive
MCMC simulation for Bayesian logistic regression model.
http://arxiv.org/abs/0911.1164
Author(s): Marjorie G. Hahn and Xinxin Jiang and Sabir Umarov
Abstract: The q-Gaussians are discussed from the point of view of variance mixtures of
normals and exchangeability. For each q< 3, there is a q-Gaussian distribution
that maximizes the Tsallis entropy under suitable constraints. This paper shows
that q-Gaussian random variables can be represented as variance mixtures of
normals. These variance mixtures of normals are the attractors in central limit
theorems for sequences of exchangeable random variables; thereby, providing a
possible model that has been extensively studied in probability theory. The
formulation provided has the additional advantage of yielding process versions
which are naturally q-Brownian motions. Explicit mixing distributions for
q-Gaussians should facilitate applications to areas such as option pricing. The
model might provide insight into the study of superstatistics.
http://arxiv.org/abs/0911.1176
Author(s): Herold Dehling and Martin Wendler
Abstract: The law of the iterated logarithm for partial sums of weakly dependent
processes was intensively studied by Walter Philipp in the late 1960s and
1970s. In this paper, we aim to extend these results to nondegenerate
U-statistics of data that are strongly mixing or functionals of an absolutely
regular process.
http://arxiv.org/abs/0911.1200
Author(s): Pierre Patie
Abstract: Let X be a positive self-similar Markov process with 0 as an absorbing state.
The purpose of this paper is to describe the law of the absorption time, say T,
which might occurs continuously or by a jump. We start by showing that the
distribution function of T can be expressed in terms of an increasing invariant
function for a specific transient Ornstein-Uhlenbeck process associated to X.
Furthermore, specializing on the spectrally negative case, we suggest an
original methodology to get a power series or an integral representation of
this invariant function. Then, by means of probabilistic arguments, we deduce
some interesting analytical properties satisfied by these functions, which
include, for instance, several types of hypergeometric functions. We end the
paper by detailing some known and new examples. In particular, we offer an
alternative proof of the recent result obtained by Bernyk et al. regarding the
law of the maximum of regular spectrally positive stable processes
http://arxiv.org/abs/0911.1203
Author(s): Abdelhadi Es-Sarhir and Wilhelm Stannat
Abstract: We study regularity properties for invariant measures of semilinear
diffusions in a separable Hilbert space. Based on a pathwise estimate for the
underlying stochastic convolution, we prove a priori estimates on such
invariant measures. As an application, we combine such estimates with a new
technique to prove the $L^1$-uniqueness of the induced Kolmogorov operator,
defined on a space of cylindrical functions. Finally, examples of stochastic
Burgers equations and thin-film growth models are given to illustrate our
abstract result.
http://arxiv.org/abs/0911.1206
Author(s): Jan M. Swart and Karel Vrbensky
Abstract: The rebellious voter model, introduced by Sturm and Swart (2008), is a
variation of the standard, one-dimensional voter model, in which types that are
locally in the minority have an advantage. It is related, both through duality
and through the evolution of its interfaces, to a system of branching
annihilating random walks that is believed to belong to the
`parity-conservation' universality class. This paper presents numerical data
for the rebellious voter model and for a closely related one-sided version of
the model. Both models appear to exhibit a phase transition between
noncoexistence and coexistence as the advantage for minority types is
increased. For the one-sided model (but not for the original, two-sided
rebellious voter model), it appears that the critical point is exactly a half
and two important functions of the process are given by simple, explicit
formulas, a fact for which we have no explanation.
http://arxiv.org/abs/0911.1266
Author(s): Xiong Jin
Abstract: With the iso-H\"older sets of a function we naturally associate subsets of
the graph, range and level sets of the function. We compute the associated
Hausdorff dimension spectra for a class of statistically self-similar
multifractal functions.
http://arxiv.org/abs/0911.1289
Author(s): Pavel Buividovich and Yuri Makeenko (ITEP and Moscow)
Abstract: We investigate the properties of the path integral over reparametrizations (=
the boundary value of the Liouville field in open string theory). Discretizing
the path integral, we apply the Metropolis-Hastings algorithm to numerical
simulations of a proper (subordinator) stochastic process and find that typical
trajectories are not Brownian but rather have discontinuities of the type of
Levy's flights. We study a fractal structure of these trajectories and show
that their Hausdorff dimension is zero. We confirm thereby the discretization
and heuristic consideration of QCD scattering amplitudes by analytical and
numerical calculations. We also perform Monte Carlo simulations of the path
integral over reparametrization in the effective-string ansatz for a circular
Wilson loop and discuss their subtleties associated with the discretization of
Douglas' functional.
http://arxiv.org/abs/0911.1083
Author(s): Mark Kelbert and Yuri Suhov
Abstract: This article addresses the issue of the proof of the entropy power inequality
(EPI), an important tool in the analysis of Gaussian channels of information
transmission, proposed by Shannon. We analyse continuity properties of the
mutual entropy of the input and output signals in an additive memoryless
channel and show how this can be used for a correct proof of the entropy-power
inequality under various types of assumptions.
http://arxiv.org/abs/0911.1275
Author(s): Xavier Bardina and Khalifa Es-Sebaiy (SAMOS and CES) and Ciprian Tudor (LPP)
Abstract: We construct a family $I_{n_{\eps}}(f)_{t}$ of continuous stochastic
processes that converges in the sense of finite dimensional distributions to a
multiple Wiener-It\^o integral $I_{n}^{H}(f1^{\otimes n}_{[0,t]})$ with respect
to the fractional Brownian motion. We assume that $H>{1/2}$ and we prove our
approximation result for the integrands $f$ in a rather general class.
http://arxiv.org/abs/0911.3223
Author(s): Cyrille Lucas (MODAL'x)
Abstract: We present precise asymptotics of the odometer function for the internal
Diffusion Limited Aggregation model. These results provide a better
understanding of this function whose importance was demonstrated by Levine and
Peres. We derive a different proof of a time-scale result by Lawler, Bramson
and Griffeath.
http://arxiv.org/abs/0911.3224
Author(s): Ayala Mashiah-Yaakovi and Eilon Solan
Abstract: We prove that every two-player non-zero-sum Borel game with
lower-semi-continuous payoffs admits a subgame-perfect $\ep$-equilibrium. This
result complements Example 3 in Solan and Vieille (2003), which shows that a
subgame-perfect $\ep$-equilibrium need not exists when the payoffs are not
lower-semi-continuous.
http://arxiv.org/abs/0911.3246
Author(s): N. Dirr and P. W. Dondl and G. R. Grimmett and A. E. Holroyd and M. Scheutzow
Abstract: We prove the existence of a (random) Lipschitz function $F : \Z^{d-1}\to\Z^+$
such that, for every $x \in \Z^{d-1}$, the site $(x,F(x))$ is open in a site
percolation process on $\Z^{d}$. The Lipschitz constant may be taken to be 1
when the parameter $p$ of the percolation model is sufficiently close to 1.
http://arxiv.org/abs/0911.3384
Author(s): Qingxin Meng
Abstract: In this paper, we study a class of stochastic optimal control problem with
jumps under partial information. More precisely, the controlled systems are
described by a fully coupled nonlinear multi- dimensional forward-backward
stochastic differential equation driven by a Poisson random measure and an
independent multi-dimensional Brownian motion, and all admissible control
processes are required to be adapted to a given subfiltration of the filtration
generated by the underlying Poisson random measure and Brownian motion. For
this type of partial information stochastic optimal control problem, we give a
necessary and sufficient maximum principle. All the coefficients appearing in
the systems are allowed to depend on the control variables and the control
domain is convex.
http://arxiv.org/abs/0911.3225
Author(s): Gordon Blower
Abstract: Painleve's transcendental differential equation P_{VI} may be expressed as
the consistency condition for a pair of linear differential equations with 2 by
2 matrix coefficients with rational entries. By a construction due to Tracy and
Widom, this linear system is associated with certain kernels which give trace
class operators on Hilbert space. This paper expresses such operators in terms
of the Hankel operators \Gamma_\phi of linear systems which are realised in
terms of the Laurent coefficients of the solutions of the differential
equations. Let P_{(t infty)}:L^2(0, \infty)\to L^2(t, \infty) be the orthogonal
projection. For such, the Fredholm determinant \tau (t)=det (I-P_{(t,
\infty)}\Gamma_\phi) defines the tau function, which is here expressed in terms
of the solutions of a matrix Gelfand--Levitan equation. For suitable paramters,
solutions of the hypergeometric equation give a linear system with similar
properties. For meromorphic transfer functions \hat\phi that have poles on an
arithmetic progression, the corresponding Hankel operator has a simple form
with respect to an exponential basis in L^2(0, \infty); so \tau (t) can be
expressed in terms of finite determinants. This applies to elliptic functions
of the second kind, such as satisfy Lame's equation with \ell=1.
http://arxiv.org/abs/0911.3359
Author(s): Michel Ledoux and Brian Rider
Abstract: We establish various small deviation inequalities for the extremal (soft
edge) eigenvalues in the beta-Hermite and beta-Laguerre ensembles. In both
settings, upper bounds on the variance of the largest eigenvalue of the
anticipated order follow immediately.
http://arxiv.org/abs/0912.5040
Author(s): Wen Lv
Abstract: In this paper, we deal with a class of one-dimensional reflected backward
doubly stochastic differential equations with one continuous lower barrier. We
derive the existence and uniqueness of solutions for these equations with
Lipschitz coefficients.
http://arxiv.org/abs/0912.5060
Author(s): Akira Sakai
Abstract: The aim of this short article is to convey the basic idea of the original
paper [3], without going into too much detail, about how to derive sharp
asymptotics of the gyration radius for random walk, self-avoiding walk and
oriented percolation above the model-dependent upper critical dimension.
http://arxiv.org/abs/0912.5117
Author(s): Alessandro De Gregorio
Abstract: The aim of this paper is to analyze a class of random motions which models
the motion of a particle on the real line with random velocity and subject to
the action of the friction. The speed randomly changes when a Poissonian event
occurs. We study the characteristic and the moment generating function of the
position reached by the particle at time $t>0$. We are able to derive the
explicit probability distributions in few cases for which discuss the
connections with the random flights. The moments are also widely analyzed.
For the random motions having an explicit density law, further interesting
probabilistic interpretations emerge if we deal with them varying up a random
time. Essentially, we consider two different type of random times, namely
Bessel and Gamma times, which contain, as particular cases, some important
probability distributions (e.g. Gaussian, Exponential). In particular, for the
random processes built by means of these compositions, we derive the
probability distributions fixed the number of Poisson events.
Some remarks on the possible extensions to the random motions in higher
spaces are proposed. We focus our attention on the persistent planar random
motion.
http://arxiv.org/abs/0912.5151
Author(s): Illes Horvath and Balint Toth and Balint Veto
Abstract: The self-repelling Brownian polymer model (SRBP) initiated by Durrett and
Rogers in [Durrett-Rogers (1992)] is the continuous space-time counterpart of
the myopic (or 'true') self-avoiding walk model (MSAW) introduced in the
physics literature by Amit, Parisi and Peliti in [Amit-Parisi-Peliti (1983)].
In both cases, a random motion in space is pushed towards domains less visited
in the past by a kind of negative gradient of the occupation time measure.
We investigate the asymptotic behaviour of SRBP in the non-recurrent
dimensions. First, extending 1-dimensional results from [Tarres-Toth-Valko
(2009)], we identify a natural stationary (in time) and ergodic distribution of
the environment (essentially, smeared-out occupation time measure of the
process), as seen from the moving particle. As main result we prove that in
three and more dimensions, in this stationary (and ergodic) regime, the
displacement of the moving particle scales diffusively and its finite
dimensional distributions converge to those of a Wiener process. This result
settles part of the conjectures (based on non-rigorous renormalization group
arguments) in [Amit-Parisi-Peliti (1983)].
The main tool is the non-reversible version of the Kipnis--Varadhan-type CLT
for additive functionals of ergodic Markov processes and the graded sector
condition of [Sethuraman-Varadhan-Yau (2000)].
http://arxiv.org/abs/0912.5174
Author(s): Jesse Goodman
Abstract: In invasion percolation, the edges of successively maximal weight (the
outlets) divide the invasion cluster into a chain of ponds separated by
outlets. On the regular tree, the ponds are shown to grow exponentially, with
law of large numbers, central limit theorem and large deviation results. The
tail asymptotics for a fixed pond are also studied and are shown to be related
to the asymptotics of a critical percolation cluster, with a logarithmic
correction.
http://arxiv.org/abs/0912.5205
Author(s): Jiagang Ren and Siyan Xu and Xicheng Zhang
Abstract: We prove a large deviation principle of Freidlin-Wentzell's type for the
multivalued stochastic differential equations with monotone drifts, which in
particular contains a class of SDEs with reflection in a convex domain.
http://arxiv.org/abs/0912.5271
Author(s): Jiheng Zhang and J.G. Dai and Bert Zwart
Abstract: We consider a processor sharing queue where the number of jobs served at any
time is limited to K, with the excess jobs waiting in a buffer. We use random
counting measures on the positive axis to model this system. The limit of this
measure-valued process is obtained under diffusion scaling and heavy traffic
conditions. As a consequence, the limit of the system size process is proved to
be a piece-wise reflected Brownian motion.
http://arxiv.org/abs/0912.5306
Author(s): Guus Balkema and Paul Embrechts and Natalia Nolde
Abstract: The paper focuses on a class of light-tailed multivariate probability
distributions. These are obtained via a transformation of the marginals from a
heavy-tailed original distribution. This class was introduced in Balkema et al.
(2009). As shown there, for the light-tailed meta distribution the sample
clouds, properly scaled, converge onto a deterministic set. The shape of the
limit set gives a good description of the relation between extreme observations
in different directions. This paper investigates how sensitive the limit shape
is to changes in the underlying heavy-tailed distribution. Copulas fit in well
with multivariate extremes. By Galambos's Theorem existence of directional
derivatives in the upper endpoint of the copula is necessary and sufficient for
convergence of the multivariate extremes provided the marginal maxima converge.
The copula of the max-stable limit distribution does not depend on the
marginals. So marginals seem to play a subsidiary role in multivariate
extremes. The theory and examples presented in this paper cast a different
light on the significance of marginals. For light-tailed meta distributions the
asymptotic behaviour is very sensitive to perturbations of the underlying
heavy-tailed original distribution, it may change drastically even when the
asymptotic behaviour of the heavy-tailed density is not affected.
http://arxiv.org/abs/0912.5337
Author(s): Krzysztof Burdzy and Zhen-Qing Chen and Soumik Pal
Abstract: We consider a family of hard core objects moving as independent Brownian
motions confined to a vessel by reflection. These are subject to gravitational
forces modeled by drifts. The stationary distribution for the process has many
interesting implications, including an illustration of the Archimedes'
principle. The analysis rests on constructing reflecting Brownian motion with
drift in a general open connected domain and studying its stationary
distribution. In dimension two we utilize known results about sphere packing.
http://arxiv.org/abs/0912.5398
Author(s): Marta Tyran-Kaminska
Abstract: We study functional limit theorems for linear type processes with short
memory under the assumption that the innovations are dependent identically
distributed random variables with infinite variance and in the domain of
attraction of non-normal stable laws.
http://arxiv.org/abs/0912.5512
Author(s): Fabian Crocce and Ernesto Mordecki
Abstract: Each of two players, by turns, rolls a dice several times accumulating the
successive scores until he decides to stop, or he rolls an ace. When stopping,
the accumulated turn score is added to the player account and the dice is given
to his opponent. If he rolls an ace, the dice is given to the opponent without
adding any point. In this paper we formulate this game in the framework of
competitive Markov decision processes (also known as stochastic games), show
that the game has a value, provide an algorithm to compute the optimal minimax
strategy, and present results of this algorithm in three different variants of
the game.
http://arxiv.org/abs/0912.5518
Author(s): Jason Miller and Yuval Peres
Abstract: We show that the measure on markings of $\Z_n^d$, $d \geq 3$, with elements
of $\{0,1\}$ given by iid fair coin flips on the range $\CR$ of a random walk
$X$ run until time $T$ and 0 otherwise becomes indistinguishable from the
uniform measure on such markings at the threshold $T =
\tfrac{1}{2}T_\cov(\Z_n^d)$. As a consequence of our methods, we show that the
total variation mixing time of the random walk on the lamplighter graph $\Z_2
\wr \Z_n^d$, $d \geq 3$, has a cutoff with threshold $\tfrac{1}{2}
T_\cov(\Z_n^d)$. We give a general criterion under which both of these results
hold; other examples for which this applies include bounded degree expander
families, the intersection of an infinite supercritical percolation cluster
with an increasing family of balls, the hypercube, and the Caley graph of the
symmetric group generated by transpositions. The proof also yields precise
asymptotics for the decay of correlation in the uncovered set.
http://arxiv.org/abs/0912.5523
Author(s): Alexandre Belloni and Robert L. Winkler
Abstract: This paper focuses on generalizing quantiles from the ordering point of view.
We propose the concept of {\it partial quantiles} based on a given partial
order. We establish that partial quantiles are equivariant under
partial-order-preserving transformations of the data, display a concentration
of measure phenomenon, generalize the concept of efficient frontier, and can
measure dispersion from the partial order perspective.
We also study several statistical aspects of partial quantiles. We provide
estimators, associated rates of convergence, and asymptotic distributions that
hold uniformly over a continuum of quantile indices. Furthermore, we provide
procedures that can restore monotonicity properties that might have been
disturbed by estimation error, and establish computational complexity bounds.
Finally, we illustrate the concepts by discussing several theoretical examples
and simulations. Empirical applications to compare intake nutrients within
diets and to evaluate the performance of investment funds are presented.
http://arxiv.org/abs/0912.5489
Author(s): Johannes Lengler
Abstract: In number theory, great efforts have been undertaken to study the
Cohen-Lenstra probability measure on the set of all finite abelian $p$-groups.
On the other hand, group theorists have studied a probability measure on the
set of all partitions induced by the probability that a randomly chosen
$n\times n$-matrix over $\FF_p$ is contained in a conjucagy class associated
with this partitions, for $n \to \infty$.
This paper shows that both probability measures are identical. As a
consequence, a multitide of results can be transferred from each theory to the
other one. The paper contains a survey about the known methods to study the
probability measure and about the results that have been obtained so far, from
both communities.
http://arxiv.org/abs/0912.4975
Author(s): Johannes Lengler
Abstract: The Cohen-Lenstra heuristic is a universal principle that assigns to each
group a probability that tells how often this group should occur "in nature".
The most important, but not the only, applications are sequences of class
groups, which behave like random sequences of groups with respect to the
so-called Cohen-Lenstra probability measure.
So far, it was only possible to define this probability measure for finite
abelian $p$-groups. We prove that it is also possible to define an analogous
probability measure on the set of \emph{all} finite abelian groups when
restricting to the $\Sigma$-algebra on the set of all finite abelian groups
that is generated by uniform properties, thereby solving a problem that was
open since 1984.
http://arxiv.org/abs/0912.4977
Author(s): Matt Bond and Alexander Volberg
Abstract: In this paper we get an estimate of Favard length of an arbitrary
neighbourhood of an arbitrary self-similar Cantor set. Consider $L$ closed
disjoint discs of radius $1/L$ inside the unit disc. By using linear maps of
smaller disc onto the unit disc we can generate a self-similar Cantor set $G$.
Then $\G=\bigcap_n\G_n$. One may then ask the rate at which the Favard length
-- the average over all directions of the length of the orthogonal projection
onto a line in that direction -- of these sets $\G_n$ decays to zero as a
function of $n$. The quantitative results for the Favard length problem were
obtained by Peres--Solomyak and Tao; in the latter paper a general way of
making a quantitative statement from the Besicovitch theorem is considered. But
being rather general, this method does not give a good estimate for
self-similar structures such as $\G_n$. Indeed, vastly improved estimates have
been proven in these cases: in the paper of Nazarov--Peres--Volberg, it was
shown that for 1/4 corner Cantor set one has $p<1/6$, such that
$Fav(\K_n)\leq\frac{c_p}{n^{p}}$, and in Laba--Zhai and Bond--Volberg the same
type power estimate was proved for the product Cantor sets (with an extra
tiling property) and for the Sierpinski gasket $S_n$ for some other $p>0$. In
the present work we give an estimate that works for {\it any} Besicovitch set
which is self-similar. However estimate is worse than the power one. The power
estimate still appears to be related to a certain regularity property of zeros
of a corresponding linear combination of exponents (we call this property {\it
analytic tiling}).
http://arxiv.org/abs/0912.5111
Author(s): E. Pardoux and A. Piatnitski
Abstract: The paper studies homogenization problem for a non-autonomous parabolic
equation with a large random rapidly oscillating potential in the case of one
dimensional spatial variable. We show that if the potential is a statistically
homogeneous rapidly oscillating function of both temporal and spatial variables
then, under proper mixing assumptions, the limit equation is deterministic and
the convergence in probability holds. To the contrary, for the potential having
a microstructure only in one of these variables, the limit problem is
stochastic and we only prove the convergence in law.
http://arxiv.org/abs/0912.5277
Author(s): Julien Berestycki and Andreas E. Kyprianou and Antonio Murillo
Abstract: We develop an idea of Evans and O'Connell, Englander and Pinsky and Duquesne
and Winkel by giving a pathwise construction of the so called `backbone'
decomposition for supercritical superprocesses. Our results also complement a
related result for critical $(1+\beta)$-superprocesses given in Etheridge and
Williams \cite{EW}. Our approach takes an analytical point of view which is
more in the spirit of the original Evans and O'Connell paper.
http://arxiv.org/abs/0912.4736
Author(s): Andreas E. Kyprianou and Juan Carlos Pardo and Kees van Schaik
Abstract: We develop a new method for simulating the joint law of the position and
running maximum at a fixed time of a general L\'evy process with a view to
application in insurance and financial mathematics. Although different, our
method takes lessons from Carr's so-called `Canadization' technique as well as
Doney's method of stochastic bounds for L\'evy processes. We rely fundamentally
on the Wiener-Hopf decomposition for L\'evy processes as well as taking
advantage of recent developments in factorisation techniques of the latter
theory due to Vigon and Kuznetsov. We illustrate our Wiener-Hopf Monte Carlo
method on a number of different processes, including a new family of L\'evy
processes called hypergeometric L\'evy processes.
http://arxiv.org/abs/0912.4743
Author(s): Martin T. Barlow and Robert Masson
Abstract: We study simple random walk on the uniform spanning tree on Z^2 . We obtain
estimates for the transition probabilities of the random walk, the distance of
the walk from its starting point after n steps, and exit times of both
Euclidean balls and balls in the intrinsic graph metric. In particular, we
prove that the spectral dimension of the uniform spanning tree on Z^2 is 16/13
almost surely.
http://arxiv.org/abs/0912.4765
Author(s): Joseph Najnudel and Ashkan Nikeghbali
Abstract: In a previous paper, we proved that for any submartingale $(X_t)_{t \geq 0}$
of class $(\Sigma)$, defined on a filtered probability space $(\Omega,
\mathcal{F}, \mathbb{P}, (\mathcal{F}_t)_{t \geq 0})$, which satisfies some
technical conditions, one can construct a $\sigma$-finite measure $\mathcal{Q}$
on $(\Omega, \mathcal{F})$, such that for all $t \geq 0$, and for all events
$\Lambda_t \in \mathcal{F}_t$: $$ \mathcal{Q} [\Lambda_t, g\leq t] =
\mathbb{E}_{\mathbb{P}} [\mathds{1}_{\Lambda_t} X_t]$$ where $g$ is the last
hitting time of zero of the process $X$. Some particular cases of this
construction are related with Brownian penalisation or mathematical finance. In
this note, we give a simpler construction of $\mathcal{Q}$, and we show that an
analog of this measure can also be defined for discrete-time submartingales.
http://arxiv.org/abs/0912.4768
Author(s): Serge Cohen and Mark M. Meerschaert and Jan Rosinski
Abstract: Self-similar processes are useful in modeling diverse phenomena that exhibit
scaling properties. Operator scaling allows a different scale factor in each
coordinate. This paper develops practical methods for modeling and simulating
stochastic processes with operator scaling. A simulation method for operator
stable Levy processes is developed, based on a series representation, along
with a Gaussian approximation of the small jumps. Several examples are given to
illustrate practical applications. A classification of operator stable Levy
processes in two dimensions is provided according to their exponents and
symmetry groups. We conclude with some remarks and extensions to general
operator self-similar processes.
http://arxiv.org/abs/0912.4784
Author(s): K. Lin and G. Reinert
Abstract: In a random graph, counts for the number of vertices with given degrees will
typically be dependent. We show via a multivariate normal and a Poisson process
approximation that, for graphs which have independent edges, with a possibly
inhomogeneous distribution, only when the degrees are large can we reasonably
approximate the joint counts as independent. The proofs are based on Stein's
method and the Stein-Chen method with a new size-biased coupling for such
inhomogeneous random graphs, and hence bounds on distributional distance are
obtained. Finally we illustrate that apparent (pseudo-) power-law type
behaviour can arise in such inhomogeneous networks despite not actually
following a power-law degree distribution.
http://arxiv.org/abs/0912.4812
Author(s): James Kuelbs and Anand N. Vidyashankar
Abstract: We establish limit theorems involving weak convergence of multiple
generations of critical and supercritical branching processes. These results
arise naturally when dealing with the joint asymptotic behavior of functionals
defined in terms of several generations of such processes. Applications of our
main result include a functional central limit theorem (CLT), a Darling-Erd\"os
result, and an extremal process result. The limiting process for our functional
CLT is an infinite dimensional Brownian motion with sample paths in the
infinite product space $(C_0[0,1])^{\infty}$, with the product topology, or in
Banach subspaces of $(C_0[0,1])^{\infty}$ determined by norms related to the
distribution of the population size of the branching process. As an application
of this CLT we obtain a central limit theorem for ratios of weighted sums of
generations of a branching processes, and also to various maximums of these
generations. The Darling-Erd\"os result and the application to extremal
distributions also include infinite dimensional limit laws. Some branching
process examples where the CLT fails are also included.
http://arxiv.org/abs/0912.4909
Author(s): Xiaoming Xu
Abstract: Anticipated backward stochastic differential equation (ABSDE) studied the
first time in 2007 is a new type of stochastic differential equations. In this
paper, we establish a general comparison theorem for 1-dimensional ABSDEs with
the generators depending on the anticipated term of $Z$.
http://arxiv.org/abs/0911.0507
Author(s): Matti Vihola
Abstract: The Adaptive Metropolis (AM) algorithm is based on the symmetric random-walk
Metropolis algorithm. The proposal distribution has the following
time-dependent covariance matrix at step $n+1$ \[
S_n = Cov(X_1,...,X_n) + \epsilon I, \] that is, the sample covariance matrix
of the history of the chain plus a (small) constant $\epsilon>0$ multiple of
the identity matrix $I$. The lower bound on the eigenvalues of $S_n$ induced by
the factor $\epsilon I$ is theoretically convenient, but practically
cumbersome, as a good value for the parameter $\epsilon$ may not always be easy
to choose. This article considers variants of the AM algorithm that do not
explicitly bound the eigenvalues of $S_n$ away from zero. The behaviour of
$S_n$ is studied in detail, indicating that the eigenvalues of $S_n$ do not
tend to collapse to zero in general.
http://arxiv.org/abs/0911.0522
Author(s): D. J. Aldous and J. R. Ong and W. Zhou
Abstract: We introduce a stochastic model in which adjacent planar regions $A, B$ merge
stochastically at some rate $\lambda(A,B)$, and observe analogies with the
well-studied topics of mean-field coagulation and of bond percolation. Do
infinite regions appear in finite time? We give a simple condition on $\lambda$
for this {\em hegemony} property to hold, and another simple condition for it
to not hold, but there is a large gap between these conditions, which includes
the case $\lambda(A,B) \equiv 1$. For this case, a non-rigorous analytic
argument and simulations suggest hegemony.
http://arxiv.org/abs/0911.0601
Author(s): Yizao Wang and Parthanil Roy and Stilian A. Stoev
Abstract: We establish characterization results for the ergodicity of symmetric
$\alpha$-stable (S$\alpha$S) and $\alpha$-Frechet max-stable stationary random
fields. We first show that the result of Samorodnitsky(2005) remains valid in
the multiparameter setting, i.e., a stationary S$\alpha$S ($0<\alpha<2$) random
field is ergodic (or equivalently, weakly mixing) if and only if it is
generated by a null group action. The similarity of the spectral
representations for sum- and max-stable random fields yields parallel
characterization results in the max-stable setting. By establishing
multiparameter versions of Stochastic and Birkhoff Ergodic Theorems, we give a
criterion for ergodicity of these random fields which is valid for all
dimensions and new even in the one-dimensional case. We also prove the
equivalence of ergodicity and weak mixing for the general class of positively
dependent random fields.
http://arxiv.org/abs/0911.0610
Author(s): Fran\c{c}ois Gautero and Fr\'ed\'eric Math\'eus
Abstract: We give a geometric description of the Poisson boundaries of certain
extensions of free and hyperbolic groups. In particular, we get a full
description of the Poisson boundaries of free-by-cyclic groups. We rely upon
the description of Poisson boundaries by means of a topological
compactification as developed by Kaimanovich. All the groups studied here share
the property of admitting a sufficiently complicated action on some real tree.
http://arxiv.org/abs/0911.0616
Author(s): A. Deya and M. Gubinelli and S. Tindel
Abstract: This article is devoted to define and solve an evolution equation of the form
$dy_t=\Delta y_t dt+ dX_t(y_t)$, where $\Delta$ stands for the Laplace operator
on a space of the form $L^p(\mathbb{R}^n)$, and $X$ is a finite dimensional
noisy nonlinearity whose typical form is given by $X_t(\varphi)=\sum_{i=1}^N
x^{i}_t f_i(\varphi)$, where each $x=(x^{(1)},...,x^{(N)})$ is a
$\gamma$-H\"older function generating a rough path and each $f_i$ is a smooth
enough function defined on $L^p(\mathbb{R}^n)$. The generalization of the usual
rough path theory allowing to cope with such kind of systems is carefully
constructed.
http://arxiv.org/abs/0911.0618
Author(s): Wolfgang Koenig and Patrick Schmid
Abstract: We construct the conditional versions of a multidimensional random walk given
that it does not leave the Weyl chambers of type C and of type D, respectively,
in terms of a Doob h-transform. Furthermore, we prove functional limit theorems
for the rescaled random walks. This is an extension of recent work by
Eichelsbacher and Koenig who studied the analogous conditioning for the Weyl
chamber of type A. Our proof follows recent work by Denisov and Wachtel who
used martingale properties and a strong approximation of random walks by
Brownian motion. Therefore, we are able to keep minimal moment assumptions.
Finally, we present an alternate function that is amenable to an h-transform in
the Weyl chamber of type C.
http://arxiv.org/abs/0911.0631
Author(s): R. Liptser
Abstract: We propose a new sufficient condition for Girsanov's exponential
$mathfrak{z}_t = \exp(\int_0^t \alpha(\omega,s)dB_s - {1/2}\int_0^t
\alpha^2(\omega,s)ds)$ to be the martingale ($\E \mathfrak{z}_t\equiv1$), where
$B_t$ is Brownian motion and a random process $\alpha(\omega,t)$ is defined on
the same filtered probability space. We show that $|\alpha(\omega,t)|^2\le
\text{\rm const.} [1 + \sup_{s\in[0,t]}B^2_s], \forall t>0 \Rightarrow
\E\mathfrak{z}_t\equiv 1.$
http://arxiv.org/abs/0911.0641
Author(s): Marcus Isaksson and Guy Kindler and Elchanan Mossel
Abstract: We prove a quantitative version of the Gibbard-Satterthwaite theorem. We show
that a uniformly chosen voter profile for a neutral function $f$ of $q \geq 4$
alternatives and $n$ voters will be manipulable with probability at least
$10^{-7} \eps^2 (1-\eps)^2 n^{-3} q^{-32}$, where $\eps$ is the statistical
distance between $f$ and a dictator function.
Our proof is geometric. More specifically it extends the method of cannonical
paths to show that the measure of the profiles that lie on the interface of 3
or more outcomes is large. To the best of our knowledge our result is the first
isoperemetric result to establish interface of more than two bodies.
http://arxiv.org/abs/0911.0517
Author(s): Roberto Imbuzeiro Oliveira
Abstract: Consider any random graph model where potential edges appear independently,
with possibly different probabilities, and assume that the minimum expected
degree is omega(ln n). We prove that the adjacency matrix and the Laplacian of
that random graph are concentrated around the corresponding matrices of the
weighted graph whose edge weights are the probabilities in the random model.
While this may seem surprising, we will see that this matrix concentration
phenomenon is a generalization of known results about the Er\"{o}s-R\'{e}nyi
model. In particular, we will argue that matrix concentration is implicit the
theory of quasi-random graph properties.
We present two main applications of the main result. In bond percolation over
a graph G, we show that the Laplacian of the random subgraph is typically very
close to the Laplacian of G. As a corollary, we improve upon a bound for the
spectral gap due to Chung and Horn that was derived via much more complicated
methods.
In inhomogeneous random graphs, there are points X_1,...,X_n uniformly
distributed on the interval [0,1] and each pair is connected with probability p
kappa(X_i,X_j). We show that if \ln n/n<< p<< 1 and kappa is bounded, then the
adjacency matrix of the random graph is close to an integral operator defined
in terms of kappa.
Our main proof tool is a new concentration inequality for matrix martingales
that generalizes Freedman's inequality for the standard scalar setting.
http://arxiv.org/abs/0911.0600
Author(s): Paul Lescot (LMRS)
Abstract: Bernstein processes are Brownian diffusions that appear in Euclidean Quantum
Mechanics. Knowledge of the symmetries of the Hamilton-Jacobi-Bellman equation
associated with these processes allows one to obtain relations between
stochastic processes (Lescot-Zambrini, Progress in Probability, vols 58 and
59). More recently it has appeared that each one--factor affine interest rate
model (in the sense of Leblanc-Scaillet) could be described using such a
Bernstein process.
http://arxiv.org/abs/0911.2757
Author(s): Lihu Xu and Boguslaw Zegarlinski
Abstract: We study an infinite white $\alpha$-stable systems with unbounded
interactions, proving the existence by Galerkin approximation and ergodicity by
an $\alpha$-stable version of gradient bounds.
http://arxiv.org/abs/0911.2866
Author(s): Lihu Xu
Abstract: We proved the existence of an infinite dimensional stochastic system driven
by white $\alpha$-stable noises ($1<\alpha \leq 2$), and prove this system is
strongly mixing. Our method is by perturbing Ornstein-Uhlenbeck $\alpha$-stable
processes.
http://arxiv.org/abs/0911.2868
Author(s): Richard C. Bradley
Abstract: A strictly stationary sequence of random variables is constructed with the
following properties: (i) the random variables take the values -1 and +1 with
probability 1/2 each, (ii) every five of the random variables are independent,
(iii) the sequence is "causal" in a certain sense, (iv) the sequence has a
trivial double tail sigma-field, and (v) regardless of the normalization used,
the partial sums do not converge to a (nondegenerate) normal law. The example
has some features in common with a recent construction (for an arbitrary fixed
positive integer N), by Alexander Pruss and the author, of a strictly
stationary N-tuplewise independent counterexample to the central limit theorem.
http://arxiv.org/abs/0911.2905
Author(s): Olivier Ledoit and Sandrine P\'ech\'e
Abstract: We consider sample covariance matrices $S_N=\frac{1}{p}\Sigma_N^{1/2}X_NX_N^*
\Sigma_N^{1/2}$ where $X_N$ is a $N \times p$ real or complex matrix with
i.i.d. entries with finite $12^{\rm th}$ moment and $\Sigma_N$ is a $N \times
N$ positive definite matrix. In addition we assume that the spectral measure of
$\Sigma_N$ almost surely converges to some limiting probability distribution as
$N \to \infty$ and $p/N \to \gamma >0.$ We quantify the relationship between
sample and population eigenvectors by studying the asymptotics of functionals
of the type $\frac{1}{N} \text{Tr} (g(\Sigma_N) (S_N-zI)^{-1})),$ where $I$ is
the identity matrix, $g$ is a bounded function and $z$ is a complex number.
This is then used to compute the asymptotically optimal bias correction for
sample eigenvalues, paving the way for a new generation of improved estimators
of the covariance matrix and its inverse.
http://arxiv.org/abs/0911.3010
Author(s): M. Bloznelis (1) and F. G\"otze (2) and J. Jaworski (3) ((1) Vilnius University, Vilnius; (2) Bielefeld University, Bielefeld; (3) Adam Mickiewicz
University, Poznan)
Abstract: We present and investigate a general model for inhomogeneous random digraphs
with labeled vertices, where the arcs are generated independently, and the
probability of inserting an arc depends on the labels of its endpoints and its
orientation. For this model the critical point for the emergence of a giant
component is determined via a branching process approach.
http://arxiv.org/abs/0911.3013
Author(s): Emmanuelle Clement (LAMA) and Vlad Bally (LAMA)
Abstract: We establish an integration by parts formula in an abstract framework in
order to study the regularity of the law for processes solution of stochastic
differential equations with jumps, including equations with discontinuous
coefficients for which the Malliavin calculus developed by Bismut and
Bichteler, Gravereaux and Jacod fails.
http://arxiv.org/abs/0911.3017
Author(s): George T. Gilbert
Abstract: We consider the Independent Chip Model (ICM) for expected value in poker
tournaments. Our first result is that participating in a fair bet with one
other player will always lower one's expected value under this model. Our
second result is that the expected value for players not participating in a
fair bet between two players always increases. We show that neither result
necessarily holds for a fair bet among three or more players.
http://arxiv.org/abs/0911.3100
Author(s): Daniela Bertacchi and Nicolas Lanchier and Fabio Zucca
Abstract: We introduce spatially explicit stochastic processes to model multispecies
hostsymbiont interactions. The host environment is static, modeled by the
infinite percolation cluster of site percolation. Symbionts evolve on the
infinite cluster through contact or voter type interactions, where each host
may be infected by a colony of symbionts. In the presence of a single symbiont
species, the condition for invasion as a function of the density of the habitat
of hosts and the maximal size of the colonies is investigated in details. In
the presence of multiple symbiont species, it is proved that the community of
symbionts clusters in two dimensions whereas symbiont species may coexist in
higher dimensions.
http://arxiv.org/abs/0911.3107
Author(s): Scott A. McKinley and Lea Popovic and Michael C. Reed
Abstract: In this paper we develop a probabilistic micro-scale model and use it to
study macro-scale properties of axonal transport, the processes by which
materials are moved in the axons of neurons. By directly modeling the smallest
scale interactions, we can use recent microscopic experimental observations to
infer all the parameters of the model. Then using techniques from queueing
theory, we can predict macroscopic behavior in order to investigate three
important biological questions: (1) How homogeneous are axons at stochastic
equilibrium? (2) How quickly can axons return to stochastic equilibrium after
large local perturbations? (3) How inhomogeneous does deposition and turnover
make the axon?
http://arxiv.org/abs/0911.2722
Author(s): L. Cha\^ari and J.-C. Pesquet and J.-Y. Tourneret and Ph. Ciuciu and A. Benazza-Benyahia
Abstract: In many signal processing problems, it may be fruitful to represent the
signal under study in a frame. If a probabilistic approach is adopted, it
becomes then necessary to estimate the hyper-parameters characterizing the
probability distribution of the frame coefficients. This problem is difficult
since in general the frame synthesis operator is not bijective. Consequently,
the frame coefficients are not directly observable. This paper introduces a
hierarchical Bayesian model for frame representation. The posterior
distribution of the frame coefficients and model hyper-parameters is derived.
Hybrid Markov Chain Monte Carlo algorithms are subsequently proposed to sample
from this posterior distribution. The generated samples are then exploited to
estimate the hyper-parameters and the frame coefficients of the target signal.
Validation experiments show that the proposed algorithms provide an accurate
estimation of the frame coefficients and hyper-parameters. Application to
practical problems of image denoising show the impact of the resulting Bayesian
estimation on the recovered signal quality.
http://arxiv.org/abs/0911.2888
Author(s): Zakhar Kabluchko
Abstract: Let $\xi_i$, $i\in\mathbb N$, be independent copies of a L\'evy process
$\{\xi(t),t\geq 0\}$. Motivated by the results obtained previously in the
context of the random energy model, we prove functional limit theorems for the
process $$ Z_N(t)=\sum_{i=1}^N e^{\xi_i(s_N+t)} $$ as $N\to\infty$, where $s_N$
is a non-negative sequence converging to $+\infty$. The limiting process
depends heavily on the growth rate of the sequence $s_N$. If $s_N$ grows slowly
in the sense that $\liminf_{N\to\infty \log N/s_N>\lambda_2$ for some critical
value $\lambda_2>0$, then the limit is an Ornstein--Uhlenbeck process. However,
if $\lambda:=\lim_{N\to\infty}\log N/s_N\in(0,\lambda_2)$, then the limit is a
certain completely asymmetric $\alpha$-stable process $Y_{\alpha;\xi}$. We
prove that the process $Y_{\alpha;\xi}$ is stationary ($\alpha\neq 1$) and that
it shares a number of properties of the Gaussian Ornstein--Uhlenbeck process.
http://arxiv.org/abs/0911.4139
Author(s): Sebastian Jaimungal and Alex Kreinin and Angelo Valov
Abstract: In this article we study a problem related to the first passage and inverse
first passage time problems for Brownian motions originally formulated by
Jackson, Kreinin and Zhang (2009). Specifically, define $\tau_X = \inf\{t>0:W_t
+ X \le b(t) \}$ where $W_t$ is a standard Brownian motion, then given a
boundary function $b:[0,\infty) \to \RR$ and a target measure $\mu$ on
$[0,\infty)$, we seek the random variable $X$ such that the law of $\tau_X$ is
given by $\mu$. We characterize the solutions, prove uniqueness and existence
and provide several key examples associated with the linear boundary.
http://arxiv.org/abs/0911.4165
Author(s): Louis-Pierre Arguin and Michael Damron and Charles Newman and Daniel Stein
Abstract: We consider the Edwards-Anderson Ising spin glass model on the half-plane $Z
\times Z^+$ with zero external field and a wide range of choices, including
mean zero Gaussian, for the common distribution of the collection J of i.i.d.
nearest neighbor couplings. The infinite-volume joint distribution
$K(J,\alpha)$ of couplings J and ground state pairs $\alpha$ with periodic
(respectively, free) boundary conditions in the horizontal (respectively,
vertical) coordinate is shown to exist without need for subsequence limits. Our
main result is that for almost every J, the conditional distribution
$K(\alpha|J)$ is supported on a single ground state pair.
http://arxiv.org/abs/0911.4201
Author(s): Vyacheslav M. Abramov
Abstract: This paper studies large dam models where the difference between lower and
upper levels $L$ is assumed to be large. Passage across the levels leads to
damage, and the damage costs of crossing the lower or upper level are
proportional to the large parameter $L$. Input stream of water is described by
compound Poisson process, and the water cost depends upon current level of
water in the dam. The aim of the paper is to choose the parameters of output
stream (specifically defined in the paper) minimizing the long-run expenses.
The particular problem, where input stream is Poisson and water costs are not
taken into account has been studied in [Abramov, \emph{J. Appl. Prob.}, 44
(2007), 249-258]. The present paper partially answers the question \textit{How
does the structure of water costs affect the optimal solution?} In particular
the case of linear costs is studied.
http://arxiv.org/abs/0911.4228
Author(s): Scott A McKinley
Abstract: We consider the anomalous sub-diffusion of a class of Gaussian processes that
can be expressed in terms of sums of Ornstein-Uhlenbeck processes. As a generic
class of processes, we introduce a single parameter such that for any $\nu \in
(0,1)$ the process can be tuned to produce a mean-squared displacement with
$\E{x^2(t)} \sim t^\nu$ for large $t$.
The motivation for the specific structure of these sums of OU processes comes
from the Rouse chain model from polymer kinetic theory. We generalize the model
by studying the general dynamics of individual particles in networks of
thermally fluctuating beads connected by Hookean springs. Such a set-up is
similar to the study of Kac-Zwanzig heat bath models. Whereas the existing heat
bath literature places its assumptions on the spectrum of the Laplacian matrix
associated to the spring connection graph, we study explicit graph structures.
In this setting we prove a notion of universality for the Rouse chain's
well-known $\E{x^2(t)} \sim t^{1/2}$ scaling behavior. Subsequently we
demonstrate the existence of other anomalous behavior by changing the dimension
of the connection graph or by allowing repulsive forces among the beads.
http://arxiv.org/abs/0911.4293
Author(s): Joseph Najnudel and Ashkan Nikeghbali
Abstract: Is this paper we study penalisations of diffusions satisfying some technical
conditions, generalizing a result obtained by Najnudel, Roynette and Yor. If
one of these diffusions has probability distribution $\mathbb{P}$, then our
result can be described as follows: for a large class of families of
probability measures $(\mathbb{Q}_t)_{t \geq 0}$, each of them being absolutely
continuous with respect to $\mathbb{P}$, there exists a probability
$\mathbb{Q}_{\infty}$ such that for all events $\Lambda$ depending only on the
canonical trajectory up to a fixed time, $\mathbb{Q}_t (\Lambda)$ tends to
$\mathbb{Q}_{\infty} (\Lambda)$ when $t$ goes to infinity. In the cases we
study here, the limit measure $\mathbb{Q}_{\infty}$ is absolutely continous
with respect to a sigma-finite measure $\mathcal{Q}$, which does not depend on
the choice of the family of probabilities $(\mathbb{Q}_t)_{t \geq 0}$, but only
on $\mathbb{P}$. The relation between $\mathbb{P}$ and $\mathcal{Q}$ is
obtained in a very general framework by the authors of this paper.
http://arxiv.org/abs/0911.4365
Author(s): Kojiro Oshima and Josef Teichmann and Dejan Veluscek
Abstract: We review Fujiwara's scheme, a sixth order weak approximation scheme for the
numerical approximation of SDEs, and embed it into a general method to
construct weak approximation schemes of order $ 2m $ for $ m \in \mathbf{N} $.
Those schemes cannot be seen as cubature schemes, but rather as universal ways
how to extrapolate from a lower order weak approximation scheme, namely the
Ninomiya-Victoir scheme, for higher orders.
http://arxiv.org/abs/0911.4380
Author(s): Marco Oesting
Abstract: Generalized Brown-Resnick processes form a flexible class of stationary
max-stable processes based on Gaussian random fields. With regard to
applications fast and accurate simulation of these processes is an important
issue. In fact, Brown-Resnick processes that are generated by a dissipative
flow do not allow for good finite approximations using the definition of the
processes. On large intervals we get either huge approximation errors or very
long operating times. Looking for solutions of this problem, we give different
representations of the generalized Brown-Resnick processes - including random
shifting and a mixed moving maxima representation - and derive various kinds of
finite approximations that can be used for simulation purposes. Furthermore,
error bounds are calculated in the case of the original process by Brown and
Resnick (1977). For a one-paramatric class of Brown-Resnick processes based on
the fractional Brownian motion we perform a simulation study and compare the
results of the different methods concerning their approximation quality. The
presented simulation techniques turn out to provide remarkable improvements.
http://arxiv.org/abs/0911.4389
Author(s): Jonathan Farfan and Alexandre B. Simas and Fabio J. Valentim
Abstract: We study the equilibrium fluctuations for a gradient exclusion process with
conductances in random environments, which can be viewed as a central limit
theorem for the empirical distribution of particles when the system starts from
an equilibrium measure.
http://arxiv.org/abs/0911.4394
Author(s): Alexandre B. Simas
Abstract: We prove the hydrodynamic limit for a particle system in which particles may
have different velocities. We assume that we have two infinite reservoirs of
particles at the boundary: this is the so-called boundary driven process. The
dynamics we considered consists of a weakly asymmetric simple exclusion process
with collision among particles having different velocities.
http://arxiv.org/abs/0911.4423
Author(s): Jonathan Farfan and Alexandre B. Simas and Fabio J. Valentim
Abstract: We prove the dynamical large deviations for a particle system in which
particles may have different velocities. We assume that we have two infinite
reservoirs of particles at the boundary: this is the so-called boundary driven
process. The dynamics we considered consists of a weakly asymmetric simple
exclusion process with collision among particles having different velocities.
http://arxiv.org/abs/0911.4425
Author(s): Bruce Hajek
Abstract: A tight upper bound is given involving the maximum of a supermartingale.
Specifically, it is shown that if $Y$ is a semimartingale with initial value
zero and quadratic variation process $[Y,Y]$ such that $Y + [Y,Y]$ is a
supermartingale, then the probability the maximum of $Y$ is greater than or
equal to a positive constant $a$ is less than or equal to $1/(1+a).$ The proof
is inspired by dynamic programming. Complements and extensions are also given.
http://arxiv.org/abs/0911.4444
Author(s): Jan Kallsen and Johannes Muhle-Karbe
Abstract: We consider the problem of maximizing expected utility from terminal wealth
in models with stochastic factors. Using martingale methods and a conditioning
argument, we determine the optimal strategy for power utility under the
assumption that the increments of the asset price are independent conditionally
on the factor process.
http://arxiv.org/abs/0911.3608
Author(s): Eyal Lubetzky and Benny Sudakov and Van Vu
Abstract: A random $n$-lift of a base graph $G$ is its cover graph $H$ on the vertices
$[n]\times V(G)$, where for each edge $u v$ in $G$ there is an independent
uniform bijection $\pi$, and $H$ has all edges of the form $(i,u),(\pi(i),v)$.
A main motivation for studying lifts is understanding Ramanujan graphs, and
namely whether typical covers of such a graph are also Ramanujan.
Let $G$ be a graph with largest eigenvalue $\lambda_1$ and let $\rho$ be the
spectral radius of its universal cover. Friedman (2003) proved that every "new"
eigenvalue of a random lift of $G$ is $O(\rho^{1/2}\lambda_1^{1/2})$ with high
probability, and conjectured a bound of $\rho+o(1)$, which would be tight by
results of Lubotzky and Greenberg (1995). Linial and Puder (2008) improved
Friedman's bound to $O(\rho^{2/3}\lambda_1^{1/3})$. For $d$-regular graphs,
where $\lambda_1=d$ and $\rho=2\sqrt{d-1}$, this translates to a bound of
$O(d^{2/3})$, compared to the conjectured $2\sqrt{d-1}$.
Here we analyze the spectrum of a random $n$-lift of a $d$-regular graph
whose nontrivial eigenvalues are all at most $\lambda$ in absolute value. We
show that with high probability the absolute value of every nontrivial
eigenvalue of the lift is $O((\lambda \vee \rho) \log \rho)$. This result is
tight up to a logarithmic factor, and for $\lambda \leq d^{2/3-\epsilon}$ it
substantially improves the above upper bounds of Friedman and of Linial and
Puder. In particular, it implies that a typical $n$-lift of a Ramanujan graph
is nearly Ramanujan.
http://arxiv.org/abs/0911.4148
Author(s): Alexandre B. Simas and Fabio J. Valentim
Abstract: Fix strictly increasing right continuous functions with left limits $W_i:\bb
R \to \bb R$, $i=1,...,d$, and let $W(x) = \sum_{i=1}^d W_i(x_i)$ for $x\in\bb
R^d$. We construct the $W$-Sobolev spaces, which consist of functions $f$
having weak generalized gradients $\nabla_W f = (\partial_{W_1}
f,...,\partial_{W_d} f)$. Several properties, that are analogous to classical
results on Sobolev spaces, are obtained. $W$-generalized elliptic and parabolic
equations are also established, along with results on existence and uniqueness
of weak solutions of such equations. Homogenization results of suitable random
operators are investigated. Finally, as an application of all the theory
developed, we prove a hydrodynamic limit for gradient processes with
conductances (induced by $W$) in random environments.
http://arxiv.org/abs/0911.4177
Author(s): Jan Maas and Jan van Neerven
Abstract: Let (P(t)) be the Ornstein-Uhlenbeck semigroup associated with the stochastic
Cauchy problem dU(t) = AU(t)dt + dW_H(t), where A is the generator of a
C_0-semigroup (S(t)) on a Banach space E, H is a Hilbert subspace of E, and
(W_H(t)) is an H-cylindrical Brownian motion. Assuming that (S(t)) restricts to
a C_0-semigroup on H, we obtain L^p-bounds for the gradient D_H P(t). We show
that if (P(t)) is analytic, then the invariance assumption is fulfilled. As an
application we determine the L^p-domain of the generator of (P(t)) explicitly
in the case where (S(t)) restricts to a C_0-semigroup on H which is similar to
an analytic contraction semigroup.
http://arxiv.org/abs/0911.4336
Author(s): Sonny Ben-Shimon and Michael Krivelevich and Benny Sudakov
Abstract: For an increasing monotone graph property $\mP$ the \emph{local resilience}
of a graph $G$ with respect to $\mP$ is the minimal $r$ for which there exists
of a subgraph $H\subseteq G$ with all degrees at most $r$ such that the removal
of the edges of $H$ from $G$ creates a graph that does not possess $\mP$. This
notion, which was implicitly studied for some ad-hoc properties, was recently
treated in a more systematic way in a paper by Sudakov and Vu. Most research
conducted with respect to this distance notion focused on the random graph
model $\GNP$ and some families of pseudo-random graphs with respect to several
graph properties such as containing a perfect matching and being Hamiltonian,
to name a few. In this paper we continue to explore the local resilience
notion, but turn our attention to random and pseudo-random \emph{regular}
graphs of constant degree. We investigate the local resilience of the typical
random $d$-regular graph with respect to edge and vertex connectivity,
containing a perfect matching, and being Hamiltonian. In particular we prove
that for every positive $\varepsilon$ and large enough values of $d$ with high
probability the local resilience of the random $d$-regular graph with respect
to being Hamiltonian is at least $(1-\varepsilon)d/6$. Using the same ideas we
also prove a similar result for the Binomial random graph model, namely that
for every positive $\varepsilon$ and large enough values of $K$ if $p\geq
\frac{K\ln n}{n}$ then with hight probability the local resilience of $\GNP$
with respect to being Hamiltonian is at least $(1-\varepsilon)np/6$.
http://arxiv.org/abs/0911.4351
Author(s): Katarzyna Lubnauer and Andrzej {\L}uczak
Abstract: We characterise the class of probability operators belonging to the domain of
attraction of Gaussian limits in the setup which is a slight generalisation of
Urbanik's scheme of noncommutative probability limit theorems.
http://arxiv.org/abs/0911.4426
Author(s): Raluca Balan and Ciprian Tudor (LPP)
Abstract: We consider the linear stochastic wave equation with spatially homogenous
Gaussian noise, which is fractional in time with index $H>1/2$. We show that
the necessary and sufficient condition for the existence of the solution is a
relaxation of the condition obtained in \cite{dalang99}, when the noise is
white in time. Under this condition, we show that the solution is
$L^2(\Omega)$-continuous. Similar results are obtained for the heat equation.
Unlike the white noise case, the necessary and sufficient condition for the
existence of the solution in the case of the heat equation is {\em different}
(and more general) than the one obtained for the wave equation.
http://arxiv.org/abs/0912.3865
Author(s): Balazs Rath and Laszlo Szakacs
Abstract: We define the edge reconnecting model, a random multigraph evolving in time.
At each time step we change one endpoint of a uniformly chosen edge: the new
endpoint is chosen by linear preferential attachment. We consider a sequence of
edge reconnecting models where the sequence of initial multigraphs is
convergent in a sense which is a natural generalization of the Lovasz-Szegedy
notion of convergence of dense graph sequences. We investigate how the limit
objects evolve under the edge reconnecting dynamics if we rescale time
properly: we give the complete characterization of the time evolution of the
limiting object from its initial state up to the stationary state using the
theory of exchangeable arrays, the Polya urn model, queuing and diffusion
processes.
http://arxiv.org/abs/0912.3904
Author(s): Panayotis Mertikopoulos and Aris L. Moustakas
Abstract: We consider the problem of routing traffic in a network whose users try to
minimise their delays by adhering to a simple learning scheme inspired by the
replicator dynamics of evolutionary game theory. Such users indeed converge to
Wardrop equilibrium if their information is accurate, but a key part in the
process is played by the redundancy of the network (a new concept which
measures the "linear dependence" of the users' paths).
On the other hand, a major challenge occurs when the users' delays fluctuate
unpredictably due to (random) external factors. In that case, we show that
strict Wardrop equilibria remain (stochastically) stable, irrespective of the
fluctuations' magnitude. In fact, if the network has no redundancy and the
learning rate of the users is sufficiently slow, we show that the long-term
average of the users' traffic distribution converges to the vicinity of an
equilibrium and estimate the corresponding stationary distribution.
http://arxiv.org/abs/0912.4012
Author(s): Jeffrey Matayoshi
Abstract: For a random polynomial with standard normal coefficients, two cases of the
K-level crossings have been considered by Farahmand. When the coefficients are
independent, Farahmand was able to derive an asymptotic value for the expected
number of level crossings, even if K is allowed to grow to infinity.
Alternatively, it was shown that when the coefficients have a constant
covariance, the expected number of level crossings is reduced by half. In this
paper we are interested in studying the behavior for dependent standard normal
coefficients where the covariance is decaying and no longer constant. Using
techniques similar to those of Farahmand, we will be able to show that for a
wide range of covariance functions behavior similar to the independent case can
be expected.
http://arxiv.org/abs/0912.4065
Author(s): Hugo Duminil-Copin and Cl\'ement Hongler and Pierre Nolin
Abstract: We prove Russo-Seymour-Welsh-type uniform bounds on crossing probabilities
for the FK Ising model at criticality, independent of the boundary conditions.
Our proof relies mainly on Smirnov's fermionic observable for the FK Ising
model, which allows us to get precise estimates on boundary connection
probabilities. It remains purely discrete, in particular we do not make use of
any continuum limit, and it can be used to derive directly several noteworthy
results - some new and some not - among which the fact that there is no
spontaneous magnetization at criticality, tightness properties for the
interfaces, and the existence of several critical exponents, in particular the
half-plane one-arm exponent.
http://arxiv.org/abs/0912.4253
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