[PAS] Probability Abstracts 113

[Probability Abstract Service]

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.

This letter can be also found on line at
http://pas.imstat.org/Letters/letter_113.shtml

Happy 2010!

stefano



9333. The critical random barrier for the survival of branching random walk with absorption
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 


9334. Bernstein processes, Euclidean Quantum Mechanics and Interest Rate Models
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 


9335. Limit Theorems for Beta-Jacobi Ensembles
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 


9336. Diffusivity bounds for 1d Brownian polymers
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 


9337. Law of large numbers for a class of random walks in dynamic random environments
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 


9338. Critical Branching Random Walks with Small Drift
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 


9339. Time irregularity of generalized Ornstein--Uhlenbeck processes
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 


9340. Critically loaded multi-server queues with abandonments, retrials, and time-varying parameters
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 


9341. Scaling for a one-dimensional directed polymer with constrained endpoints
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 


9342. The infinite volume limit of Ford's alpha model
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 


9343. Exit times of diffusions with incompressible drift
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 


9344. Explicit identities for L\'evy processes associated to symmetric stable processes
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 


9345. Priors for the Bayesian star paradox
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 


9346. Fleming-Viot Processes in an Environment
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 


9347. An integral test on time dependent local extinction for super-coalescing Brownian motion with Lebesgue initial measure
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 


9348. Occupation times of subcritical branching immigration systems with Markov motion, clt and deviations principles
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 


9349. Arm exponents in high dimensional percolation
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 


9350. Tail asymptotics for the total progeny of the critical killed branching random walk
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 


9351. The number of small blocks in exchangeable random partitions
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 


9352. Large Deviations for Deterministic Walks in Random Environments
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 


9353. Limit theorems for Markov processes indexed by continuous time Galton-Watson trees
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 


9354. Limits of determinantal processes near a tacnode
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 


9355. Optimal Paths on the Space-Time SINR Random Graph
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 


9356. Survival probability of the branching random walk killed below a linear boundary
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 


9357. Existence and uniqueness of the stationary measure in the continuous Abelian sandpile
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 


9358. Desordre et phenomenes de localisation
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 


9359. Random walks with occasionally modified transition probabilities
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 


9360. A stochastic maximum principle via Malliavin calculus
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 


9361. $\gamma$-Radonifying operators -- a survey
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 


9362. Probabilistic one-player Ramsey games via deterministic two-player games
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 


9363. Asymptotic independence for unimodal densities
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 


9364. Brownian limits, local limits, extreme value and variance asymptotics for convex hulls in the ball
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 


9365. On some strong ratio limit theorems for heat kernels
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 


9366. A note on large deviations for the stable marriage of Poisson and Lebesgue with random appetites
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 


9367. Probability Bracket Notation: the Unified Expressions of Conditional Expectation and Conditional Probability in Quantum Modeling
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 


9368. Formulas for the Laplace Transform of Stopping Times based on Drawdowns and Drawups
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 


9369. Geometric Influences
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 


9370. Harnack Inequalities and Applications for SDEs with Non-Additive Noise and Neumann Semigroup on Non-Convex Manifold
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 


9371. On the critical point of the Random Walk Pinning Model in dimension d=3
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 


9372. Deviation inequalities for sums of weakly dependent time series
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 


9373. Quantum Markov fields on graphs
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 


9374. A note on loglog distances in a power law random intersection graph
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 


9375. Borel type bounds for the self-avoiding walk connective constant
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 


9376. Travelling waves and homogeneous fragmentation
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 


9377. The Euler scheme for Feller processes
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 


9378. A note on the recurrence of edge reinforced random walks
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 


9379. Which Connected Spatial Networks on Random Points have Linear Route-Lengths?
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 


9380. The Shape Theorem for Route-lengths in Connected Spatial Networks on Random Points
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 


9381. Variable Length Coding of Two-Sided Asymptotically Mean Stationary Measures
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 


9382. Asynchronous CDMA Systems with Random Spreading-Part II: Design Criteria
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 


9383. Regularity of the Exercise Boundary for American Put Options on Assets with Discrete Dividends
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 


9384. The almost-sure population growth rate in branching Brownian motion with a quadratic breeding potential
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 


9385. Consistent Minimal Displacement of Branching Random Walks
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 


9386. Harmonic functions, h-transform and large deviations for random walks in random environments in dimensions four and higher
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 


9387. Stein's method and stochastic orderings
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 


9388. The Symbol Associated with the Solution of a Stochastic Differential Equation
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 


9389. Transition density estimates for a class of L\'evy and L\'evy-type processes
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 


9390. Local time and Tanaka formula for G-Brownian Motion
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 


9391. Conditional Distribution of Heavy Tailed Random Variables on Large Deviations of their Sum
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 


9392. Explicit Conditions for the Convergence of Point Processes Associated to Stationary Arrays
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 


9393. Topological Properties of an Exponential Random Geometric Graph Process
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 


9394. Improved mixing time bounds for the Thorp shuffle
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 


9395. Sharp Green Function Estimates for $\Delta + \Delta^{\alpha/2}$ in $C^{1,1}$ Open Sets and Their Applications
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 


9396. An improvement of the Berry--Esseen inequality with applications to Poisson and mixed Poisson random sums
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 


9397. The Bivariate Normal Copula
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 


9398. The $G/GI/N$ queue in the Halfin--Whitt regime
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 


9399. Martingale approximation and optimality of some conditions for the central limit theorem
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 


9400. Existence of the stationary regime of a Non-Markovian Stochastic Differential Equation
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 


9401. Characteristic Polynomials of Sample Covariance Matrices: The Non-Square Case
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 


9402. Laplce, Fourier, and stochastic diffusion
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 


9403. Stochastic characterization of harmonic sections and a Liouville theorem
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 


9404. Markov processes on time-like graphs
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 


9405. Invasion percolation on the Poisson-weighted infinite tree
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 


9406. Correlation Decay in Random Decision Networks
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 


9407. Fractal and Smoothness Properties of Space-Time Gaussian Models
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 


9408. Brownian motion and the Dirichlet problem at infinity on two-dimensional Cartan-Hadamard manifolds
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 


9409. Variance Optimal Hedging for continuous time processes with independent increments and applications
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 


9410. Painleve functions in statistical physics
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 


9411. Almost sure central limit theorems on the Wiener space
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 


9412. Central limit theorem for the modulus of continuity of the Brownian local time in $L^3(\mathbb{R})$
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 


9413. Gaussian Covariance faithful Markov Trees
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 


9414. Combinatorial approach to the interpolation method and scaling limits in sparse random graphs
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 


9415. A $d$-dimensional Brownian motion as a weak limit from a one-dimensional Poisson process
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 


9416. Universality in the bulk of the spectrum for complex sample covariance matrices
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 


9417. Distributed-order fractional Cauchy problems on bounded domains
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 


9418. Lp solutions of backward stochastic Volterra integral equations
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 (1
<2), which extends the results in [15]. The unique solvability of adapted solutions of BSVIEs in Lp (p>1) is also considered, which also generalize the results in [6] and [11].

http://arxiv.org/abs/0912.2567 


9419. Global Alignment of Molecular Sequences via Ancestral State Reconstruction
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 


9420. Renewal theory in analysis of tries and strings
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 


9421. A sufficient condition for bifurcation in random dynamical systems
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 


9422. A Pluzhnikov's Theorem, Brownian motions and Martingales in Lie Group with skew-symmetric connections
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 


9423. Ergodicity and Gaussianity for Spherical Random Fields
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 


9424. Random Subnetworks of Random Sorting Networks
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 


9425. On some properties of a universal sigma-finite measure associated with a remarkable class of submartingales
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 


9426. Linear State space theory in the white noise space setting
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 


9427. Rate of escape and central limit theorem for the supercritical Lamperti problem
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 


9428. Regularization properties of the 2D homogeneous Boltzmann equation without cutoff
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 


9429. Regularity of probability laws using the Riesz transform and Sobolev spaces techniques
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 


9430. One Dimensional Quantum Walks with Memory
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 


9431. Susceptibility of random graphs with given vertex degrees
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 


9432. Maximum GCD Among Pairs of Random Integers
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 


9433. Coalescing systems of Brownian particles on the Sierpinski gasket and stable particles on the line or circle
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 


9434. Deterministic Thinning of Finite Poisson Processes
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 


9435. Random Schrodinger operators on long boxes, noise explosion and the GOE
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 


9436. Some applications of duality for L\'evy processes in a half-line
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 


9437. Continuous-time quantum walk on integer lattices and homogeneous trees
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 


9438. From U-bounds to isoperimetry with applications to H-type groups
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 


9439. Reflected BSDE with stochastic Lipschitz coefficient
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 


9440. Malliavin calculus for fractional delay equations
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 


9441. Reflected BSDEs and continuous solutions of parabolic obstacle problem for semilinear PDEs in divergence form
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 


9442. Estimates on the tail behavior of Gaussian polynomials. The discussion of a result of Latala
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 


9443. An equivalence between harmonic sections and sections that are harmonic maps
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 


9444. Conformally invariant scaling limits in planar critical percolation
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 


9445. Notes on the Cauchy Problem for Backward Stochastic Partial Differential Equations
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 


9446. Free Probability Theory
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 


9447. Perfect Matchings as IID Factors on Non-Amenable Groups
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 


9448. Multidimensional $q$-Normal and related distributions
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 


9449. One-dimensional long-range diffusion-limited aggregation III -- The limit aggregate
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 


9450. A PDE for Nonintersecting Brownian Motions and Applications
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 


9451. Asymptotic expansion of semi-Markov random evolutions
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 


9452. Asymptotic expansion of Markov random evolution
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 


9453. On weak convergence for stochastic evolutionary systems in average principle
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 


9454. Distribution Function of Markovian Random Evolution in Rn
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 


9455. Levy Approximation of Impulsive Recurrent Process with Markov Switching
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 


9456. The conflict interaction between two complex systems. Cyclic migration
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 


9457. Fluctuation limits of the super-Brownian motion with a single point catalyst
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 


9458. Limit theorems for some adaptive MCMC algorithms with subgeometric
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 


9459. Harmonic maps on amenable groups and a diffusive lower bound for random walks
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 


9460. Non-Linear Evolution Equations Driven by Rough Paths
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 


9461. Log-Harnack Inequality for Stochastic Differential Equations in Hilbert Spaces and its Consequences
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 


9462. Bounds on the Speed and on Regeneration Times for Certain Processes on Regular Trees
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 


9463. The simple harmonic urn
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 


9464. Laws of large numbers and nearest neighbor distances
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 


9465. Local Times of Multidimensional $\alpha$-time Fractional Brownian Motion
Author(s): Erkan Nane and Dongsheng Wu and Yimin Xiao

Abstract: For $0<\alpha \leq 2$ and $0<1$, an $\alpha$-time fractional Brownian motion is an iterated process $Z = \{Z(t)=W(Y(t)), t \ge 0\}$ obtained by taking a fractional Brownian motion $\{W(t), t\in \RR{R} \}$ with Hurst index $0<1$ and replacing the time parameter with a strictly $\alpha$-stable L\'evy process $\{Y(t), t\geq 0 \}$ in $\RR{R}$ independent of $W(t)$. Define $d$-dimensional $\alpha$-time fractional Brownian motion $X = \{X(t), t \in \R_+$\} by $$ X(t)=\big(X_{1}(t),..., X_{d}(t) \big) (t\geq 0), $$ where $X_{1},..., X_{d}$ are independent copies of $Z$. We investigate the existence, joint continuity and sharp H\"older conditions in the set variable of the local time $$ L=\big\{L(x,B): x\in \RR{R}^{d}, B\in \mathcal{B}(\RR{R}_{+}) \big\} $$ of $X$, where $\mathcal{B}(\RR{R}_{+})$ is the Borel $\sigma$-algebra of $\RR{R}_{+}$. Our methods rely on the strong local nondeterminism of fractional Brownian motion.

http://arxiv.org/abs/0911.0357 


9466. Analyticity of the Wiener-Hopf factors and valuation of exotic options in L\'evy models
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 


9467. Approximating the moments of marginals of high dimensional distributions
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 


9468. Context-free pairs of groups. II - cuts, tree sets, and random walks
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 


9469. Entropy sensitivity of languages defined by infinite automata, via Markov chains with forbidden transitions
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 


9470. Phenotypic diversity and population growth in fluctuating environment: a MBPRE approach
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 


9471. A local limit theorem for random walks on the chambers of $\tilde{A}_2$ buildings
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 


9472. Hydrodynamic limits for the free Kawasaki dynamics of continuous particle systems
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 


9473. Random covariance matrices: Universality of local statistics of eigenvalues
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 


9474. Computable de Finetti measures
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 


9475. Combinatorics of the three-parameter PASEP partition function
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 


9476. Gaussian and non-Gaussian processes of zero power variation
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 


9477. On the LSL for random fields
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 


9478. Evaluating Point Forecasts
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 


9479. Random Convex Hulls and Extreme Value Statistics
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 


9480. Persistence of Activity in Threshold Contact Processes, an "Annealed Approximation" of Random Boolean Networks
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 


9481. From Stein Identities to Moderate Deviations
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 


9482. Poisson Thickening
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 


9483. Minimising the time to a decision
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 


9484. Solutions of semilinear wave equation via stochastic cascades
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 


9485. On the probability that integrated random walks stay positive
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 


9486. A Cluster Limit Theorem for Infinitely Divisible Point Processes
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 


9487. Asymptotic and spectral properties of exponentially \phi-ergodic Markov processes
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 


9488. Some Darling-Siegert relationships connected with random flights
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 


9489. Large deviation principle for one-dimensional random walk in dynamic random environment: attractive spin-flips and simple symmetric exclusion
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 


9490. Wave Propagation in Shallow-Water Acoustic Random Waveguides
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 


9491. Restricted exchangeable partitions and embedding of associated hierarchies in continuum random trees
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 


9492. Scaling Limits for Random Walks on Long Range Percolation Clusters
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 


9493. The L\'evy-Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups
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 


9494. Central limit theorem for first-passage percolation time across thin cylinders
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 


9495. Asynchronous CDMA Systems with Random Spreading-Part I: Fundamental Limits
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 


9496. Asymptotic fluctuations of representations of the unitary groups
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 


9497. Tightness for a stochastic Allen--Cahn equation
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 


9498. Scaling limits for shortest path lengths along the edges of stationary tessellations - Supplementary material
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 


9499. Generalized Gamma Process: some results about composition and subordination
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 


9500. Truncated Variation, Upward Truncated Variation and Downward Truncated Variation of Brownian Motion with Drift - their Characteristics and Applications
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 


9501. Cucker-Smale Flocking Under Hierarchical Leadership and Random Interactions
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 


9502. Excursions and local limit theorems for Bessel-like random walks
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 


9503. Invariance principle for Mott variable range hopping and other walks on point processes
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 


9504. Asymptotic Expansions for the Heat Kernel and the Trace of a Stochastic Geodesic Flow
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 


9505. Strong approximations in a charged-polymer model
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 


9506. Sequential optimizing strategy in multi-dimensional bounded forecasting games
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 


9507. Almost sure multifractal spectrum for the tip of an SLE curve
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 


9508. On the rate of convergence of loop-erased random walk to SLE(2)
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 


9509. On averages of randomized class functions on the symmetric groups and their asymptotics
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 


9510. Dissipative stochastic evolution equations driven by general Gaussian and non-Gaussian noise
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 


9511. Visible parts of fractal percolation
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 


9512. The Ghirlanda-Guerra identities without averaging
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 


9513. Sampling the Fermi statistics and other conditional product measures
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 


9514. Regeneration for interacting particle systems with interactions of infinite range
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 


9515. Quasi-invariance and integration by parts for determinantal and permanental processes
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 


9516. Weak order for the discretization of the stochastic heat equation driven by impulsive noise
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 


9517. Exact asymptotics for a distribution density of certain Levy functionals
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 


9518. Stochastic integral characterizations of semi-selfdecomposable distributions and related Ornstein-Uhlenbeck type processes
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 


9519. G-L\'{e}vy Processes under Sublinear Expectations
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 


9520. Asymptotic behaviour of a family of time-inhomogeneous diffusions
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 


9521. Riemannian Median and Its Estimation
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 


9522. The cycle structure of compositions of random involutions
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 


9523. On subhamonicity for symmetric Markov processes
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 


9524. On spectral representations of tensor random fields on the sphere
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 


9525. U-statistics and random subgraph counts: Multivariate normal approximation via exchangeable pairs and embedding
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 


9526. Multiple defaults and contagion risks
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 


9527. A characterization of Einstein manifolds
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 


9528. Viscosity and Principal-Agent Problem
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 


9529. Bonds with volatilities proportional to forward rates
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 


9530. Closeness to the Diagonal for Longest Common Subsequences
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 


9531. Central Binomial Tail Bounds
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 


9532. On the relation between plausibility logic and the maximum-entropy
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 


9533. Ergodicity of a stress release point process seismic model with aftershocks
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 


9534. A "bang-bang" principle for predicting the supremum of a random walk or Le'vy process
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 


9535. Totally Asymmetric Zero-Range process in the Rarefaction Fan
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 


9536. The weak limit of Ising models on locally tree-like graphs
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 


9537. New estimates of the convergence rate in the Lyapunov theorem
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 


9538. Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures
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 


9539. Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation
Author(s): Rapha\"el Rossignol and Marie Th\'eret

Abstract: Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random capacities. 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 


9540. Optimal Gaussian density estimates for a class of stochastic equations with additive noise
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 


9541. A simple proof for monotone CLT
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 


9542. Influence of spatial correlation for directed polymers
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 


9543. SLE(6) and the geometry of diffusion fronts
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 


9544. Geometric Representations of Hypergraphs for Prior Specification and Posterior Sampling
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 


9545. On Prekopa-Leindler inequalities on metric-measure spaces
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 


9546. Exchangeability and sets of desirable gambles
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 


9547. A central limit theorem for a two-dimensional random walk in a cone
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 


9548. On the Rate of Approximation in Finite-Alphabet Longest Increasing Subsequence Problems
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 


9549. On $\ell_1$-regularized estimation for nonlinear models that have sparse underlying linear structures
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 


9550. Jucys-Murphy Elements and Unitary Matrix Integrals
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 


9551. The spectrum of random k-lifts of large graphs (with possibly large k)
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 


9552. Quasi Ornstein-Uhlenbeck Processes
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 


9553. Estimates for moments of supremum of reflected fractional Brownian motion
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 


9554. Variations and Hurst index estimation for a Rosenblatt process using longer filters
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 


9555. Representation formulae for the fractional Brownian motion
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 


9556. Long time behavior of diffusions with Markov switching
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 


9557. On the Laplace transform of perpetuities with thin tails
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 


9558. Mod-Gaussian convergence and the value distribution of $\zeta(1/2+it)$ and related quantities
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 


9559. Mod-discrete expansions
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 


9560. The first passage event for sums of dependent L\'evy processes with applications to insurance risk
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 


9561. Conditional limit theorems for regulated fractional Brownian motion
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 


9562. Concentration of measure and spectra of random matrices: Applications to correlation matrices, elliptical distributions and beyond
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 


9563. One-Dimensional Diffusions That Eventually Stop Down-Crossing
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 


9564. Approximation of projections of random vectors
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 


9565. A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods
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 


9566. Disorder relevance for the random walk pinning model in d=3
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 


9567. A remarkable $\sigma$-finite measure associated with last passage times and penalisation problems
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 


9568. Perpetuities with thin tails revisited
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 


9569. Contact processes on random graphs with power law degree distributions have critical value 0
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 


9570. Time-Homogeneous Diffusions with a Given Marginal at a Random Time
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 


9571. Nonlinear stochastic wave equations: Blow-up of second moments in $L^2$-norm
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 


9572. Zero Range Condensation at Criticality
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 


9573. A duality approach to the worst case value at risk for a sum of dependent random variables with known covariances
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 


9574. Functions of random walks on hyperplane arrangements
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 


9575. Determinantal point processes
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 


9576. Kernel estimators of asymptotic variance for adaptive Markov Chain Monte Carlo
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 


9577. On q-Gaussians and Exchangeability
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 


9578. Law of the Iterated Logarithm for U-Statistics of Weakly Dependent Observations
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 


9579. Law of the absorption time of positive self-similar Markov processes
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 


9580. Improved moment estimates for invariant measures of semilinear diffusions in Hilbert spaces and applications
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 


9581. Numerical analysis of the rebellious voter model
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 


9582. The Graph, Range and Level Sets Dimension Spectra of Signed Random Cascades
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 


9583. Path Integral over Reparametrizations: Levy Flights versus Random Walks
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 


9584. Continuity of mutual entropy in the large signal-to-noise ratio limit
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 


9585. Approximation of the finite dimensional distributions of multiple fractional integrals
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 


9586. Asymptotics of the odometer function for the internal Diffusion Limited Aggregation model
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 


9587. Borel Games with Lower-Semi-Continuous Payoffs
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 


9588. Lipschitz percolation
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 


9589. Optimal control problem of fully coupled forward-backward stochastic systems with Poisson jumps under partial information
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 


9590. On linear systems and tau functions associated with Lame's equation and Painleve's equation VI
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 


9591. Small deviations for beta ensembles
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 


9592. $L^p$-solutions of Reflected Backward Doubly Stochastic Differential Equations
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 


9593. Large-time asymptotics of the gyration radius for long-range statistical-mechanical models
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 


9594. Stochastic velocity motions and processes with random time
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 


9595. Diffusive limit for self-repelling Brownian polymers in three and more dimensions
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 


9596. Exponential growth of ponds in invasion percolation on regular trees
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 


9597. Large Deviations for Multi-valued Stochastic Differential Equations
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 


9598. Diffusion Limits of Limited Processor Sharing Queues
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 


9599. Sensitivity of the asymptotic behaviour of meta distributions
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 


9600. Archimedes' principle for Brownian liquid
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 


9601. Functional limit theorems for linear processes in the domain of attraction of stable laws
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 


9602. Optimal minimax strategy in a dice game
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 


9603. Uniformity of the Uncovered Set of Random Walk and Cutoff for Lamplighter Chains
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 


9604. On multivariate quantiles under partial ordering
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 


9605. The Cohen-Lenstra Heuristic: Methodology and Results
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 


9606. The Global Cohen-Lenstra Heuristic
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 


9607. Buffon's needle landing near Besicovitch irregular self-similar sets
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 


9608. Homogenization of a singular random one dimensional parabolic PDE with time varying coefficients
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 


9609. The prolific backbone for supercritical superdiffusions
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 


9610. A Wiener-Hopf Monte Carlo simulation technique for L\'evy processes
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 


9611. Spectral dimension and random walks on the two dimensional uniform spanning tree
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 


9612. A new construction of the $\sigma$-finite measures associated with submartingales of class $(\Sigma)$
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 


9613. Modeling and simulation with operator scaling
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 


9614. Joint Vertex Degrees in an Inhomogeneous Random Graph Model
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 


9615. Weak Convergence Results for Multiple Generations of a Branching Process
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 


9616. A general comparison theorem for 1-dimensional anticipated BSDEs
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 


9617. Can the Adaptive Metropolis Algorithm Collapse Without the Covariance Lower Bound?
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 


9618. Empires and Percolation: Stochastic Merging of Adjacent Regions
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 


9619. Ergodic Properties of Sum- and Max- Stable Stationary Random Fields via Null and Positive Group Actions
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 


9620. Poisson boundary of groups acting on real trees
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 


9621. Non-linear Rough Heat Equations
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 


9622. Random walks conditioned to stay in Weyl chambers of type C and D
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 


9623. The Girsanov exponential martingale
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 


9624. The Geometry of Manipulation - a Quantitative Proof of the Gibbard Satterthwaite Theorem
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 


9625. Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges
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 


9626. A note on affine interest rate models
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 


9627. Existence and Ergodicity of infinite white $\alpha$-stable Systems with unbounded interactions
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 


9628. Ergodicity of infinite white $\alpha$-stable Systems with linear and bounded interactions
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 


9629. A strictly stationary, "causal," 5-tuplewise independent counterexample to the central limit theorem
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 


9630. Eigenvectors of some large sample covariance matrix ensembles
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 


9631. Birth of a strongly connected giant in an inhomogeneous random digraph
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 


9632. Integration by parts formula and applications to equations with jumps
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 


9633. The Independent Chip Model and Risk Aversion
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 


9634. Contact and voter processes on the infinite percolation cluster as models of host-symbiont interactions
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 


9635. A Stochastic Compartmental Model for Fast Axonal Transport
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 


9636. A Hierarchical Bayesian Model for Frame Representation
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 


9637. Functional limit theorems for sums of independent geometric L\'evy processes
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 


9638. Randomized First Passage Times
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 


9639. Uniqueness of Ground States for Short-Range Spin Glasses in the Half-Plane
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 


9640. Optimal control of a large dam, taking into account the water costs [New Edition]
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 


9641. Anomalous diffusion of distinguished particles in bead-spring networks
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 


9642. On penalisation results related with a remarkable class of submartingales
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 


9643. A new extrapolation method for weak approximation schemes with applications
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 


9644. Simulationsverfahren fuer Brown-Resnick-Prozesse (Simulation Techniques for Brown-Resnick Processes)
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 


9645. Equilibrium fluctuations for gradient exclusion processes with conductances in random environments
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 


9646. Hydrodynamic limit for a boundary driven stochastic lattice gas model with many conserved quantities
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 


9647. Dynamical large deviations for a boundary driven stochastic lattice gas model with many conserved quantities
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 


9648. A Maximal Inequality for Supermartingales
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 


9649. Utility maximization in models with conditionally independent increments
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 


9650. Spectra of lifted Ramanujan graphs
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 


9651. $W$-Sobolev spaces: Theory, Homogenization and Applications
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 


9652. Gradient estimates and domain identification for analytic Ornstein-Uhlenbeck operators
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 


9653. On the Local Resilience of Random Regular Graphs
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 


9654. Domain of attraction of Gaussian probability operators in quantum limit theory
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 


9655. The Stochastic Wave Equation with Fractional Noise: a random field approach
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 


9656. Time evolution of dense multigraph limits under edge-conservative preferential attachment dynamics
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 


9657. Balancing Traffic in Networks: Redundancy, Learning and the Effect of Stochastic Fluctuations
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 


9658. The K-level crossings of a random algebraic polynomial with dependent coefficients
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 


9659. Connection probabilities and RSW-type bounds for the FK Ising model
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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Stefano M. Iacus
Department of Economics,
Business and Statistics
University of Milan
Via Conservatorio, 7
I-20123 Milan - Italy
Ph.: +39 02 50321 461
Fax: +39 02 50321 505
http://www.economia.unimi.it/iacus
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