From pas at lists.imstat.org Fri Jan 1 00:02:50 2010 From: pas at lists.imstat.org (Probability Abstract Service) Date: Fri, 1 Jan 2010 15:02:50 +0900 Subject: [PAS] Probability Abstracts 113 Message-ID: <21E1D48B-0983-4FC7-9214-12D3D510FBCE@unimi.it> 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 ----------------------------------- Stefano M. Iacus Department of Economics, Business and Statistics University of Milan Via Conservatorio, 7 I-20123 Milan - Italy Ph.: +39 02 50321 461 Fax: +39 02 50321 505 http://www.economia.unimi.it/iacus ------------------------------------------------------------------------------------ Please don't send me Word or PowerPoint attachments if not absolutely necessary. See: http://www.gnu.org/philosophy/no-word-attachments.html From pas at lists.imstat.org Mon Mar 1 04:13:38 2010 From: pas at lists.imstat.org (Probability Abstract Service) Date: Mon, 01 Mar 2010 11:13:38 +0100 Subject: [PAS] Probability Abstracts 114 Message-ID: <92EECB99-298C-4068-9202-241552CBBD99@unimi.it> Probability Abstracts 114 This document contains abstracts 9660-9932 from Jan-1-2010 to Feb-28-2010. They have been mailed on Mar 1st, 2010. 9660. Multiplicative strong unimodality for positive stable laws Author(s): Thomas Simon (LPP) Abstract: It is known that real Non-Gaussian stable distributions are unimodal, not additive strongly unimodal, and multiplicative strongly unimodal in the symmetric case. By a theorem of Cuculescu-Theodorescu, the only remaining relevant situation for the multiplicative strong unimodality of stable laws is the one-sided. In this paper, we show that positive $\alpha-$stable laws are multiplicative strongly unimodal iff $\alpha\le 1/2.$ http://arxiv.org/abs/1002.4977 9661. Functional macroscopic behavior of weighted random ball model Author(s): Jean-Christophe Breton and Clement Dombry Abstract: We consider a generalization of the weighted random ball model. The model is driven by a random Poisson measure with a product heavy tailed intensity measure. Such a model typically represents the transmission of a network of stations with a fading effect. In a previous article, the authors proved the convergence of the finite-dimensional distributions of related generalized random fields under various scalings and in the particular case when the fading function is the indicator function of the unit ball. In this paper, tightness and functional convergence are investigated. Using suitable moment estimates, we prove functional convergences for some parametric classes of configurations under the so-called large ball scaling and intermediate ball scaling. Convergence in the space of distributions is also discussed. http://arxiv.org/abs/1002.4985 9662. On the size of a finite vacant cluster of random interlacements with small intensity Author(s): Augusto Teixeira Abstract: In this paper we establish some properties of percolation for the vacant set of random interlacements, for d at least 5 and small intensity u. The model of random interlacements was first introduced by A.S. Sznitman in arXiv:0704.2560. It is known that, for small u, almost surely there is a unique infinite connected component in the vacant set left by the random interlacements at level u, see arXiv:0808.3344 and arXiv:0805.4106. We estimate here the distribution of the diameter and the volume of the vacant component at level u containing the origin, given that it is finite. This comes as a by-product of our main theorem, which proves a stretched exponential bound on the probability that the interlacement set separates two macroscopic connected sets in a large cube. As another application, we show that with high probability, the unique infinite connected component of the vacant set is `ubiquitous' in large neighborhoods of the origin. http://arxiv.org/abs/1002.4995 9663. The largest component in an inhomogeneous random intersection graph with clustering Author(s): Mindaugas Bloznelis Abstract: Given b>0, integers n, m=bn and a probability measure Q on {0, 1,..., m}, consider the random intersection graph on the vertex set [n]={1, ..., n}, where i and j are declared adjacent whenever S(i) and S(j) intersect. Here S(1), ..., S(n) denote iid random subsets of [m] such that P(|S(i)|=k)=Q(k). For sparse random intersection graphs we establish a first order asymptotic for the order of the largest connected component N=n(1-Q(0))g+o(n) in probability. Here g is an average of nonextinction probabilities of a related multi-type Poisson branching process. http://arxiv.org/abs/1002.4649 9664. The martingale approach to disorder irrelevance for pinning models Author(s): Hubert Lacoin Abstract: This paper presents a very simple and self-contained proof of disorder irrelevance for inhomogeneous pinning models with return exponent \alpha\in (0,1/2). We also give a new upper bound for the contact fraction of the disordered model at criticality. http://arxiv.org/abs/1002.4753 9665. Stochastic monotonicity and duality for one-dimensional Markov processes Author(s): Vassili Kolokoltsov Abstract: Stochastic monotonicity and the related duality are well studied for one-dimensional diffusions and discrete Markov chains. In this note we extend the theory to arbitrary one-dimensional Markov Feller processes. This seems to be relevant in connection with the recent increase of interest to the analysis of general processes containing jumps, in particular in financial mathematics. http://arxiv.org/abs/1002.4773 9666. Brownian semistationary processes and conditional full support Author(s): Mikko S. Pakkanen Abstract: In this note, we study the infinite-dimensional conditional laws of Brownian semistationary processes. Motivated by the fact that these processes are typically not semimartingales, we present sufficient conditions ensuring that a Brownian semistationary process has conditional full support, a property introduced by Guasoni, R\'asonyi, and Schachermayer [Ann. Appl. Probab., 18 (2008) pp. 491--520]. By the results of Guasoni, R\'asonyi, and Schachermayer, this property has two important implications. It ensures, firstly, that the process admits no free lunches under proportional transaction costs, and secondly, that it can be approximated pathwise (in the sup norm) by semimartingales that admit equivalent martingale measures. http://arxiv.org/abs/1002.4774 9667. Transformations of one-dimensional Gibbs measures with infinite range interaction Author(s): Frank Redig and Feijia Wang Abstract: We study single-site stochastic and deterministic transforma- tions of one-dimensional Gibbs measures in the uniqueness regime with infinite-range interactions. We prove conservation of Gibbsianness and give quantitative estimates on the decay of the transformed potential. As examples, we consider exponentially decaying potentials, and potentials decaying as a power-law. http://arxiv.org/abs/1002.4796 9668. On the neighborhood radius estimation in Variable-neighborhood Markov Random Fields Author(s): Enza Orlandi and Eva Loecherbach Abstract: We consider Markov Random Fields defined by finite-region conditional probabilities depending on a neighborhood of the region which changes with the boundary conditions. The formal definition of these models requires partitions of the set of configurations according to their projections on finite neighborhoods of each lattice site. Each of these projections is called a context for the site. This framework is a natural extension, to d-dimensional fields, of the notion of variable-length Markov chains introduced by Rissanen (1983) in his classical paper. We define an algorithm to estimate the radius of the smallest ball containing the context based on a realization of the field. We prove the consistency of this estimator when the Dobrushin uniqueness condition for the one point conditional probabilities holds. Our proofs are constructive and yield explicit upper bounds for the probability of wrong estimation of the radius of the context. http://arxiv.org/abs/1002.4850 9669. Generically stable and smooth measures in NIP theories Author(s): Ehud Hrushovski and Anand Pillay and Pierre Simon Abstract: We study stable like behaviour in first order theories without the independence property. We introduce generically stable measures, give characterizatiions, and show their ubiquity. We also introduce generic compact domination. We also prove the approximate definability of arbitrary Borel probability measures on definable sets in the real and p-adic fields. http://arxiv.org/abs/1002.4763 9670. Cumulants and convolutions via Abel polynomials Author(s): E. Di Nardo and P. Petrullo and D. Senato Abstract: We provide an unifying polynomial expression giving moments in terms of cumulants, and viceversa, holding in the classical, boolean and free setting. This is done by using a symbolic treatment of Abel polynomials. As a by-product, we show that in the free cumulant theory the volume polynomial of Pitman and Stanley plays the role of the complete Bell exponential polynomial in the classical theory. Moreover via generalized Abel polynomials we construct a new class of cumulants, including the classical, boolean and free ones, and the convolutions linearized by them. Finally, via an umbral Fourier transform, we state a explicit connection between boolean and free convolution. http://arxiv.org/abs/1002.4803 9671. Asymptotic distribution of singular values of powers of random matrices Author(s): Nikita Alexeev (Saint-Petersburg State University and Russia) and Friedrich G\"otze (University of Bielefeld, Germany), and Alexander Tikhomirov (Syktyvkar State University, Russia) Abstract: Let $x$ be a complex random variable such that ${\E {x}=0}$, ${\E |x|^2=1}$, ${\E |x|^{4} < \infty}$. Let $x_{ij}$, $i,j \in \{1,2,...\}$ be independet copies of $x$. Let ${\Xb=(N^{-1/2}x_{ij})}$, $1\leq i,j \leq N$ be a random matrix. Writing $\Xb^*$ for the adjoint matrix of $\Xb$, consider the product $\Xb^m{\Xb^*}^m$ with some $m \in \{1,2,...\}$. The matrix $\Xb^m{\Xb^*}^m$ is Hermitian positive semi-definite. Let $\lambda_1,\lambda_2,...,\lambda_N$ be eigenvalues of $\Xb^m{\Xb^*}^m$ (or squared singular values of the matrix $\Xb^m$). In this paper we find the asymptotic distribution function \[ G^{(m)}(x)=\lim_{N\to\infty}\E{F_N^{(m)}(x)} \] of the empirical distribution function \[ {F_N^{(m)}(x)} = N^{-1} \sum_{k=1}^N {\mathbb{I}{\{\lambda_k \leq x\}}}, \] where $\mathbb{I} \{A\}$ stands for the indicator function of event $A$. The moments of $G^{(m)}$ satisfy \[ M^{(m)}_p=\int_{\mathbb{R}}{x^p dG^{(m)}(x)}=\frac{1}{mp+1}\binom{mp+p}{p}. \] In Free Probability Theory $M^{(m)}_p$ are known as Fuss--Catalan numbers. With $m=1$ our result turns to a well known result of Marchenko--Pastur 1967. http://arxiv.org/abs/1002.4442 9672. Phi-entropy inequalities and Fokker-Planck equations Author(s): Fran\c{c}ois Bolley (CEREMADE) and Ivan Gentil (CEREMADE) Abstract: We present new $\Phi$-entropy inequalities for diffusion semigroups under the curvature-dimension criterion. They include the isoperimetric function of the Gaussian measure. Applications to the long time behaviour of solutions to Fokker-Planck equations are given. http://arxiv.org/abs/1002.4478 9673. Nonlinear Expectations and Stochastic Calculus under Uncertainty Author(s): Shige Peng Abstract: In this book, we introduce a new approach of sublinear expectation to deal with the problem of probability and distribution model uncertainty. We a new type of (robust) normal distributions and the related central limit theorem under sublinear expectation. We also present a new type of Brownian motion under sublinear expectations and the related stochastic calculus of Ito's type. The results provide robust tools for the problem of probability model uncertainty arising from financial risk management, statistics and stochastic controls. http://arxiv.org/abs/1002.4546 9674. Dynamics of the supermarket model Author(s): I. M. MacPhee and M. V. Menshikov and M. Vachkovskaia Abstract: We consider the long term behaviour of a Markov chain \xi(t) on \Z^N based on the N station supermarket model. Different routing policies for the supermarket model give different Markov chains. We show that for a general class of local routing policies, "join the least weighted queue" (JLW), the N one-dimensional components \xi_i(t) can be partitioned into disjoint clusters C_k. Within each cluster C_k the "speed" of each component \xi_j converges to a constant V_k and under certain conditions \xi is recurrent in shape on each cluster. To establish these results we have assembled methods from two distinct areas of mathematics, semi-martingale techniques used for showing stability of Markov chains together with the theory of optimal flows in networks. As corollaries to our main result we obtain the stability classification of the supermarket model under any JLW policy and can explicitly compute the C_k and V_k for any instance of the model and specific JLW policy. http://arxiv.org/abs/1002.4570 9675. Heavy traffic on a controlled motorway Author(s): F. P. Kelly and R. J. Williams Abstract: Unlimited access to a motorway network can, in overloaded conditions, cause a loss of capacity. Ramp metering (signals on slip roads to control access to the motorway) can help avoid this loss of capacity. The design of ramp metering strategies has several features in common with the design of access control mechanisms in communication networks. Inspired by models and rate control mechanisms developed for Internet congestion control, we propose a Brownian network model as an approximate model for a controlled motorway and consider it operating under a proportionally fair ramp metering policy.We present an analysis of the performance of this model. http://arxiv.org/abs/1002.4591 9676. Nonparametric Estimation and On-Line Prediction for General Stationary Ergodic Sources Author(s): Joe Suzuki Abstract: We proposed a learning algorithm for nonparametric estimation and on-line prediction for general stationary ergodic sources. We prepare histograms each of which estimates the probability as a finite distribution, and mixture them with weights to construct an estimator. The whole analysis is based on measure theory. The estimator works whether the source is discrete or continuous. If it is stationary ergodic, then the measure theoretically given Kullback-Leibler information divided by the sequence length $n$ converges to zero as $n$ goes to infinity. In particular, for continuous sources, the method does not require existence of a probability density function. http://arxiv.org/abs/1002.4453 9677. On a new probabilistic representation for the solution of the heat equation Author(s): Paolo Da Pelo and Alberto Lanconelli Abstract: We obtain a new probabilistic representation for the solution of the heat equation in terms of a product for smooth random variables which is introduced and studied in this paper. This multiplication, expressed in terms of the Hida-Malliavin derivatives of the random variables involved, exhibits many useful properties which are to some extents opposite to some peculiar features of the Wick product. http://arxiv.org/abs/1002.4269 9678. Crossing random walks and stretched polymers at weak disorder Author(s): Dmitry Ioffe and Yvan Velenik Abstract: We consider a model of a polymer in Z^{d+1}, constrained to join 0 and a hyperplane at distance N. The polymer is subject to a quenched non-negative random environment. Alternatively, the model describes crossing random walks in a random potential (see Chapter 5 of [Sznitman] for the original Brownian motion formulation). It was recently shown, by Flury and by Zygouras, that, in such a setting, the quenched and annealed free energies coincide in the limit N to infinity, when d is at least 3 and the temperature is sufficiently high. We first strengthen this result by proving that, under somewhat weaker assumptions on the distribution of disorder which, in particular, enable a small probability of traps, the ratio of quenched and annealed partition functions actually converges. We then conclude that, in this case, the polymer obeys a diffusive scaling, with the same diffusivity constant as the annealed model. http://arxiv.org/abs/1002.4289 9679. Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients Author(s): Xicheng Zhang Abstract: In this article we prove the existence and uniqueness for degenerate stochastic differential equations with Sobolev (possibly singular) drift and diffusion coefficients in a generalized sense. In particular, our result covers the classical DiPerna-Lions flows and, we also obtain the well-posedness for degenerate Fokker-Planck equations with irregular coefficients. Moreover, a large deviation principle of Freidlin-Wenzell type for this type of SDEs is established. http://arxiv.org/abs/1002.4297 9680. Coupling time distribution asymptotics for some couplings of the Levy stochastic area Author(s): Wilfrid S. Kendall Abstract: We exhibit some explicit co-adapted couplings for n-dimensional Brownian motion and all its Levy stochastic areas. In the two-dimensional case we show how to derive exact asymptotics for the coupling time under various mixed coupling strategies, using Dufresne's formula for the distribution of exponential functionals of Brownian motion. This yields quantitative asymptotics for the distributions of random times required for certain simultaneous couplings of stochastic area and Brownian motion. The approach also applies to higher dimensions, but will then lead to upper and lower bounds rather than exact asymptotics. http://arxiv.org/abs/1002.4348 9681. Weak disorder asymptotics in the stochastic mean-field model of distance Author(s): Shankar Bhamidi and Remco van der Hofstad Abstract: In the recent past, there has been a concerted effort to develop mathematical models for real-world networks and analyze various dynamics on these models. One particular problem of significant importance is to understand the effect of random edge lengths or costs on the geometry and flow transporting properties of the network. Two different regimes are of great interest, the weak disorder regime where optimality of a path is determined by the sum of edge weights on the path and the strong disorder regime where optimality of a path is determined by the maximal edge weight on the path. In the context of the stochastic mean-field model of distance, we provide the first mathematically tractable model of weak disorder and show that no transition occurs at finite temperature. Indeed we show that for all fixed finite temperatures, the number of edges on the minimal weight path (i.e the hopcount) is always $\Theta(\log{n})$ and satisfies a central limit theorem with asymptotic means and variances of order $\Theta(\log{n})$, with limiting constants expressible in terms of the Malthusian rate of growth and the mean of the stable-age distribution of the associated continuous-time branching process. More precisely, we take independent and identically distributed edge weights with distribution $E^s$ for some parameter $s>0$, where $E$ is an exponential random variable with mean 1. Then, the asymptotic mean and variance of the central limit theorem for the hopcount are $s\log{n}$ and $s^2 \log{n}$ respectively. We also find limiting distributional asymptotics for the value of the minimal weight path in terms of extreme value distributions, Cox processes and martingale limits of branching processes. http://arxiv.org/abs/1002.4362 9682. Regularity partitions and the topology of graphons Author(s): L\'aszl\'o Lov\'asz and Bal\'azs Szegedy Abstract: We highlight a topological aspect of the graph limit theory. Graphons are limit objects for convergent sequences of dense graphs. We introduce the representation of a graphon on a unique metric space and we relate the dimension of this metric space to the size of regularity partitions. We prove that if a graphon has an excluded induced sub-bigraph then the underlying metric space is compact and has finite packing dimension. It implies in particular that such graphons have regularity partitions of polynomial size. http://arxiv.org/abs/1002.4377 9683. A characterization of freeness by invariance under quantum spreading Author(s): Stephen Curran Abstract: We construct spaces of quantum increasing sequences, which give quantum families of maps in the sense of Soltan. We then introduce a notion of quantum spreadability for a sequence of noncommutative random variables, by requiring their joint distribution to be invariant under taking quantum subsequences. Our main result is a free analogue of a theorem of Ryll-Nardzewski: for an infinite sequence of noncommutative random variables, quantum spreadability is equivalent to free independence and identical distribution with respect to a conditional expectation. http://arxiv.org/abs/1002.4390 9684. Psi-series method in random trees and moments of high orders Author(s): Hua-Huai Chern and Hsien-Kuei Hwang and Conrado Mart\'inez Abstract: An unusual and surprising expansion of the form \[ p_n = \rho^{-n-1}(6n +\tfrac{18}5+ \tfrac{336}{3125} n^{-5}+\tfrac{1008}{3125} n^{-6} +\text{smaller order terms}), \] as $n\to\infty$, is derived for the probability $p_n$ that two randomly chosen binary search trees are identical (in shape and in labels of all corresponding nodes). A quantity arising in the analysis of phylogenetic trees is also proved to have a similar asymptotic expansion. Our method of proof is new in the literature of discrete probability and analysis of algorithms, and based on the psi-series expansions for nonlinear differential equations. Such an approach is very general and applicable to many other problems involving nonlinear differential equations; many examples are discussed and several attractive phenomena are discovered. http://arxiv.org/abs/1002.3859 9685. A sharper threshold for bootstrap percolation in two dimensions Author(s): Janko Gravner and Alexander E. Holroyd and Robert Morris Abstract: Two-dimensional bootstrap percolation is a cellular automaton in which sites become 'infected' by contact with two or more already infected nearest neighbors. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n x n square, with sites initially infected independently with probability p. The critical probability p_c is the smallest p for which the probability that the entire square is eventually infected exceeds 1/2. Holroyd determined the sharp first-order approximation: p_c \sim \pi^2/(18 log n) as n \to \infty. Here we sharpen this result, proving that the second term in the expansion is -(log n)^{-3/2+ o(1)}, and moreover determining it up to a poly(log log n)-factor. http://arxiv.org/abs/1002.3881 9686. Almost sure asymptotics for the random binary search tree Author(s): Matthew I. Roberts Abstract: We consider a (random permutation model) binary search tree with n nodes and give asymptotics on the loglog scale for the height H_n and saturation level h_n of the tree as n\to\infty, both almost surely and in probability. We then consider the number F_n of particles at level H_n at time n, and show that F_n is unbounded almost surely. http://arxiv.org/abs/1002.3896 9687. Parameter estimations for SPDEs with multiplicative fractional noise Author(s): Igor Cialenco Abstract: We study parameter estimation problem for diagonalizable parabolic stochastic partial differential equations driven by a multiplicative fractional noise with any Hurst parameter $H\in(0,1)$. Two classes of estimates are investigated: traditional maximum likelihood type estimates, and a new class called closed-form exact estimates. Finally several examples are discussed, including statistical inference for stochastic heat equation driven by a fractional Brownian motion. http://arxiv.org/abs/1002.3911 9688. Large number of queues in tandem: Scaling properties under back-pressure algorithm Author(s): Alexander Stolyar Abstract: We consider a system with N unit-service-rate queues in tandem, with exogenous arrivals of rate lambda at queue 1, under a back-pressure (MaxWeight) algorithm: service at queue n is blocked unless its queue length is greater than that of next queue n+1. The question addressed is how steady-state queues scale as N goes to infinity. We show that the answer depends on whether lambda is below or above the critical value 1/4: in the former case queues remain uniformly stochastically bounded, while otherwise they grow to infinity. The problem is essentially reduced to the behavior of the system with infinite number of queues in tandem, which is studied using tools from interacting particle systems theory. In particular, the criticality of load 1/4 is closely related to the fact that this is the maximum possible flux (flow rate) of a stationary totally asymmetric simple exclusion process. http://arxiv.org/abs/1002.3940 9689. Variations on the Berry-Esseen theorem Author(s): Bo'az Klartag and Sasha Sodin Abstract: We analyze the quality of the gaussian approximation to linear combinations of n independent, identically-distributed random variables with finite fourth moments. It turns out that there exist universal, simple linear combinations that perform better than the sum of the variables. We also investigate the case in which the random variables are independent, yet they are not necessarily identically distributed. http://arxiv.org/abs/1002.3970 9690. Moments of Gamma type and the Brownian supremum process area Author(s): Svante Janson Abstract: We study positive random variables whose moments can be expressed by products and quotients of Gamma functions; this includes many standard distributions. General results are given on existence, series expansion and asymptotics of density functions. It is shown that the integral of the supremum process of Brownian motion has moments of this type, as well as a related random variable occuring in the study of hashing with linear displacement, and the general results are applied to these variables. http://arxiv.org/abs/1002.4135 9691. A limit theorem for particle current in the symmetric exclusion process Author(s): Alexander Vandenberg-Rodes Abstract: Using the recently discovered strong negative dependence properties of the symmetric exclusion process, we derive general conditions for when the normalized current of particles between regions converges to the Gaussian distribution. The main novelty is that the results do not assume any translation invariance, and hold for most initial configurations. http://arxiv.org/abs/1002.4148 9692. Particle systems with quasi-homogeneous initial states and their occupation time fluctuations Author(s): Tomasz Bojdecki and Luis G. Gorostiza and Anna Talarczyk Abstract: Occupation time fluctuation limits of particle systems in R^d with independent motions (symmetric stable Levy process, with or without critical branching) have been studied assuming initial distributions given by Poisson random measures (homogeneous and some inhomogeneous cases). In this paper, with d=1 for simplicity, we extend previous results to a wide class of initial measures obeying a quasi-homogeneity property, which includes as special cases homogeneous Poisson measures and many deterministic measures (simple example: one atom at each point of Z), by means of a new unified approach. In previous papers, in the homogeneous Poisson case, for the branching system in "low" dimensions, the limit was characterized by a long-range dependent Gaussian process called sub-fractional Brownian motion (sub-fBm), and this effect was attributed to the branching because it had appeared only in that case. An unexpected finding in this paper is that sub-fBm is more prevalent than previously thought. Namely, it is a natural ingredient of the limit process in the non-branching case (for "low" dimension), as well. On the other hand, fractional Brownian motion is not only related to systems in equilibrium (e.g., non-branching system with initial homogeneous Poisson measure), but it also appears here for a wider class of initial measures of quasi-homogeneous type. http://arxiv.org/abs/1002.4152 9693. Bifractional Brownian motion with parameter $K\in(1,2)$ Author(s): Xavier Bardina and Khalifa Es-Sebaiy (SAMM) Abstract: In this paper we introduce and study a self-similar Gaussian process that is the bifractional Brownian motion $B^{H,K}$ with parameters $H\in(0,1)$ and $K\in(1,2)$ such that $HK\in(0,1)$. A remarkable difference between the case $K\in(0,1)$ and our situation is that this process is a semimartingale when $2HK=1$. http://arxiv.org/abs/1002.3680 9694. Scaling Limits for Random Quadrangulations of Positive Genus Author(s): J\'er\'emie Bettinelli (LM-Orsay) Abstract: We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given $g$, we consider, for every $n \ge 1$, a random quadrangulation $\q_n$ uniformly distributed over the set of all rooted bipartite quadrangulations of genus $g$ with $n$ faces. We view it as a metric space by endowing its set of vertices with the graph distance. We show that, as $n$ tends to infinity, this metric space, with distances rescaled by the factor $n^{-1/4}$, converges in distribution, at least along some subsequence, toward a limiting random metric space. This convergence holds in the sense of the Gromov-Hausdorff topology on compact metric spaces. We show that, regardless of the choice of the subsequence, the Hausdorff dimension of the limiting space is almost surely equal to 4. Our main tool is a bijection introduced by Chapuy, Marcus, and Schaeffer between the quadrangulations we consider and objects they call well-labeled $g$-trees. An important part of our study consists in determining the scaling limits of the latter. http://arxiv.org/abs/1002.3682 9695. First-passage percolation with exponential times on a ladder Author(s): Henrik Renlund Abstract: We consider first-passage percolation on a ladder, i.e. the graph {0,1,...}*{0,1} where nodes at distance 1 are joined by an edge, and the times are exponentially i.i.d. with mean 1. We find an appropriate Markov chain to calculate an explicit expression for the time constant whose numerical value is approximately 0.6827. This time constant is the long-term average inverse speed of the process. We also calculate the average residual time. http://arxiv.org/abs/1002.3709 9696. Generalized Polya urns via stochastic approximation Author(s): Henrik Renlund Abstract: We collect, survey and develop methods of (one-dimensional) stochastic approximation in a framework that seems suitable to handle fairly broad generalizations of Polya urns. To show the applicability of the results we determine the limiting fraction of balls in an urn with balls of two colors. We consider two models generalizing the Polya urn, in the first one ball is drawn and replaced with balls of (possibly) both colors according to which color was drawn. In the second, two balls are drawn simultaneously and replaced along with balls of (possibly) both colors according to what combination of colors were drawn. http://arxiv.org/abs/1002.3716 9697. Singular perturbations to semilinear stochastic heat equations Author(s): Martin Hairer Abstract: We consider a class of singular perturbations to the stochastic heat equation or semilinear variations thereof. The interesting feature of these perturbations is that, as the small parameter epsilon tends to zero, their solutions converge to the 'wrong' limit, i.e. they do not converge to the solution obtained by simply setting epsilon = 0. A similar effect is also observed for some (formally) small stochastic perturbations of a deterministic semilinear parabolic PDE. Our proofs are based on a detailed analysis of the spatially rough component of the equations, combined with a judicious use of Gaussian concentration inequalities. http://arxiv.org/abs/1002.3722 9698. Optimal stopping, Appell polynomials and Wiener-Hopf factorization representations of excessive functions of L\'evy processes Author(s): Paavo Salminen Abstract: In this paper we study the optimal stopping problem for L\'evy processes studied by Novikov and Shiryayev, Stochastics, 2007 In particular, we are interested in finding the representing measure of the value function. It is seen that that this can be expressed in terms of the Appell polynomials. An important tool in our approach and computations is the Wiener-Hopf factorization. http://arxiv.org/abs/1002.3746 9699. Non-equilibrium dynamics of stochastic point processes: a dead-time approach Author(s): Moritz Deger and Moritz Helias and Stefano Cardanobile and Fatihcan M. Atay and Stefan Rotter Abstract: The Poisson process with dead time (PPD) is a widely used model for time series of events. Here we analyze non-equilibrium properties of an ensemble of PPDs. We derive a delay differential equation that describes the dynamics of the state of the ensemble. Analytical solutions are obtained for the time-dependent ensemble output rate in response to a step input. We also derive the mapping of periodic input to steady-state output, which we solve specifically for sinusoidal inputs. We are able to generalize the dynamics of the PPD to the case of random dead times, by which the method becomes applicable to a much larger class of stochastic point processes. Transient properties of the PPD are a recurring theme in many quantitative sciences, since a dead time after event detection is a feature of most technical counting devices. Our results are also relevant for the neurosciences because refractoriness is a characteristic of trains of action potentials emitted by nerve cells. http://arxiv.org/abs/1002.3798 9700. On the unimodality of inverse positive stable laws Author(s): Thomas Simon (LPP) Abstract: We observe that the function $F_\alpha (x) = (1+ \alpha x^\alpha)e^{-x^\alpha}$ is completely monotone iff $\alpha \le \alpha_0$ for some $\alpha_0 \in ]2/3, 3/4[.$ This property is equivalent to the unimodality of the inverse positive $\alpha$-stable law. The random variable associated with $F_\alpha$ appears then in two different factorizations of the positive $\alpha$-stable distribution. Furthermore, it is infinitely divisible iff $\alpha \le \alpha_1$ for some $\alpha_1 \in ]2/3, \alpha_0[$ and self-decomposable iff $\alpha \le \alpha_2$ for some $\alpha_2 \in ]2/3, \alpha_1[.$ http://arxiv.org/abs/1002.3813 9701. Risk assessment for uncertain cash flows: Model ambiguity, discounting ambiguity, and the role of bubbles Author(s): Beatrice Acciaio and Hans Foellmer and Irina Penner Abstract: We study the risk assessment of uncertain cash flows in terms of dynamic convex risk measures for processes as introduced in Cheridito, Delbaen, and Kupper (2006). These risk measures take into account not only the amounts but also the timing of a cash flow. We discuss their robust representation in terms of suitably penalized probability measures on the optional sigma-field. This yields an explicit analysis both of model and discounting ambiguity. We focus on supermartingale criteria for different notions of time consistency. In particular we show how bubbles may appear in the dynamic penalization, and how they cause a breakdown of asymptotic safety of the risk assessment procedure. http://arxiv.org/abs/1002.3627 9702. Interpolation and $\Phi$-moment inequalities of noncommutative martingales Author(s): Turdebek N. Bekjan and Zeqian Chen Abstract: This paper is devoted to the study of $\Phi$-moment inequalities for noncommutative martingales. In particular, we prove the noncommutative $\Phi$-moment analogues of martingale transformations, Stein's inequalities, Khintchine's inequalities for Rademacher's random variables, and Burkholder-Gundy's inequalities. The key ingredient is a noncommutative version of Marcinkiewicz type interpolation theorem for Orlicz spaces which we establish in this paper. http://arxiv.org/abs/1002.3670 9703. Optimal investment with bounded VaR for power utility functions Author(s): Behamar Chouaf and Serguei Pergamenchtchikov (LMRS) Abstract: We consider the optimal investment problem for Black-Scholes type financial market with bounded VaR measure on the whole investment interval $[0,T]$. The explicit form for the optimal strategies is found. http://arxiv.org/abs/1002.3681 9704. Dynamic risk measures Author(s): Beatrice Acciaio and Irina Penner Abstract: This paper gives an overview of the theory of dynamic convex risk measures for random variables in discrete time setting. We summarize robust representation results of conditional convex risk measures, and we characterize various time consistency properties of dynamic risk measures in terms of acceptance sets, penalty functions, and by supermartingale properties of risk processes and penalty functions. http://arxiv.org/abs/1002.3794 9705. Energy efficiency of consecutive fragmentation processes Author(s): Joaquin Fontbona and Nathalie Krell (IRMAR) and Servet Martinez Abstract: We present a ?rst study on the energy required to reduce a unit mass fragment by consecutively using several devices, as it happens in the mining industry. Two devices are considered, which we represent as different stochastic fragmentation processes. Following the self-similar energy model introduced by Bertoin and Martinez, we compute the average energy required to attain a size x with this two-device procedure. We then asymptotically compare, as x goes to 0 or 1, its energy requirement with that of individual fragmentation processes. In particular, we show that for certain range of parameters of the fragmentation processes and of their energy cost-functions, the consecutive use of two devices can be asymptotically more efficient than using each of them separately, or conversely. http://arxiv.org/abs/1002.3460 9706. A short proof of Cramer's theorem in R Author(s): Raphael Cerf and Pierre Petit Abstract: We expose here a short proof of Cramer's theorem in R based on convex duality. http://arxiv.org/abs/1002.3496 9707. Asymptotic risks of Viterbi segmentation Author(s): Kristi Kuljus and J\"uri Lember Abstract: We consider the maximum likelihood (Viterbi) alignment of a hidden Markov model (HMM). In an HMM, the underlying Markov chain is usually hidden and the Viterbi alignment is often used as the estimate of it. This approach will be referred to as the Viterbi segmentation. The goodness of the Viterbi segmentation can be measured by several risks. In this paper, we prove the existence of asymptotic risks. Being independent of data, the asymptotic risks can be considered as the characteristics of the model that illustrate the long-run behavior of the Viterbi segmentation. http://arxiv.org/abs/1002.3509 9708. Multi-type TASEP in discrete time Author(s): James Martin and Philipp Schmidt Abstract: The TASEP (totally asymmetric simple exclusion process) is a basic model for an one-dimensional interacting particle system with non-reversible dynamics. Despite the simplicity of the model it shows a very rich and interesting behaviour. In this paper we study some aspects of the TASEP in discrete time and compare the results to the recently obtained results for the TASEP in continuous time. In particular we focus on stationary distributions for multi-type models, speeds of second-class particles, collision probabilities and the "speed process". In discrete time, jump attempts may occur at different sites simultaneously, and the order in which these attempts are processed is important; we consider various natural update rules. http://arxiv.org/abs/1002.3539 9709. Approximation by Log-Concave Distributions with Applications to Regression Author(s): Lutz Duembgen and Richard Samworth and Dominic Schuhmacher Abstract: We study the approximation of arbitrary distributions P on d-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback-Leibler type functional. We show that such an approximation exists if, and only if, P has finite first moments and is not concentrated on some hyperplane. Furthermore we show that this approximation depends continuously on P with respect to Mallows' distance D_1. This result implies consistency of the maximum likelihood estimator of a log-concave density under fairly general conditions. It also allows us to prove existence and consistency of estimators in regression models with a response Y = m(X) + E, where X and E are independent, m(.) belongs to a certain class of regression functions while E is a random error with log-concave density. http://arxiv.org/abs/1002.3448 9710. Double scaling limits of random matrices and minimal (2m,1) models: the merging of two cuts in a degenerate case Author(s): Olivier Marchal and Mattia Cafasso Abstract: In this article, we show that the double scaling limit correlation functions of a random matrix model when two cuts merge with degeneracy $2m$ (i.e. when $y\sim x^{2m}$ for arbitrary values of the integer $m$) are the same as the determinental formulae defined by conformal $(2m,1)$ models. Our approach follows the one developped by Berg\`{e}re and Eynard in \cite{BergereEynard} and uses a Lax pair representation of the conformal $(2m,1)$ models (giving Painleve II integrable hierarchy) as suggested by Bleher and Eynard in \cite{BleherEynard}. In particular we define Baker-Akhiezer functions associated to the Lax pair to construct a kernel which is then used to compute determinental formulae giving the correlation functions of the double scaling limit of a matrix model near the merging of two cuts. http://arxiv.org/abs/1002.3347 9711. On monotone convolution and monotone infinite divisivility Author(s): Takahiro Hasebe Abstract: This article is focused on properties of monotone convolutions. A criterion for infinite divisibility and time evolution of convolution semigroups are mainly studied. In particular, we clarify that many analogues of the classical results of L\'{e}vy processes hold such as characterizations of subordinators and strictly stable distributions. http://arxiv.org/abs/1002.3430 9712. Rumor Spreading on Random Regular Graphs and Expanders Author(s): Nikolaos Fountoulakis and Konstantinos Panagiotou Abstract: Broadcasting algorithms are important building blocks of distributed systems. In this work we investigate the typical performance of the classical and well-studied push model. Assume that initially one node in a given network holds some piece of information. In each round, every one of the informed nodes chooses independently a neighbor uniformly at random and transmits the message to it. In this paper we consider random networks where each vertex has degree d, which is at least 3, i.e., the underlying graph is drawn uniformly at random from the set of all d-regular graphs with n vertices. We show that with probability 1 - o(1) the push model broadcasts the message to all nodes within (1 + o(1))C_d ln n rounds, where C_d = 1/ ln(2(1-1/d)) - 1/(d ln(1 - 1/d)). In particular, we can characterize precisely the effect of the node degree to the typical broadcast time of the push model. Moreover, we consider pseudo-random regular networks, where we assume that the degree of each node is very large. There we show that the broadcast time is (1+o(1))C ln n with probability 1 - o(1), where C= 1/ ln 2 + 1, is the limit of C_d as d grows. http://arxiv.org/abs/1002.3518 9713. Uniform Large deviations for infinite dimensional stochastic systems with jumps Author(s): Vasileios Maroulas Abstract: Uniform large deviation principles for positive functionals of all equivalent types of infinite dimensional Brownian motions acting together with a Poisson random measure are established. The core of our approach is a variational representation formula which for an infinite sequence of i.i.d real Brownian motions and a Poisson random measure was shown in [5]. http://arxiv.org/abs/1002.3290 9714. Coarse-grained modeling of multiscale diffusions: the p-variation estimates Author(s): Anastasia Papavasiliou Abstract: We study the problem of estimating parameters of the limiting equation of a multiscale diffusion in the case of averaging and homogenization, given data from the corresponding multiscale system. First, we review some recent results that make use of the maximum likelihood of the limiting equation. In particular, it has been shown that in the averaging case, the MLE will be asymptotically consistent in the limit while in the homogenization case, the MLE will be asymptotically consistent only if we subsample the data. Then, we focus on the problem of estimating the diffusion coefficient. We suggest a novel approach that makes use of the total $p$-variation, as defined in the theory of rough paths and avoids the subsampling step. The method is applied to a multiscale OU process. http://arxiv.org/abs/1002.3241 9715. A discrete time neural network model with spiking neurons II. Dynamics with noise Author(s): B. Cessac Abstract: We provide rigorous and exact results characterizing the statistics of spike trains in a network of leaky integrate and fire neurons, where time is discrete and where neurons are submitted to noise, without restriction on the synaptic weights. We show the existence and uniqueness of an invariant measure of Gibbs type and discuss its properties. We also discuss Markovian approximations and relate them to the approaches currently used in computational neuroscience to analyse experimental spike trains statistics. http://arxiv.org/abs/1002.3275 9716. Distribution of Relative Edge Density of the Underlying Graphs Based on a Random Digraph Family Author(s): Elvan Ceyhan Abstract: The data-random graphs called proximity catch digraphs (PCDs) have been introduced recently and have applications in pattern recognition and spatial pattern analysis. A PCD is a random directed graph (i.e., digraph) which is constructed from data using the relative positions of the points from various classes. Different PCDs result from different definitions of the proximity region associated with each data point. We consider the underlying graphs based on a family of PCDs which is determined by a family of parameterized proximity maps called proportional-edge proximity map. The graph invariant we investigate is the relative edge density of the underlying graphs. We demonstrate that, properly scaled, relative edge density of the underlying graphs is a U-statistic, and hence obtain the asymptotic normality of the relative edge density for data from any distribution that satisfies mild regulatory conditions. By detailed probabilistic and geometric calculations, we compute the explicit form of the asymptotic normal distribution for uniform data on a bounded region. We also compare the relative edge densities of the two types of the underlying graphs and the relative arc density of the PCDs. The approach presented here is also valid for data in higher dimensions. http://arxiv.org/abs/1002.2957 9717. Khasminskii-Type Theorem and LaSalle-Type Theorem for Stochastic Evolution Delay Equations Author(s): Jianhai Bao and Xuerong Mao and Chenggui Yuan Abstract: In this paper we study the well-known Khasminskii-Type Theorem, i.e. the existence and uniqueness of solutions of stochastic evolution delay equations, under local Lipschitz condition, but without linear growth condition. We then establish one stochastic LaSalle-type theorem for asymptotic stability analysis of strong solutions. Moreover, several examples are established to illustrate the power of our theories. http://arxiv.org/abs/1002.3116 9718. Subgraphs of dense random graphs with specified degrees Author(s): Brendan D McKay Abstract: Let d = (d1, d2, ..., dn) be a vector of non-negative integers with even sum. We prove some basic facts about the structure of a random graph with degree sequence d, including the probability of a given subgraph or induced subgraph. Although there are many results of this kind, they are restricted to the sparse case with only a few exceptions. Our focus is instead on the case where the average degree is approximately a constant fraction of n. Our approach is the multidimensional saddle-point method. This extends the enumerative work of McKay and Wormald (1990) and is analogous to the theory developed for bipartite graphs by Greenhill and McKay (arXiv:math/0701600, 2009). http://arxiv.org/abs/1002.3018 9719. A Sharp Liouville Theorem for Elliptic Operators Author(s): Enrico Priola and Feng-Yu Wang Abstract: We introduce a new condition on elliptic operators $L= {1/2}\triangle + b \cdot \nabla $ which ensures the validity of the Liouville property for bounded solutions to $Lu=0$ on $\R^d$. Such condition is sharp when $d=1$. We extend our Liouville theorem to more general second order operators in non-divergence form assuming a Cordes type condition. http://arxiv.org/abs/1002.3055 9720. Random Strict Partitions and Determinantal Point Processes Author(s): Leonid Petrov Abstract: In this note we present new examples of determinantal point processes with infinitely many particles. The particles live on the half-lattice {1,2,...} or on the open half-line (0,+\infty). The main result is the computation of the correlation kernels. They have integrable form and are expressed through the Euler gamma function (the lattice case) and the classical Whittaker functions (the continuous case). Our processes are obtained via a limit transition from a model of random strict partitions introduced by Borodin (1997) in connection with the problem of harmonic analysis for projective characters of the infinite symmetric group. http://arxiv.org/abs/1002.2714 9721. Long Strange Segments, Ruin Probabilities and the Effect of Memory on Moving Average Processes Author(s): Souvik Ghosh and Gennady Samorodnitsky Abstract: We obtain the rate of growth of long strange segments and the rate of decay of infinite horizon ruin probabilities for a class of infinite moving average processes with exponentially light tails. The rates are computed explicitly. We show that the rates are very similar to those of an i.i.d. process as long as moving average coefficients decay fast enough. If they do not, then the rates are significantly different. This demonstrates the change in the length of memory in a moving average process associated with certain changes in the rate of decay of the coefficients. http://arxiv.org/abs/1002.2751 9722. On the trace of branching random walk Author(s): Itai Benjamini and Sebastian M\"uller Abstract: We study branching random walk on Cayley graphs. A first result is that the trace of a transient branching random walk on a Cayley graph is a.s. transient for simple random walk. In addition, it has a.s. critical percolation probability less than one and exponential volume growth. The proofs rely on the fact that the trace induces an invariant percolation on the family tree of the branching random walk. Furthermore, we prove that the trace is a.s. strongly recurrent for any branching random walk. This follows from the observation that the trace, after appropriate biasing of the root, defines a unimodular measure. All the results hold more generally for branching random walk on unimodular random graphs. http://arxiv.org/abs/1002.2781 9723. Competing particle systems evolving by interacting Levy processes Author(s): Mykhaylo Shkolnikov Abstract: We consider finite and infinite systems of particles on the real line and half-line evolving in continuous time. Hereby, the particles are driven by i.i.d. Levy processes endowed with rank-dependent drift and diffusion coefficients. In the finite systems we show that the processes of gaps in the respective particle configurations possess unique invariant distributions and prove the convergence of the gap processes to the latter in the total variation distance, assuming a bound on the jumps of the Levy processes. In the infinite case we show that the gap process of the particle system on the half-line is tight for appropriate initial conditions and same drift and diffusion coefficients for all particles. Applications of such processes include the modelling of capital distributions among the ranked participants in a financial market, the stability of certain stochastic queueing and storage networks and the study of the Sherrington-Kirkpatrick model of spin glasses. http://arxiv.org/abs/1002.2811 9724. Analysis on Path Spaces over Riemmannian Manifolds with Boundary Author(s): Feng-Yu Wang Abstract: By using Hsu's multiplicative functional for the Neumann heat equation, a natural damped gradient operator is defined for the reflecting Brownian motion on compact manifolds with boundary. This operator is linked to quasi-invariant flows in terms of a integration by parts formula, which leads to the standard log-Sobolev inequality for the associated Dirichlet form on the path space. http://arxiv.org/abs/1002.2887 9725. Coupling and Strong Feller Property for Ornstein-Uhlenbeck Jump Processes Author(s): Feng-Yu Wang Abstract: Coupling and strong Feller property are investigated for the linear SDE on $\R^d$: $$\d X_t= A X_t\d t+ \d L_t,$$ where $A$ is a $d\times d$ real matrix and $L_t$ is a L\'evy process with L\'evy measure $\nu$ on $\R^d$. Assume that $\nu(\d z)\ge \rr_0(z)\d z$ for some $\rr_0\ge 0$. If $A \le 0$ and $\int_{B(x_0,\vv)} \rr_0(z)^{-1}\d z<\infty$ holds for some $x_0\in \R^d$ and some $\vv>0$, then the associated Markov transition probability $P_t(x,\d y)$ satisfies $$\|P_t (x, \cdot)- P_t (y, \cdot)\|_{var} \le \ff{C(1+|x-y|)}{\ss t}, x,y\in \R^d, t>0$$ for some constant $C>0$, which is sharp for large $t$ and implies that the process has successful couplings. If $\rr_0\in C(\R^d\setminus \{0\})$ with $\int_{\R^d}\rr_0(z)\d z=\infty$, then the process is strong Feller. http://arxiv.org/abs/1002.2890 9726. Distance statistics in quadrangulations with no multiple edges and the geometry of minbus Author(s): J. Bouttier and E. Guitter Abstract: We present a detailed calculation of the distance-dependent two-point function for quadrangulations with no multiple edges. Various discrete observables measuring this two-point function are computed and analyzed in the limit of large maps. For large distances and in the scaling regime, we recover the same universal scaling function as for general quadrangulations. We then explore the geometry of "minimal neck baby universes" (minbus), which are the outgrowths to be removed from a general quadrangulation to transform it into a quadrangulation with no multiple edges, the "mother universe". We give a number of distance-dependent characterizations of minbus, such as the two-point function inside a minbu or the law for the distance from a random point to the mother universe. http://arxiv.org/abs/1002.2552 9727. Characteristic functions of affine processes via calculus of their operator symbols Author(s): Joerg Kampen Abstract: The characteristic functions of multivariate Feller processes with generator of affine type, and with smooth symbol functions have an explicit representation in terms of power series with rational number coefficients and with monmoms consisting of powers of the the symbol functions and formal derivatives of the symbol functions. The power series repesentations are convergent globally in time and on bounded domains of arbitrary size. Generalized symbol functions can be derived leading to power series expansions which are convergent on arbitrary domains in special cases. The rational number coefficients can be efficiently computed by an integer recursion. As a numerical consequence characteristic functions of multivariate affine processes can be efficiently computed from the symbol function avoiding computation of the generalized Riccati equations (an observation first made recently in a more general context). http://arxiv.org/abs/1002.2764 9728. Optimal consumption and investment with bounded downside risk for power utility functions Author(s): Claudia Kluppelberg and Serguei Pergamenchtchikov (LMRS) Abstract: We investigate optimal consumption and investment problems for a Black-Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall. We formulate various utility maximization problems, which can be solved explicitly. We compare the optimal solutions in form of optimal value, optimal control and optimal wealth to analogous problems under additional uniform risk bounds. Our proofs are partly based on solutions to Hamilton-Jacobi-Bellman equations, and we prove a corresponding verification theorem. This work was supported by the European Science Foundation through the AMaMeF programme. http://arxiv.org/abs/1002.2487 9729. Functional Ito calculus and stochastic integral representation of martingales Author(s): Rama Cont and David-Antoine Fournie Abstract: We develop a non-anticipative calculus for functionals of a continuous semimartingale, using a notion of pathwise functional derivative. A functional extension of the Ito formula is derived and used to obtain a constructive martingale representation theorem for a class of continuous martingales verifying a regularity property. By contrast with the Clark-Haussmann-Ocone formula, this representation involves non-anticipative quantities which can be computed pathwise. These results are used to construct a weak derivative acting on square-integrable martingales, which is shown to be the inverse of the Ito integral, and derive an integration by parts formula for Ito stochastic integrals. We show that this weak derivative may be viewed as a non-anticipative "lifting" of the Malliavin derivative. Regular functionals of an Ito martingale which have the local martingale property are characterized as solutions of a functional differential equation, for which a uniqueness result is given. http://arxiv.org/abs/1002.2446 9730. Plaquettes, Spheres, and Entanglement Author(s): Geoffrey R. Grimmett and Alexander E. Holroyd Abstract: The high-density plaquette percolation model in d dimensions contains a surface that is homeomorphic to the (d-1)-sphere and encloses the origin. This is proved by a path-counting argument in a dual model. When d=3, this permits an improved lower bound on the critical point p_e of entanglement percolation, namely p_e >= \mu^-2 where \mu is the connective constant for self-avoiding walks on Z^3. Furthermore, when the edge density p is below this bound, the radius of the entanglement cluster containing the origin has an exponentially decaying tail. http://arxiv.org/abs/1002.2623 9731. Optimal consumption and investment with bounded downside risk measures for logarithmic utility functions Author(s): Claudia Kluppelberg and Serguei Pergamenchtchikov (LMRS) Abstract: We investigate optimal consumption problems for a Black-Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall for logarithmic utility functions. We find the solutions in terms of a dynamic strategy in explicit form, which can be compared and interpreted. This paper continues our previous work, where we solved similar problems for power utility functions. http://arxiv.org/abs/1002.2486 9732. Optimal consumption and investment with bounded downside risk for power utility functions Author(s): Claudia Kluppelberg and Serguei Pergamenchtchikov (LMRS) Abstract: We investigate optimal consumption and investment problems for a Black-Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall. We formulate various utility maximization problems, which can be solved explicitly. We compare the optimal solutions in form of optimal value, optimal control and optimal wealth to analogous problems under additional uniform risk bounds. Our proofs are partly based on solutions to Hamilton-Jacobi-Bellman equations, and we prove a corresponding verification theorem. This work was supported by the European Science Foundation through the AMaMeF programme. http://arxiv.org/abs/1002.2487 9733. Geometric ergodicity for families of homogeneous Markov chains Author(s): Leonid Galtchouk (IRMA) and Serguei Pergamenchtchikov (LMRS) Abstract: In this paper we find nonasymptotic exponential upper bounds for the deviation in the ergodic theorem for families of homogeneous Markov processes. We find some sufficient conditions for geometric ergodicity uniformly over a parametric family. We apply this property to the nonasymptotic nonparametric estimation problem for ergodic diffusion processes. http://arxiv.org/abs/1002.2341 9734. Duality theory for Markov processes: Part 1 Author(s): Ronald Getoor Abstract: This is the first part of a possible monograph on the duality of Markov processes. It contains a proof of Fitzsimmons' existence theorem of a moderate Markov dual process relative to an excessive measure, m, together with the necessary preliminary material. Then this is applied to prove the correspondence between optional copredictable homogenous random measures and sigma finite measures not charging m-exceptional sets again following Fitzsimmons. The second part which may never be written would deal with duality proper including results from, but not limited to, my joint paper with P. J. Fitzsimmons"Potential Theory of Moderate Markov Dual Processes" which appeared in Potential Anal.(2009) 31:275-310. Complete proofs of all results not appearing in standard books are given with the one exception of Dellacherie's result characterizing semipolar sets. http://arxiv.org/abs/1002.2399 9735. Viability, Invariance and Reachability for Controlled Piecewise Deterministic Markov Processes Associated to Gene Networks Author(s): D. Goreac Abstract: We aim at characterizing viability, invariance and some reachability properties of controlled piecewise deterministic Markov processes (PDMPs). Using analytical methods from the theory of viscosity solutions, we establish criteria for viability and invariance in terms of the first order normal cone. We also investigate reachability of arbitrary open sets. The method is based on viscosity techniques and duality for some associated linearized problem. The theoretical results are applied to general On/Off systems, Cook's model for haploinssuficiency, and a stochastic model for bacteriophage lambda. http://arxiv.org/abs/1002.2242 9736. Forward-convex convergence of sequences in $\mathbb{L}^0_+$ Author(s): Constantinos Kardaras and Gordan Zitkovic Abstract: For a sequence in $\mathbb{L}^0_+$, we provide simple necessary and sufficient conditions to ensure that each sequence of its forward convex combinations converges to the same limit. These conditions correspond to a measure-free version of the notion of uniform integrability and are related to the numeraire problem of mathematical finance. http://arxiv.org/abs/1002.1889 9737. Impulse control problem with switching technology Author(s): Rim Amami Abstract: We consider an impulse control problem in infinite horizon applied with switching technology. We suppose that the firm decides at certain moments (impulse moments) to switch technology, leading to a jump of the firm value. We show that the value function for such problems satisfies a dynamic programming principle version. Our objective is to look for an optimal strategy which maximizes the value function associated with a switching problem. http://arxiv.org/abs/1002.2086 9738. Applying coupon-collecting theory to computer-aided assessments Author(s): Charles M. Goldie (University of Sussex) and Rosie Cornish (University of Bristol), Carol L. Robinson (Loughborough University) Abstract: Computer-based tests with randomly generated questions allow a large number of different tests to be generated. Given a fixed number of alternatives for each question, the number of tests that need to be generated before all possible questions have appeared is surprisingly low. http://arxiv.org/abs/1002.2114 9739. The fractional Poisson measure in infinite dimensions Author(s): Maria Joao Oliveira and Habib Ouerdiane and Jose Luis da Silva and R. Vilela Mendes Abstract: The Mittag-Leffler function $E_{\alpha}$ being a natural generalization of the exponential function, an infinite-dimensional version of the fractional Poisson measure would have a characteristic functional \[ C_{\alpha}(\varphi) :=E_{\alpha}(\int (e^{i\varphi(x)}-1)d\mu (x)) \] which we prove to fulfill all requirements of the Bochner-Minlos theorem. The identity of the support of this new measure with the support of the infinite-dimensional Poisson measure ($\alpha =1$) allows the development of a fractional infinite-dimensional analysis modeled on Poisson analysis through the combinatorial harmonic analysis on configuration spaces. This setting provides, in particular, explicit formulas for annihilation, creation, and second quantization operators. In spite of the identity of the supports, the fractional Poisson measure displays some noticeable differences in relation to the Poisson measure, which may be physically quite significant. http://arxiv.org/abs/1002.2124 9740. Bernstein type's concentration inequalities for symmetric Markov processes Author(s): Fuqing Gao and Arnaud Guillin and Liming Wu Abstract: Using the method of transportation-information inequality introduced in \cite{GLWY}, we establish Bernstein type's concentration inequalities for empirical means $\frac 1t \int_0^t g(X_s)ds$ where $g$ is a unbounded observable of the symmetric Markov process $(X_t)$. Three approaches are proposed : functional inequalities approach ; Lyapunov function method ; and an approach through the Lipschitzian norm of the solution to the Poisson equation. Several applications and examples are studied. http://arxiv.org/abs/1002.2163 9741. The Ghirlanda-Guerra identities for mixed p-spin model Author(s): Dmitry Panchenko Abstract: We show that, under the conditions known to imply the validity of the Parisi formula, if the generic Sherrington-Kirkpatrick Hamiltonian contains a $p$-spin term then the Ghirlanda-Guerra identities for the $p$th power of the overlap hold in a strong sense without averaging. This implies strong version of the extended Ghirlanda-Guerra identities for mixed $p$-spin models than contain terms for all even $p\geq 2$ and $p=1.$ http://arxiv.org/abs/1002.2190 9742. On the strict comparison theorem for $G$-expectations Author(s): Xinpeng Li Abstract: In this paper, we prove two forms of strict comparison theorem for $X,Y\in L_G^1(\Omega)$. Furthermore, if $X,Y\in Lip(\Omega)$ and $\x>0$, we give a necessary and sufficient condition under which the strict comparison theorem holds. http://arxiv.org/abs/1002.1765 9743. On isoperimetric sets of radially symmetric measures Author(s): Alexander V. Kolesnikov and Roman I. Zhdanov Abstract: We study the isoperimetric problem for the radially symmetric measures. Applying the spherical symmetrization procedure and variational arguments we reduce this problem to a one-dimensional ODE of the second order. Solving numerically this ODE we get an empirical description of isoperimetric regions of the planar radially symmetric exponential power laws. We also prove some isoperimetric inequalities for the log-convex measures. We show, in particular, that the symmetric balls of large size are isoperimetric sets for strictly log-convex and radially symmetric measures. We also establish a comparison theorem for the products of the one-dimensional log-convex measures. http://arxiv.org/abs/1002.1829 9744. A local limit theorem for random walks in random scenery and on randomly oriented lattices Author(s): Fabienne Castell (LATP) and Nadine Guillotin-Plantard (UCB and ICJ) and Fran\c{c}oise P\`ene (LM), Bruno Schapira (LM-Orsay) Abstract: Random walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^n\xi_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and $(\xi_y,y\in\mathbb Z)$ are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index $\alpha\in (0,2]$ and $\beta\in (0,2]$ respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when $\alpha\neq 1$ and as $n\to \infty$, of $n^{-\delta}Z_n$, for some suitable $\delta>0$ depending on $\alpha$ and $\beta$. Here we are interested in the convergence, as $n\to \infty$, of $n^\delta{\mathbb P}(Z_n=\lfloor n^{\delta} x\rfloor)$, when $x\in \RR$ is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results. http://arxiv.org/abs/1002.1878 9745. Percolation in invariant Poisson graphs with i.i.d. degrees Author(s): Maria Deijfen and Olle Haggstrom and and Alexander E. Holroyd Abstract: Let each point of a homogeneous Poisson process in R^d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution mu on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution mu. Leaving aside degenerate cases, we prove that for any mu there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme that is a natural extension of Gale-Shapley stable marriage, we give sufficient conditions on mu for the absence and presence of infinite components. http://arxiv.org/abs/1002.1943 9746. Stochastic Analysis of Non-slotted Aloha in Wireless Ad-Hoc Networks Author(s): Bartek Blaszczyszyn (INRIA Rocquencourt) and Paul Muhlethaler (INRIA Rocquencourt) Abstract: In this paper we propose two analytically tractable stochastic models of non-slotted Aloha for Mobile Ad-hoc NETworks (MANETs): one model assumes a static pattern of nodes while the other assumes that the pattern of nodes varies over time. Both models feature transmitters randomly located in the Euclidean plane, according to a Poisson point process with the receivers randomly located at a fixed distance from the emitters. We concentrate on the so-called outage scenario, where a successful transmission requires a Signal-to-Interference-and-Noise Ratio (SINR) larger than a given threshold. With Rayleigh fading and the SINR averaged over the duration of the packet transmission, both models lead to closed form expressions for the probability of successful transmission. We show an excellent matching of these results with simulations. Using our models we compare the performances of non-slotted Aloha to previously studied slotted Aloha. We observe that when the path loss is not very strong both models, when appropriately optimized, exhibit similar performance. For stronger path loss non-slotted Aloha performs worse than slotted Aloha, however when the path loss exponent is equal to 4 its density of successfully received packets is still 75% of that in the slotted scheme. This is still much more than the 50% predicted by the well-known analysis where simultaneous transmissions are never successful. Moreover, in any path loss scenario, both schemes exhibit the same energy efficiency. http://arxiv.org/abs/1002.1629 9747. Increasing the chromatic number of a random graph Author(s): N. Alon and B. Sudakov Abstract: What is the minimum number of edges that have to be added to the random graph $G=G_{n,0.5}$ in order to increase its chromatic number $\chi=\chi(G)$ by one percent ? One possibility is to add all missing edges on a set of $1.01 \chi$ vertices, thus creating a clique of chromatic number $1.01 \chi$. This requires, with high probability, the addition of $\Omega(n^2/\log^2 n)$ edges. We show that this is tight up to a constant factor, consider the question for more general random graphs $G_{n,p}$ with $p=p(n)$, and study a local version of the question as well. The question is motivated by the study of the resilience of graph properties, initiated by the second author and Vu, and improves one of their results. http://arxiv.org/abs/1002.1748 9748. How to lose as little as possible Author(s): Vittorio Addona and Stan Wagon and and Herb Wilf Abstract: Suppose Alice has a coin with heads probability $q$ and Bob has one with heads probability $p>q$. Now each of them will toss their coin $n$ times, and Alice will win iff she gets more heads than Bob does. Evidently the game favors Bob, but for the given $p,q$, what is the choice of $n$ that maximizes Alice's chances of winning? The problem of determining the optimal $N$ first appeared in \cite{wa}. We show that there is an essentially unique value $N(q,p)$ of $n$ that maximizes the probability $f(n)$ that the weak coin will win, and it satisfies $\frac{1}{2(p-q)}-\frac12\le N(q,p)\le \frac{\max{(1-p,q)}}{p-q}$. The analysis uses the multivariate form of Zeilberger's algorithm to find an indicator function $J_n(q,p)$ such that $J>0$ iff $n http://arxiv.org/abs/1002.1763 9749. Forward-convex convergence of sequences in $\mathbb{L}^0_+$ Author(s): Constantinos Kardaras and Gordan Zitkovic Abstract: For a sequence in $\mathbb{L}^0_+$, we provide simple necessary and sufficient conditions to ensure that each sequence of its forward convex combinations converges to the same limit. These conditions correspond to a measure-free version of the notion of uniform integrability and are related to the numeraire problem of mathematical finance. http://arxiv.org/abs/1002.1889 9750. The Bismut-Elworthy-Li type formulae for stochastic differential equations with jumps Author(s): Atsushi Takeuchi Abstract: Consider jump-type stochastic differential equations with the drift, diffusion and jump terms. Logarithmic derivatives of densities for the solution process are studied, and the Bismut-Elworthy-Li type formulae can be obtained under the uniformly elliptic condition on the coefficients of the diffusion and jump terms. Our approach is based upon the Kolmogorov backward equation by making full use of the Markovian property of the process. http://arxiv.org/abs/1002.1384 9751. Stochastic Power Law Fluids:the Existence and the Uniqueness of the Weak Solution Author(s): Nobuo Yoshida and Yutaka Terasawa Abstract: We consider a SPDE (stochastic partial differential equation) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random perturbation. Here, the extra stress tensor of the fluid is given by a polynomial of degree $p-1$ of the deformation rate tensor, while the colored noise is considered as the random perturbation. We investigate the existence and the uniqueness of the weak solution to this SPDE. http://arxiv.org/abs/1002.1431 9752. Extension of the Olkin and Rubin Characterization to the Wishart distribution on homogeneous cones Author(s): Imen Boutouria and Abdelhamid Hassairi and Helene Massam Abstract: The Wishart distribution on an homogeneous cone is a generalization of the Riesz distribution on a symmetric cone which corresponds to a given graph. The paper extends to this distribution, the famous Olkin and Rubin characterization of the ordinary Wishart distribution on symmetric matrices. http://arxiv.org/abs/1002.1451 9753. Hitting densities for spectrally positive stable processes Author(s): Thomas Simon (LPP) Abstract: A multiplicative identity in law connecting the hitting times of completely asymmetric $\alpha-$stable L\'evy processes in duality is established. In the spectrally positive case, this identity allows with an elementary argument to compute fractional moments and to get series representations for the density. We also prove that the hitting times are unimodal as soon as $\alpha\le 3/2.$ Analogous results are obtained, in a much simplified manner, for the first passage time across a positive level. http://arxiv.org/abs/1002.1540 9754. Generalized Dirichlet distributions on the ball and moments Author(s): Franck Barthe and Fabrice Gamboa and Li-Vang Lozada-Chang and Alain Rouault Abstract: The geometry of unit $N$-dimensional $\ell_{p}$ balls has been intensively investigated in the past decades. A particular topic of interest has been the study of the asymptotics of their projections. Apart from their intrinsic interest, such questions have applications in several probabilistic and geometric contexts (Barthe et al. 2005). In this paper, our aim is to revisit some known results of this flavour with a new point of view. Roughly speaking, we will endow the ball with some kind of Dirichlet distribution that generalizes the uniform one and will follow the method developed in Skibinsky (1967), Chang et al. (1993) in the context of the randomized moment space. The main idea is to build a suitable coordinate change involving independent random variables. Moreover, we will shed light on a nice connection between the randomized balls and the randomized moment space. http://arxiv.org/abs/1002.1544 9755. Space-time correlations of a Gaussian interface Author(s): Francois M. Dunlop Abstract: The serial harness introduced by Hammersley is equivalent, in the Gaussian case, to the Gaussian Solid-On-Solid interface model with parallel heat bath dynamics. Here we consider sub-lattice parallel dynamics, and give exact results about relaxation dynamics, based on the equivalence to the infinite time limit of a time periodic random field. We also give a numerical comparison to the harness process in continuous time studied by Hsiao and by Ferrari, Niederhauser and Pechersky. http://arxiv.org/abs/1002.1604 9756. Adaptive LASSO-type estimation for ergodic diffusion processes Author(s): A. De Gregorio and S.M. Iacus Abstract: The LASSO is a widely used statistical methodology for simultaneous estimation and variable selection. In the last years, many authors analyzed this technique from a theoretical and applied point of view. We introduce and study the adaptive LASSO problem for discretely observed ergodic diffusion processes. We prove oracle properties also deriving the asymptotic distribution of the LASSO estimator. Our theoretical framework is based on the random field approach and it applied to more general families of regular statistical experiments in the sense of Ibragimov-Hasminskii (1981). Furthermore, we perform a simulation and real data analysis to provide some evidence on the applicability of this method. http://arxiv.org/abs/1002.1312 9757. Probabilistic interpretation of the M\"obius function identity and the Riemann Hypothesis Author(s): R. M. Abrarov and R. M. Abrarov Abstract: We obtained the probabilities for the values of the M\"obius function for arbitrary numbers and found that the asymptotic densities of the squarefree integers among the odd and even numbers are $8/\pi^2$ and $4/\pi^2$, respectively. It is determined that statistics of successive outcomes of the M\"obius function for very large squarefree odd and even numbers behaves similar to statistics of heads and tails of two flipping coins. These preliminary results are giving arguments supporting the Riemann Hypothesis. Its plausibility is based on statistical phenomena for integers. http://arxiv.org/abs/1002.1682 9758. Heat Kernel Estimate for $\Delta+\Delta^{\alpha/2}$ in $C^{1,1}$ open sets Author(s): Zhen-Qing Chen and Panki Kim and Renming Song Abstract: We consider a family of pseudo differential operators $\{\Delta+ a^\alpha \Delta^{\alpha/2}; a\in (0, 1]\}$ on $\bR^d$ for every $d\geq 1$ that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/2}$, where $\alpha \in (0, 2)$. It gives rise to a family of L\'evy processes $\{X^a, a\in (0, 1]\}$ in $\bR^d$, where $X^a$ is the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with weight $a$. We establish sharp two-sided estimates for the heat kernel of $\Delta + a^{\alpha} \Delta^{\alpha/2}$ with zero exterior condition in a family of open subsets, including bounded $C^{1, 1}$ (possibly disconnected) open sets. This heat kernel is also the transition density of the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with weight $a$ in such open sets. Our result is the first sharp two-sided estimates for the transition density of a Markov process with both diffusion and jump components in open sets. Moreover, our result is uniform in $a$ in the sense that the constants in the estimates are independent of $a\in (0, 1]$ so that it recovers the Dirichlet heat kernel estimates for Brownian motion by taking $a\to 0$. Integrating the heat kernel estimates in time $t$, we recover the two-sided sharp uniform Green function estimates of $X^a$ in bounded $C^{1,1}$ open sets in $\bR^d$, which were recently established in \cite{CKSV2} by using a completely different approach. http://arxiv.org/abs/1002.1121 9759. Random sampling of lattice paths with constraints, via transportation Author(s): Lucas Gerin (MODAL'x) Abstract: We discuss a Monte Carlo Markov Chain (MCMC) procedure for the random sampling of some one-dimensional lattice paths with constraints, for various constraints. We show that an approach inspired by optimal transport allows us to bound efficiently the mixing time of the associated Markov chain. The algorithm is robust and easy to implement, and samples an "almost" uniform path of length $n$ in $n^{3+\eps}$ steps. This bound makes use of a certain contraction property of the Markov chain, and is also used to derive a bound for the running time of Propp-Wilson's CFTP algorithm. http://arxiv.org/abs/1002.1183 9760. Universal Gaussian fluctuations of non-Hermitian matrix ensembles: from weak convergence to almost sure CLTs Author(s): Ivan Nourdin (PMA) and Giovanni Peccati Abstract: In the paper [25], written in collaboration with Gesine Reinert, we proved a universality principle for the Gaussian Wiener chaos. In the present work, we aim at providing an original example of application of this principle in the framework of random matrix theory. More specifically, by combining the result in [25] with some combinatorial estimates, we are able to prove multi-dimensional central limit theorems for the spectral moments (of arbitrary degrees) associated with random matrices with real-valued i.i.d. entries, satisfying some appropriate moment conditions. Our approach has the advantage of yielding, without extra effort, bounds over classes of smooth (i.e., thrice differentiable) functions, and it allows to deal directly with discrete distributions. As a further application of our estimates, we provide a new "almost sure central limit theorem", involving logarithmic means of functions of vectors of traces. http://arxiv.org/abs/1002.1212 9761. A New Phase Transition for Local Delays in MANETs Author(s): Fran\c{c}ois Baccelli (INRIA Rocquencourt) and Bartek Blaszczyszyn (INRIA Rocquencourt) Abstract: We consider Mobile Ad-hoc Network (MANET) with transmitters located according to a Poisson point in the Euclidean plane, slotted Aloha Medium Access (MAC) protocol and the so-called outage scenario, where a successful transmission requires a Signal-to-Interference-and-Noise (SINR) larger than some threshold. We analyze the local delays in such a network, namely the number of times slots required for nodes to transmit a packet to their prescribed next-hop receivers. The analysis depends very much on the receiver scenario and on the variability of the fading. In most cases, each node has finite-mean geometric random delay and thus a positive next hop throughput. However, the spatial (or large population) averaging of these individual finite mean-delays leads to infinite values in several practical cases, including the Rayleigh fading and positive thermal noise case. In some cases it exhibits an interesting phase transition phenomenon where the spatial average is finite when certain model parameters are below a threshold and infinite above. We call this phenomenon, contention phase transition. We argue that the spatial average of the mean local delays is infinite primarily because of the outage logic, where one transmits full packets at time slots when the receiver is covered at the required SINR and where one wastes all the other time slots. This results in the "RESTART" mechanism, which in turn explains why we have infinite spatial average. Adaptive coding offers a nice way of breaking the outage/RESTART logic. We show examples where the average delays are finite in the adaptive coding case, whereas they are infinite in the outage case. http://arxiv.org/abs/1002.0855 9762. Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation Author(s): Lung-Chi Chen and Akira Sakai Abstract: We consider random walk and self-avoiding walk whose 1-step distribution is given by D, and oriented percolation whose bond-occupation probability is proportional to D. Suppose that D(x) decays as |x|^{-d-a} with a>0. For random walk in any dimension and for self-avoiding walk and critical/subcritical oriented percolation above the common upper-critical dimension 2min{a,2}, we prove large-t asymptotics of the gyration radius, which is the average end-to-end distance of random walk/self-avoiding walk of length t or the average spatial size of an oriented-percolation cluster at time t. This proves the conjecture for long-range self-avoiding walk by Heydenreich and for long-range oriented percolation in Chen and Sakai (2009). http://arxiv.org/abs/1002.0875 9763. On the Existence and Uniqueness of Solutions to Stochastic Differential Equations Driven by G-Brownian Motion with Integral-Lipschitz Coefficients Author(s): Xuepeng Bai and Yiqing Lin Abstract: In this paper, we study the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with an integral-Lipschitz condition for the coefficients. http://arxiv.org/abs/1002.1046 9764. Crowding of Brownian spheres Author(s): Krzysztof Burdzy and Soumik Pal and Jason Swanson Abstract: We study two models consisting of reflecting one-dimensional Brownian "particles" of positive radius. We show that the stationary empirical distributions for the particle systems do not converge to the harmonic function for the generator of the individual particle process, unlike in the case when the particles are infinitely small. http://arxiv.org/abs/1002.1057 9765. Exit times in non-Markovian drifting continuous-time random walk processes Author(s): Miquel Montero and Javier Villarroel Abstract: By appealing to renewal theory we determine the equations that the mean exit time of a continuous-time random walk with drift satisfies both when the present coincides with a jump instant or when it does not. Particular attention is paid to the corrections ensuing from the non-Markovian nature of the process. We show that when drift and jumps have the same sign the relevant integral equations can be solved in closed form. The case when holding times have the classical Erlang distribution is considered in detail. http://arxiv.org/abs/1002.0571 9766. Gibbs Random Graphs Author(s): Pablo A. Ferrari and Eugene A. Pechersky and Valentin V. Sisko and Anatoly A. Yambartsev Abstract: Consider a discrete locally finite subset $\Gamma$ of $R^d$ and the complete graph $(\Gamma,E)$, with vertices $\Gamma$ and edges $E$. We consider Gibbs measures on the set of sub-graphs with vertices $\Gamma$ and edges $E'\subset E$. The Gibbs interaction acts between open edges having a vertex in common. We study percolation properties of the Gibbs distribution of the graph ensemble. The main results concern percolation properties of the open edges in two cases: (a) when the $\Gamma$ is a sample from homogeneous Poisson process and (b) for a fixed $\Gamma$ with exponential decay of connectivity. http://arxiv.org/abs/1002.0610 9767. Series representations and asymptotic expansions for the density of the supremum of a stable process Author(s): Alexey Kuznetsov Abstract: We derive explicit asymptotic expansions of the density of the supremum of a strictly stable process when the index $\alpha$ is not rational. In the case when parameters $\alpha$ and $\rho=\p(X_1>0)$ satisfy $\rho+k=l/\alpha$ for some integers $k,l \ge 1$ we prove that these asymptotic expansions are in fact convergent series representations of the density of supremum. http://arxiv.org/abs/1002.0614 9768. Small Time Chung Type LIL for L\'evy Processes at Zero Author(s): Frank Aurzada and Leif Doering and and Mladen Savov Abstract: We prove Chung-type laws of the iterated logarithm for general L\'evy processes at zero. In particular, we provide a tool to translate small deviation estimates directly into laws of the iterated logarithm without any loss of constants nor any extra conditions. This reveals new laws of the iterated logarithm for L\'evy processes at small times in many concrete examples. In some cases, exotic norming functions are derived. http://arxiv.org/abs/1002.0675 9769. Applications of the graphs to the Generalized Ornstein-Uhlenbeck process Author(s): Boubaker Smii Abstract: We consider the generalized Ornstein- Uhlenbeck equation $\partial_t X=-m X_t+\eta$. In this paper We construct the L\'evy noise $\eta$. The generalized Ornstein- Uhlenbeck process $X_t$ will be represented by a special types of graphs called rooted trees with two types of leaves. http://arxiv.org/abs/1002.0744 9770. Fractional L\'evy processes by compact interval integral transformation Author(s): Heikki Tikanm\"aki Abstract: We define fractional L\'evy processes by two different integral transformations by taking integral representation of fractional Brownian motion and replacing the driving Brownian motion by more general square integrable L\'evy process. The definition using infinitely supported transformation kernel is well known in the literature but the definition by compact interval representation is new to the best of my knowledge in this setup. We prove that the processes defined by different transformations do not have the same finite dimensional distributions in general, even though it is the case in fractional Brownian motion setup. Hovever, we prove a connection between the two concepts. We consider different properties of fractional L\'evy processes by compact interval transformation and compare them to the properties of fractional L\'evy processes by infinite interval transformation. We also consider financial applications and represent a no-arbitrage theorem for a model including fractional L\'evy processes by any of the two integral transformations. http://arxiv.org/abs/1002.0780 9771. Metastability in communication networks Author(s): D. Tibi Abstract: Two models of loss networks, introduced by Gibbens et al. and by Antunes et al., are known to exhibit a mean field limiting regime with several stable equilibria. This paper first provides an interpretation of the Lyapunov function given by Antunes et al., in terms of entropy dissipation. The argument extends to another, similar but closed model. The two main models are next reexamined in the light of Freidlin and Wentzell's large deviation approach of randomly perturbed dynamical systems. Assuming that some of their results still hold under slightly relaxed conditions, the metastability property is derived for both systems. The Lyapunov function of the second model is then identified with the quasipotential associated with a slightly modified, asymptotically reversible, Markovian perturbation of the same dynamical system. http://arxiv.org/abs/1002.0796 9772. Exit times in non-Markovian drifting continuous-time random walk processes Author(s): Miquel Montero and Javier Villarroel Abstract: By appealing to renewal theory we determine the equations that the mean exit time of a continuous-time random walk with drift satisfies both when the present coincides with a jump instant or when it does not. Particular attention is paid to the corrections ensuing from the non-Markovian nature of the process. We show that when drift and jumps have the same sign the relevant integral equations can be solved in closed form. The case when holding times have the classical Erlang distribution is considered in detail. http://www.arxiv.org 9773. Fluctuations for the Ginzburg-Landau $\nabla \phi$ Interface Model on a Bounded Domain Author(s): Jason Miller Abstract: The object of our study is the massless field on $D_n = D \cap \tfrac{1}{n} \Z^2$, where $D \subseteq \R^2$ is a bounded domain with smooth boundary, with Hamiltonian $\CH(h) = \sum_{x \sim y} \CV(h(x) - h(y))$ and $h(x) = f(x)$ if $x \in \partial D_n$ for a given continuous function $f \colon \R^2 \to \R$. The interaction $\CV$ is assumed to be symmetric, strictly convex, and have bounded second derivatives. This is a general model for a $(2+1)$-dimensional effective interface where $h$ represents the height. We prove that linear functionals of $h$ converge in the limit to a Gaussian free field on $D$, the standard Gaussian with respect to the Dirichlet inner product $(f,g)_\nabla = \int_D \nabla f \cdot \nabla g$. The main step in the proof is to establish a general estimate that serves to quantify the degree to which the presence of boundary conditions affect the behavior of the model. In particular, our estimate implies that $\E h(x)$ converges to the harmonic extension of $f$ from $\partial D$ to $D$. In a subsequent article, we will employ the tools developed here to resolve a conjecture made by Sheffield that the zero contour lines of $h$ are asymptotically described by a family of conformally invariant random curves which are variants of SLE(4). http://arxiv.org/abs/1002.0381 9774. The maximum of Brownian motion with parabolic drift Author(s): Svante Janson and Guy Louchard and Anders Martin-L\"of Abstract: We study the maximum of a Brownian motion with a parabolic drift; this is a random variable that often occurs as a limit of the maximum of discrete processes whose expectations have a maximum at an interior point. We give series expansions and integral formulas for the distribution and the first two moments, together with numerical values to high precision. http://arxiv.org/abs/1002.0497 9775. Convergence of an Adaptive Approximation Scheme for the Wiener Process Author(s): Mats Brod\'en and Magnus Wiktorsson Abstract: The problem of approximating/tracking the value of a Wiener process is considered. The discretization points are placed at times when the value of the process differs from the approximation by some amount, here denoted by eta. It is found that the limiting difference, as eta goes to 0, between the approximation and the value of the process normalized with eta converges in distribution to a triangularly distributed random variable. http://arxiv.org/abs/1002.0528 9776. On some Bayesian nonparametric estimators for species richness under two-parameter Poisson-Dirichlet priors Author(s): Annalisa Cerquetti Abstract: We present an alternative approach to the Bayesian nonparametric analysis of conditional species richness under two-parameter Poisson Dirichlet priors. We rely on a known characterization by deletion of classes property and on results for Beta-Binomial distributions. Besides leading to simplified and much more direct proofs, our proposal provides a new scale mixture representation of the conditional asymptotic law. http://arxiv.org/abs/1002.0535 9777. On Stochastic generalized functions Author(s): Pedro Catuogno and Christian Olivera Abstract: We introduced a new algebra of stochastic generalized functions which contains to the space of stochastic distributions G, [25]. As an application, we prove existence and uniqueness of the solution of a stochastic Cauchy problem involving singularities. http://arxiv.org/abs/1002.0454 9778. Wright-Fisher model with negative mutation rates Author(s): Soumik Pal Abstract: We study a family of multidimensional diffusions taking values in the unit simplex of vectors with non-negative coordinates that add up to one. The family of processes satisfy stochastic differential equations which are similar to the ones for the classical Wright-Fisher model, except that the "mutation rates" are now nonpositive. This model, suggested by Aldous, appears in the study of a conjectured diffusion limit for a Markov chain on Cladograms. The striking feature of these models is that the boundary is not reflecting, and we kill the process once it hits the boundary. We derive the explicit exit distribution from the simplex, and probabilistic bounds on the exit time. We also prove that these processes can be viewed as a "stochastic time-reversal" of a Wright-Fisher process of increasing dimensions and conditioned at a random time. A key idea in our proofs is a skew-product construction using certain one-dimensional diffusions called Bessel-square processes of negative dimensions which have been recently introduced by Going-Jaeschke and Yor. http://arxiv.org/abs/1002.0159 9779. BS$\Delta$Es and BSDEs with non-Lipschitz drivers: comparison, convergence and robustness Author(s): Patrick Cheridito and Mitja Stadje Abstract: We provide existence results and comparison principles for solutions of backward stochastic difference equations (BS$\Delta$Es) and then prove convergence of these to solutions of backward stochastic differential equations (BSDEs) when the mesh size of the time-discretizaton goes to zero. The BS$\Delta$Es and BSDEs are governed by drivers $f^N(t,\omega,y,z)$ and $f(t,\omega,y,z),$ respectively. The new feature of this paper is that they may be non-Lipschitz in $z$. For the convergence results it is assumed that the BS$\Delta$Es are based on $d$-dimensional random walks $W^N$ approximating the $d$-dimensional Brownian motion $W$ underlying the BSDE and that $f^N$ converges to $f$. Conditions are given under which for any terminal condition $\xi$, there exist terminal conditions $\xi^N$ for the sequence of BS$\Delta$Es, converging to $\xi$ in $L^2$, such that for the solutions $Y^N$ and $Y$ of the corresponding BS$\Delta$Es and the limiting BSDE one has $\sup_{0\le t\le T} |Y^N_t - Y_t| \to 0$ in $L^2$. An important special case is when $f^N(t,\omega,y,z)$ and $f(t,\omega,y,z)$ are convex in $z.$ We show that in this situation, $\sup_{0\le t\le T} |Y^N_t - Y_t| \to 0$ in $L^2$ for every sequence of discrete terminal conditions $\xi^N$ converging to $\xi$ in $L^2$. As a consequence, one obtains that the BSDE is robust in the sense that if $(W^N,\xi^N)$ is close to $(W,\xi)$ in distribution, then $Y^N$ is close to $Y$ in distribution too. http://arxiv.org/abs/1002.0175 9780. Convergence of U-statistics for interacting particle systems Author(s): P. Del Moral and F. Patras and S. Rubenthaler Abstract: The convergence of U-statistics has been intensively studied for estimators based on families of i.i.d. random variables and variants of them. In most cases, the independence assumption is crucial [Lee90, de99]. When dealing with Feynman-Kac and other interacting particle systems of Monte Carlo type, one faces a new type of problem. Namely, in a sample of N particles obtained through the corresponding algorithms, the distributions of the particles are correlated -although any finite number of them is asymptotically independent with respect to the total number N of particles. In the present article, exploiting the fine asymptotics of particle systems, we prove convergence theorems for U-statistics in this framework. http://arxiv.org/abs/1002.0224 9781. Ergodicity for infinite particle systems with locally conserved quantities Author(s): J. Inglis and M. Neklyudov and B. Zegarlinski Abstract: We analyse certain degenerate infinite dimensional sub-elliptic generators and obtain estimates on the long-time behaviour of the corresponding Markov semigroups. http://arxiv.org/abs/1002.0282 9782. Non-Equilibrium Statistical Physics of Currents in Queuing Networks Author(s): Vladimir Y.Chernyak and Michael Chertkov and David A. Goldberg and Konstantin Turitsyn Abstract: A stable open queuing network is considered as a steady non-equilibrium system of interacting particles. The network is completely specified by its underlying graphical structure, type of interaction at each node, and the Poisson transition rates between nodes. For such systems we identify two regimes in which the system may operate depending on the value of currents accumulated on the graph edges over time, large compared to the system correlation time scale. In the first regime of moderate currents, the large-deviation distribution of currents is universal (independent of the interaction details), and the system behaves in an "uncongested" mode. In the second regime of larger currents, the large-deviation current distribution is sensitive to interaction details, and the system is in a "congested" mode. The transition between the two regimes can be described as a dynamical second order phase transition. We illustrate these ideas using a simple, yet non-trivial, example of a single node with feedback. http://arxiv.org/abs/1001.5454 9783. Scalar curvature and $Q$-curvature of random metrics Author(s): Yaiza Canzani and Dmitry Jakobson and Igor Wigman Abstract: We study Gauss curvature for random Riemannian metrics on a compact surface, lying in a fixed conformal class; our questions are motivated by comparison geometry. Next, analogous questions are considered for the scalar curvature in dimension $n>2$, and for the $Q$-curvature of random Riemannian metrics. http://arxiv.org/abs/1002.0030 9784. Left and right convergence of graphs with bounded degree Author(s): Christian Borgs and Jennifer Chayes and Jeff Kahn and L\'aszl\'o Lov\'asz Abstract: The theory of convergent graph sequences has been worked out in two extreme cases, dense graphs and bounded degree graphs. One can define convergence in terms of counting homomorphisms from fixed graphs into members of the sequence (left-convergence), or counting homomorphisms into fixed graphs (right-convergence). Under appropriate conditions, these two ways of defining convergence was proved to be equivalent in the dense case by Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi. In this paper a similar equivalence is established in the bounded degree case. In terms of statistical physics, the implication that left convergence implies right convergence means that for a left-convergent sequence, partition functions of a large class of statistical physics models converge. The proof relies on techniques from statistical physics, like cluster expansion and Dobrushin Uniqueness. http://arxiv.org/abs/1002.0115 9785. Kalman-Bucy filter and SPDEs with growing lower-order coefficients in $W^{1}_{p}$ spaces without weights Author(s): N.V. Krylov Abstract: We consider divergence form uniformly parabolic SPDEs with VMO bounded leading coefficients, bounded coefficients in the stochastic part, and possibly growing lower-order coefficients in the deterministic part. We look for solutions which are summable to the $p$th power, $p\geq2$, with respect to the usual Lebesgue measure along with their first-order derivatives with respect to the spatial variable. Our methods allow us to include Zakai's equation for the Kalman-Bucy filter into the general filtering theory. http://arxiv.org/abs/1002.0306 9786. Packing and Hausdorff measures of stable trees Author(s): Thomas Duquesne (PMA) Abstract: In this paper we discuss Hausdorff and packing measures of random continuous trees called stable trees. Stable trees form a specific class of L\'evy trees (introduced by Le Gall and Le Jan in 1998) that contains Aldous's continuum random tree (1991) which corresponds to the Brownian case. We provide results for the whole stable trees and for their level sets that are the sets of points situated at a given distance from the root. We first show that there is no exact packing measure for levels sets. We also prove that non-Brownian stable trees and their level sets have no exact Hausdorff measure with regularly varying gauge function, which continues previous results from a joint work with J-F Le Gall (2006). http://arxiv.org/abs/1001.5329 9787. Universality of slow decorrelation in KPZ growth Author(s): I. Corwin and P.L. Ferrari and S. Peche Abstract: We demonstrate that, under minimal hypothesis, a wide class of growth models diplays a phenomenon known as slow decorrelation, where along certain characteristic directions the range of correlation for fluctuations of the growth surface height is much longer than other directions. We apply this result to certain models known to be in the Kardar-Parisi-Zhang (KPZ) universality class in 1+1 dimension for which the necessary hypothesis holds. These models are the totally asymmetric simple exclusion process (TASEP), last passage percolation (LPP), and the polynuclear growth (PNG) model. Utilizing the slow decorrelation of fluctuations in these models we are able to extend known fluctuation limit process results away from the fixed curves on which they were proved, to general space-time curves. Using the monotonicity of the basic coupling we additionally prove that the partially asymmetric simple exclusion process (PASEP) displays slow decorrelation. http://arxiv.org/abs/1001.5345 9788. The shortest distance in random multi-type intersection graphs Author(s): A. D. Barbour and G. Reinert Abstract: Using an associated branching process as the basis of our approximation, we show that typical inter-point distances in a multitype random intersection graph have a defective distribution, which is well described by a mixture of translated and scaled Gumbel distributions, the missing mass corresponding to the event that the vertices are not in the same component of the graph. http://arxiv.org/abs/1001.5357 9789. Long time behaviour in a model of microtubule growth Author(s): O.Hryniv and M.Menshikov Abstract: We study a continuous time stochastic process on strings made of two types of particles, whose dynamics mimics the behaviour of microtubules in a living cell; namely, the strings evolve via a competition between (local) growth/shrinking as well as (global) hydrolysis processes. We give a complete characterization of the phase diagram of the model, and derive several criteria of the transient and recurrent regimes for the underlying stochastic process. http://arxiv.org/abs/1001.5469 9790. On uniqueness of mild solutions for dissipative stochastic evolution equations Author(s): Carlo Marinelli and Michael R\"ockner Abstract: In the semigroup approach to stochastic evolution equations, the fundamental issue of uniqueness of mild solutions is often "reduced" to the much easier problem of proving uniqueness for strong solutions. This reduction is usually carried out in a formal way, without really justifying why and how one can do that. We provide sufficient conditions for uniqueness of mild solutions to a broad class of semilinear stochastic evolution equations with coefficients satisfying a monotonicity assumption. http://arxiv.org/abs/1001.5413 9791. Scalar conservation laws with stochastic forcing Author(s): Arnaud Debussche (IRMAR) and Julien Vovelle (ICJ) Abstract: We show that the Cauchy Problem for a randomly forced, periodic multi-dimensional scalar first-order conservation law with additive or multiplicative noise is well-posed: it admits a unique solution, characterized by a kinetic formulation of the problem, which is the limit of the solution of the stochastic parabolic approximation. http://arxiv.org/abs/1001.5415 9792. Hidden Regular Variation: Detection and Estimation Author(s): Abhimanyu Mitra and Sidney I. Resnick Abstract: Hidden regular variation defines a subfamily of distributions satisfying multivariate regular variation on $\mathbb{E} = [0, \infty]^d \backslash \{(0,0, ..., 0) \} $ and models another regular variation on the sub-cone $\mathbb{E}^{(2)} = \mathbb{E} \backslash \cup_{i=1}^d \mathbb{L}_i$, where $\mathbb{L}_i$ is the $i$-th axis. We extend the concept of hidden regular variation to sub-cones of $\mathbb{E}^{(2)}$ as well. We suggest a procedure of detecting the presence of hidden regular variation, and if it exists, propose a method of estimating the limit measure exploiting its semi-parametric structure. We exhibit examples where hidden regular variation yields better estimates of probabilities of risk sets. http://arxiv.org/abs/1001.5058 9793. Fluctuations of the occupation times for branching system starting from infinitely divisible point processes Author(s): Piotr Milos Abstract: In the paper the rescaled occupation time fluctuation process of a certain empirical system is investigated. The system consists of particles evolving independently according to \alpha-stable motion in R^d, \alpha<2\alpha. The particles split according to the binary critical branching law with intensity V>0. We study how the limit behaviour of the fluctuations of the occupation time depends on the \emph{initial particle configuration}. We obtain a functional central limit theorem for a vast class of infinitely divisible distributions. Our findings extend and put in a unified setting results which previously seemed to be disconnected. The limit processes form a one dimensional family of long-range dependance centred Gaussian processes. http://arxiv.org/abs/1001.5142 9794. A representation formula for the Freidlin-Wentzell functional on the one dimensional torus Author(s): A. Faggionato and D. Gabrielli Abstract: Inspired by some recent results on fluctuation theory for piecewise deterministic Markov processes, we consider a generic diffusion on the 1D torus and give a simple representation formula for the large deviation rate functional of its invariant probability measure, in the limit of vanishing noise. Previously, this rate functional had been characterized by M.I. Freidlin and A.D. Wentzell as solution of a rather complex optimization problem. We discuss this last problem in full generality and show that it leads to our formula. Finally, we discuss some geometric and regularity properties of the rate functional. In particular, we prove a universality result showing that the rate functional is a viscosity solution of the stationary Hamilton--Jacobi equation associated to any Hamiltonian H satisfying weak suitable conditions. http://arxiv.org/abs/1001.5160 9795. A Laplace principle for a stochastic wave equation in spatial dimension three Author(s): V\'ictor Ortiz-L\'opez and Marta Sanz-Sol\'e Abstract: We consider a stochastic wave equation in spatial dimension three, driven by a Gaussian noise, white in time and with a stationary spatial covariance. The free terms are nonlinear with Lipschitz continuous coefficients. Under suitable conditions on the covariance measure, Dalang and Sanz-Sol\'e [Memoirs of the AMS, Vol 199, 2009] have proved the existence of a random field solution with H\"older continuous sample paths, jointly in both arguments, time and space. By perturbing the driving noise with a multiplicative parameter $\varepsilon\in]0,1]$, a family of probability laws corresponding to the respective solutions to the equation is obtained. Using the weak convergence approach to large deviations developed in [P. Dupuis, R. S. Ellis, 1997], we prove that this family satisfies a Laplace principle in the H\"older norm. http://arxiv.org/abs/1001.5228 9796. Large deviations for slow-fast stochastic partial differential equations Author(s): Wei Wang and A. J. Roberts and Jinqiao Duan Abstract: A large deviation principle is derived for stochastic partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations. http://arxiv.org/abs/1001.4826 9797. Stochastic Approximation, Cooperative Dynamics and Supermodular Games Author(s): Michel Bena\"im (UNINE) and Mathieu Faure (UNINE) Abstract: This paper considers a stochastic approximation algorithm, with decreasing step size and martingale difference noise. Under very mild assumptions, we prove the non convergence of this process toward a certain class of repulsive sets for the associated ordinary differential equation (ODE). We then use this result to derive the convergence of the process when the ODE is cooperative in the sense of [Hirsch, 1985]. In particular, this allows us to extend significantly the main result of [Hofbauer and Sandholm, 2002] on the convergence of stochastic fictitious play in supermodular games. http://arxiv.org/abs/1001.4871 9798. The asymptotic behavior of densities related to the supremum of a stable process Author(s): R. A. Doney and M. S. Savov Abstract: If $X$ is a stable process of index $\alpha\in(0,2)$ whose L\'{e}vy measure has density $cx^{-\alpha-1}$ on $(0,\infty)$, and $S_1=\sup_{0x)\backsim A\alpha ^{-1}x^{-\alpha}$ as $x\to\infty$ and $P(S_1\leq x)\backsim B\alpha^{-1}\rho^{-1}x^{\alpha\rho}$ as $x\downarrow0$. [Here $\rho =P(X_1>0)$ and $A$ and $B$ are known constants.] It is also known that $S_1$ has a continuous density, $m$ say. The main point of this note is to show that $m(x)\backsim Ax^{-(\alpha+1)}$ as $x\to\infty$ and $m(x)\backsim Bx^{\alpha\rho-1}$ as $x\downarrow0$. Similar results are obtained for related densities. http://arxiv.org/abs/1001.4872 9799. Properties of hitting times for $G$-martingale Author(s): Yongsheng Song Abstract: In this article, we consider the properties of hitting times for $G$-martingale and the stopped processes. We prove that the stopped processes for $G$-martingales are still $G$-martingales and that the hitting times for a class of $G$-martingales including $G$-Brownian motion are quasi-continuous. http://arxiv.org/abs/1001.4907 9800. Limit theorems for the number of occupied boxes in the Bernoulli sieve Author(s): Alexander Gnedin and Alexander Iksanov and and Alexander Marynych Abstract: The Bernoulli sieve is a version of the classical `balls-in-boxes' occupancy scheme, in which random frequencies of infinitely many boxes are produced by a multiplicative renewal process, also known as the residual allocation model or stick-breaking. We focus on the number $K_n$ of boxes occupied by at least one of $n$ balls, as $n\to\infty$. A variety of limiting distributions for $K_n$ is derived from the properties of associated perturbed random walks. Refining the approach based on the standard renewal theory we remove a moment constraint to cover the cases left open in previous studies. http://arxiv.org/abs/1001.4920 9801. Limit law for some modified ergodic sums Author(s): Jean-Pierre Conze (IRMAR) and St\'ephane Le Borgne (IRMAR) Abstract: An example due to Erdos and Fortet shows that, for a lacunary sequence of integers (q_n) and a trigonometric polynomial f, the asymptotic distribution of normalized sums of f(q_k x) can be a mixture of gaussian laws. Here we give a generalization of their example interpreted as the limiting behavior of some modified ergodic sums in the framework of dynamical systems. http://arxiv.org/abs/1001.4862 9802. The McShane integral in weakly compactly generated spaces Author(s): Antonio Avil\'es and Grzegorz Plebanek and Jos\'e Rodr\'iguez Abstract: Di Piazza and Preiss asked whether every Pettis integrable function defined on [0,1] and taking values in a weakly compactly generated Banach space is McShane integrable. In this paper we answer this question in the negative. http://arxiv.org/abs/1001.4896 9803. Percolation on self-dual polygon configurations Author(s): Bela Bollobas and Oliver Riordan Abstract: Recently, Scullard and Ziff noticed that a broad class of planar percolation models are self-dual under a simple condition that, in a parametrized version of such a model, reduces to a single equation. They state that the solution of the resulting equation gives the critical point. However, just as in the classical case of bond percolation on the square lattice, self-duality is simply the starting point: the mathematical difficulty is precisely showing that self-duality implies criticality. Here we do so for a generalization of the models considered by Scullard and Ziff. In these models, the states of the bonds need not be independent; furthermore, increasing events need not be positively correlated, so new techniques are needed in the analysis. The main new ingredients are a generalization of Harris's Lemma to products of partially ordered sets, and a new proof of a type of Russo-Seymour-Welsh Lemma with minimal symmetry assumptions. http://arxiv.org/abs/1001.4674 9804. On Beta-Product Convolutions Author(s): Enkelejd Hashorva Abstract: Let R be a positive random variable independent of S which is beta distributed. In this paper we are interested on the relation between the distribution function of $R$ and that of RS. For this model we derive first some distributional properties, and then investigate the lower tail asymptotics of RS when R is regularly varying at 0, and vice-versa. The applications we present in this paper concern a) the simplicity of Dirichlet distributions, b) asymptotics of the sample minima of elliptical distributions, and c) the effect of the scaling on the asymptotics of aggregated risks. http://arxiv.org/abs/1001.4684 9805. Invariance principle for the random conductance model with unbounded conductances Author(s): M. T. Barlow and J.-D. Deuschel Abstract: We study a continuous time random walk $X$ in an environment of i.i.d. random conductances $\mu_e\in[1,\infty)$. We obtain heat kernel bounds and prove a quenched invariance principle for $X$. This holds even when ${\mathbb{E}}\mu_e=\infty$. http://arxiv.org/abs/1001.4702 9806. A shape theorem and semi-infinite geodesics for the Hammersley model with random weights Author(s): E.A. Cator and L.P.R. Pimentel Abstract: In this paper we will prove a shape theorem for the last passage percolation model on a two dimensional $F$-compound Poisson process, called the Hammersley model with random weights. We will also provide diffusive upper bounds for shape fluctuations. Finally we will indicate how these results can be used to prove existence and coalescence of semi-infinite geodesics in some fixed direction $\alpha$, following an approach developed by Newman and co-authors, and applied to the classical Hammersley process by W\"uthrich. These results will be crucial in the development of an upcoming paper on the relation between Busemann functions and equilibrium measures in last passage percolation models. http://arxiv.org/abs/1001.4706 9807. On the existence and position of the farthest peaks of a family of stochastic heat and wave equations Author(s): Daniel Conus and Davar Khoshnevisan Abstract: We study a family of non-linear stochastic heat equations in (1+1) dimensions, driven by the generator of a L\'evy process and space-time white noise. We assume that the underlying L\'evy process has finite exponential moments in a neighborhood of the origin and that the initial condition has exponential decay at infinity. Then we prove that under natural conditions on the non-linearity: (i) The absolute moments of the solution to our stochastic heat equation grow exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the non-linear stochastic heat equation. Finally, we show that these results extend to the stochastic wave equation driven by Laplacian. http://arxiv.org/abs/1001.4759 9808. Formulas for ASEP with Two-Sided Bernoulli Initial Condition Author(s): Craig A. Tracy and Harold Widom Abstract: For the asymmetric simple exclusion process on the integer lattice with two-sided Bernoulli initial condition, we derive exact formulas for the following quantities: (1) the probability that site x is occupied at time t; (2) a correlation function, the probability that site 0 is occupied at time 0 and site x is occupied at time t; (3) the distribution function for the total flux across 0 at time t and its exponential generating function. http://arxiv.org/abs/1001.4766 9809. Joint distribution of the process and its sojourn time for pseudo-processes governed by high-order heat equation Author(s): Valentina Cammarota and Aime Lachal Abstract: Consider the high-order heat-type equation $\partial u/\partial t=\pm \partial^N u/\partial x^N$ for an integer $N>2$ and introduce the related Markov pseudo-process $(X(t))_{t\ge 0}$. In this paper, we study the sojourn time $T(t)$ in the interval $[0,+\infty)$ up to a fixed time $t$ for this pseudo-process. We provide explicit expressions for the joint distribution of the couple $(T(t),X(t))$. http://arxiv.org/abs/1001.4201 9810. Optimal tuning of the Hybrid Monte-Carlo Algorithm Author(s): Alexandros Beskos and Natesh S. Pillai and Gareth O. Roberts and Jesus M. Sanz-Serna, Andrew M. Stuart Abstract: We investigate the properties of the Hybrid Monte-Carlo algorithm (HMC) in high dimensions. HMC develops a Markov chain reversible w.r.t. a given target distribution $\Pi$ by using separable Hamiltonian dynamics with potential $-\log\Pi$. The additional momentum variables are chosen at random from the Boltzmann distribution and the continuous-time Hamiltonian dynamics are then discretised using the leapfrog scheme. The induced bias is removed via a Metropolis-Hastings accept/reject rule. In the simplified scenario of independent, identically distributed components, we prove that, to obtain an $\mathcal{O}(1)$ acceptance probability as the dimension $d$ of the state space tends to $\infty$, the leapfrog step-size $h$ should be scaled as $h= l \times d^{-1/4}$. Therefore, in high dimensions, HMC requires $\mathcal{O}(d^{1/4})$ steps to traverse the state space. We also identify analytically the asymptotically optimal acceptance probability, which turns out to be 0.651 (to three decimal places). This is the choice which optimally balances the cost of generating a proposal, which {\em decreases} as $l$ increases, against the cost related to the average number of proposals required to obtain acceptance, which {\em increases} as $l$ increases. http://arxiv.org/abs/1001.4460 9811. Emergence of Randomness and Arrow of Time in Quantum Walks Author(s): Yutaka Shikano and Kota Chisaki and Etsuo Segawa and Norio Konno Abstract: Quantum walks are powerful tools not only to construct the quantum speedup algorithms but also to describe specific models in physical processes. Furthermore, the discrete time quantum walk has been experimentally realized in various setups. We apply the concept of the quantum walk to the problems in quantum foundations. We show that randomness and the arrow of time in the quantum walk gradually emerge by periodic projective measurements from the mathematically obtained limit distribution under the time scale transformation. http://arxiv.org/abs/1001.3989 9812. On the Zero-Type property and mixing of Bernoulli shifts Author(s): Zemer Kosloff Abstract: We extend the notion of zero-type to the whole class of non-singular transformations and then prove that every non-singular Bernoulli shift is either zero-type or there is an equivalent invariant probability. http://arxiv.org/abs/1001.4261 9813. On the Vershik-Kerov Conjecture Concerning the Shannon-Macmillan-Breiman Theorem for the Plancherel Family of Measures on the Space of Young Diagrams Author(s): Alexander I. Bufetov Abstract: Vershik and Kerov conjectured in 1985 that suitably normalized dimensions of irreducible representations of finite symmetric groups converge to a constant with respect to the Plancherel family of measures on the space of Young diagrams. The main result of this paper is the proof of the Vershik-Kerov conjecture. The argument is based on the methods of Borodin, Okounkov and Olshanski. http://arxiv.org/abs/1001.4275 9814. Martingale representation for Poisson processes with applications to minimal variance hedging Author(s): Guenter Last and Mathew D. Penrose Abstract: We consider a Poisson process $\eta$ on a measurable space $(\BY,\mathcal{Y})$ equipped with a partial ordering, assumed to be strict almost everwhwere with respect to the intensity measure $\lambda$ of $\eta$. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with $\eta$), which was previously known only in the special case, when $\lambda$ is the product of Lebesgue measure on $\R_+$ and a $\sigma$-finite measure on another space $\BX$. Our proof is new and based on only a few basic properties of Poisson processes and stochastic integrals. We also consider the more general case of an independent random measure in the sense of It\^o of pure jump type and show that the Clark-Ocone type representation leads to an explicit version of the Kunita-Watanabe decomposition of square integrable martingales. We also find the explicit minimal variance hedge in a quite general financial market driven by an independent random measure. http://arxiv.org/abs/1001.3972 9815. A d dimensional nucleation and growth model Author(s): Raphael Cerf and Francesco Manzo Abstract: We analyze the relaxation time of a ferromagnetic d dimensional growth model on the lattice. The model is characterized by d param- eters which represent the activation energies of a site, depending on the number of occupied nearest neighbours. This model is a natural generalisation of the model studied by Dehghanpour and Schonmann [DS97a], where the activation energy of a site with more than two occupied neighbours is zero. http://arxiv.org/abs/1001.3990 9816. Stochastic evolution equations driven by Liouville fractional Brownian motion Author(s): Zdzislaw Brzezniak and Jan van Neerven and Donna Salopek Abstract: Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of L(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motions (fBm) with arbitrary Hurst parameter in the interval (0,1). For Hurst parameters in (0,1/2) we show that a function F:(0,T)\to L(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fBm if and only if it is stochastically integrable with respect to an H-cylindrical fBm with the same Hurst parameter. As an application we show that second-order parabolic SPDEs on bounded domains in \mathbb{R}^d, driven by space-time noise which is white in space and Liouville fractional in time with Hurst parameter in (d/4,1) admit mild solution which are H\"older continuous both and space. http://arxiv.org/abs/1001.4013 9817. Spanning forests and the vector bundle laplacian Author(s): Richard Kenyon Abstract: The classical matrix-tree theorem relates the determinant of the combinatorial laplacian on a graph to the number of spanning trees. We generalize this result to laplacians on one- and two-dimensional vector bundles, giving a combinatorial interpretation of their determinants in terms of so-called cycle rooted spanning forests. We construct natural measures on CRSFs for which the edges form a determinantal process. This theory gives a natural generalization of the spanning tree process adapted to graphs embedded on surfaces. We give a number of other applications, for example we compute the probability that a loop-erased random walk on a planar graph between two vertices on the outer boundary passes left of two given faces. This probability can not be computed using the standard Laplacian alone. http://arxiv.org/abs/1001.4028 9818. Is the minimum value of an option on variance generated by local volatility? Author(s): Mathias Beiglboeck and Peter Friz and Stefan Sturm Abstract: We discuss the possibility of obtaining model-free bounds on volatility derivatives, given present market data in the form of a calibrated local volatility model. A counter-example to a wide-spread conjecture is given. http://arxiv.org/abs/1001.4031 9819. Thermophoresis as persistent random walk Author(s): A.V. Plyukhin Abstract: In a simple model of a continuous random walk a particle moves in one dimension with the velocity fluctuating between V and -V. If V is associated with the thermal velocity of a Brownian particle and allowed to be position dependent, the model accounts readily for the particle's drift along the temperature gradient and recovers basic results of the conventional thermophoresis theory. http://arxiv.org/abs/0903.3584 9820. Stochastic process leading to wave equations in dimensions higher than one Author(s): A.V. Plyukhin Abstract: Stochastic processes are proposed whose master equations coincide with classical wave, telegraph, and Klein-Gordon equations. Similar to predecessors based on the Goldstein-Kac telegraph process, the model describes the motion of particles with constant speed and transitions between discreet allowed velocity directions. A new ingredient is that transitions into a given velocity state depend on spatial derivatives of other states populations, rather than on populations themselves. This feature requires the sacrifice of the single-particle character of the model, but allows to imitate the Huygens' principle and to recover wave equations in arbitrary dimensions. http://arxiv.org/abs/1001.3821 9821. An aging phenomenon for a fragmentation-coagulation process Author(s): Jean Bertoin (PMA and Dma) Abstract: We point out that aging occurs for the following simple model of fragmentation-coagulation inspired by Pitman's coalescent random forests. For every $n\in \N$, we consider a uniform random tree with $n$ vertices, and at each step, depending on the outcome of an independent fair coin tossing, either we remove one edge chosen uniformly at random amongst the remaining edges, or we replace one edge chosen uniformly at random amongst the edges which have been removed previously. The process that records the sizes of the tree-components evolves by fragmentation and coagulation. It exhibits aging in the sense that when it is observed after $k$ steps in the regime $k\sim tn+s\sqrt n$ with $t>0$ fixed, it seems to reach a statistical equilibrium as $n\to\infty$; but different values of $t$ yield distinct pseudo-stationary distributions. The approach owes much to the construction by Aldous and Pitman of the standard additive coalescent via Poissonian cuts on the skeleton of a Continuum Random Tree. http://arxiv.org/abs/1001.3721 9822. Jump-diffusion modeling in emission markets Author(s): K. Borovkov and G. Decrouez and J. Hinz Abstract: Mandatory emission trading schemes are being established around the world. Participants of such market schemes are always exposed to risks. This leads to the creation of an accompanying market for emission-linked derivatives. To evaluate the fair prices of such financial products, one needs appropriate models for the evolution of the underlying assets, emission allowance certificates. In this paper, we discuss continuous time diffusion and jump-diffusion models, the latter enabling one to model information shocks that cause jumps in allowance prices. We show that the resulting martingale dynamics can be described in terms of non-linear partial differential and integro-differential equations and use a finite difference method to investigate numerical properties of their discretizations. The results are illustrated by a small numerical study. http://arxiv.org/abs/1001.3728 9823. Martingale Representation Theorem for the G-expectation Author(s): H.M. Soner; N. Touzi; J. Zhang Abstract: This paper considers the nonlinear theory of G-martingales as introduced by Peng. A martingale representation theorem for this theory is proved by using the techniques and the results established in an accompanying paper for the second order stochastic target problems and the second order backward stochastic differential equations. In particular, this representation provides a hedging strategy in a market with an uncertain volatility. http://arxiv.org/abs/1001.3802 9824. Limit theorem for randomly indexed sequence of random processes Author(s): Elena Permyakova Abstract: In this paper is proved the limit theorem for randomly indexed sequence of random processes in the case where sequences of random index and random processes are independent, also the estimation of convergence rate is obtained. http://arxiv.org/abs/1001.3844 9825. Stochastic firing rate models Author(s): Jonathan Touboul and Bard Ermentrout and Olivier Faugeras and Bruno Cessac Abstract: We review a recent approach to the mean-field limits in neural networks that takes into account the stochastic nature of input current and the uncertainty in synaptic coupling. This approach was proved to be a rigorous limit of the network equations in a general setting, and we express here the results in a more customary and simpler framework. We propose a heuristic argument to derive these equations providing a more intuitive understanding of their origin. These equations are characterized by a strong coupling between the different moments of the solutions. We analyse the equations, present an algorithm to simulate the solutions of these mean-field equations, and investigate numerically the equations. In particular, we build a bridge between these equations and Sompolinsky and collaborators approach (1988, 1990), and show how the coupling between the mean and the covariance function deviates from customary approaches. http://arxiv.org/abs/1001.3872 9826. Functional Regression for General Exponential Families Author(s): Wei Dou and David Pollard and and Harrison H. Zhou Abstract: The paper derives a minimax lower bound for rates of convergence for an infinite-dimensional parameter in an exponential family model. An estimator that achieves the optimal rate is constructed by maximum likelihood on finite-dimensional approximations with parameter dimension that grows with sample size. http://arxiv.org/abs/1001.3742 9827. Secure Communication in Stochastic Wireless Networks Author(s): Pedro C. Pinto and Joao Barros and Moe Z. Win Abstract: Information-theoretic security -- widely accepted as the strictest notion of security -- relies on channel coding techniques that exploit the inherent randomness of the propagation channels to significantly strengthen the security of digital communications systems. Motivated by recent developments in the field, this paper aims at a characterization of the fundamental secrecy limits of wireless networks. Based on a general model in which legitimate nodes and potential eavesdroppers are randomly scattered in space, the intrinsically secure communications graph (iS-graph) is defined from the point of view of information-theoretic security. Conclusive results are provided for the local connectivity of the Poisson iS-graph, in terms of node degrees and isolation probabilities. It is shown how the secure connectivity of the network varies with the wireless propagation effects, the secrecy rate threshold of each link, and the noise powers of legitimate nodes and eavesdroppers. Sectorized transmission and eavesdropper neutralization are explored as viable strategies for improving the secure connectivity. Lastly, the maximum secrecy rate between a node and each of its neighbours is characterized, and the case of colluding eavesdroppers is studied. The results help clarify how the spatial density of eavesdroppers can compromise the intrinsic security of wireless networks. http://arxiv.org/abs/1001.3697 9828. Stochastic Switching Games and Duopolistic Competition in Emissions Markets Author(s): Michael Ludkovski Abstract: We study optimal behavior of energy producers under a CO_2 emission abatement program. We focus on a two-player discrete-time model where each producer is sequentially optimizing her emission and production schedules. The game-theoretic aspect is captured through a reduced-form price-impact model for the CO_2 allowance price. Such duopolistic competition results in a new type of a non-zero-sum stochastic switching game on finite horizon. Existence of game Nash equilibria is established through generalization to randomized switching strategies. No uniqueness is possible and we therefore consider a variety of correlated equilibrium mechanisms. We prove existence of correlated equilibrium points in switching games and give a recursive description of equilibrium game values. A simulation-based algorithm to solve for the game values is constructed and a numerical example is presented. http://arxiv.org/abs/1001.3455 9829. On the inference of large phylogenies with long branches: How long is too long? Author(s): Elchanan Mossel and Sebastien Roch and Allan Sly Abstract: Recent work has highlighted deep connections between sequence-length requirements for high-probability phylogeny reconstruction and the related problem of the estimation of ancestral sequences. In [Daskalakis et al.'09], building on the work of [Mossel'04], a tight sequence-length requirement was obtained for the CFN model. In particular the required sequence length for high-probability reconstruction was shown to undergo a sharp transition (from $O(\log n)$ to $\hbox{poly}(n)$, where $n$ is the number of leaves) at the "critical" branch length $\critmlq$ (if it exists) of the ancestral reconstruction problem. Here we consider the GTR model. For this model, recent results of [Roch'09] show that the tree can be accurately reconstructed with sequences of length $O(\log(n))$ when the branch lengths are below $\critksq$, known as the Kesten-Stigum (KS) bound. Although for the CFN model $\critmlq = \critksq$, it is known that for the more general GTR models one has $\critmlq \geq \critksq$ with a strict inequality in many cases. Here, we show that this phenomenon also holds for phylogenetic reconstruction by exhibiting a family of symmetric models $Q$ and a phylogenetic reconstruction algorithm which recovers the tree from $O(\log n)$-length sequences for some branch lengths in the range $(\critksq,\critmlq)$. Second we prove that phylogenetic reconstruction under GTR models requires a polynomial sequence-length for branch lengths above $\critmlq$. http://arxiv.org/abs/1001.3480 9830. A framework for adaptive Monte-Carlo procedures Author(s): Bernard Lapeyre (CERMICS) and J\'er\^ome Lelong (LJK) Abstract: Adaptive Monte Carlo methods are powerful variance reduction techniques. In this work, we propose a mathematical setting which greatly relaxes the assumptions needed by for the adaptive importance sampling techniques presented by Arouna in 2003. We establish the convergence and asymptotic normality of the adaptive Monte Carlo estimator under local assumptions which are easily verifiable in practice. We present one way of approximating the optimal importance sampling parameter using a randomly truncated stochastic algorithm. Finally, we apply this technique to the valuation of financial derivatives and our numerical experiments show that the computational time needed to achieve a given accuracy is divided by a factor up to 5. http://arxiv.org/abs/1001.3551 9831. Solvability of general backward stochastic Volterra integral equation with non-Lipschitz coefficients Author(s): Tianxiao Wang and Yufeng Shi Abstract: In this paper we study the unique solvability of backward stochastic Volterra integral equations (BSVIEs in short), in terms of both the M-solutions introduced in [17] and the adapted solutions in [6], [12] or [14]. A general existence and uniqueness of M-solutions is proved under non-Lipschitz conditions by virtue of a briefer argument than the one in [17], which also extends the results in [17]. For the adapted solutions, the unique solvability of BSVIEs, under more general stochastic non-Lipschitz conditions, is obtained which generalizes the results in [6], [12] and [14]. http://arxiv.org/abs/1001.3557 9832. BSVIEs with stochastic Lipschitz coefficients and applications in finance Author(s): Tianxiao Wang Abstract: This paper is concerned with existence and uniqueness of M-solutions of backward stochastic Volterra integral equations (BSVIEs for short), which Lipschitz coefficients are allowed to be random, which generalize the results in [15]. Then a class of continuous time dynamic dynamic coherent risk measures is derived, allowing the riskless interest rate to be random, which is different from the case in [15]. http://arxiv.org/abs/1001.3558 9833. The measurability of hitting times Author(s): Richard F. Bass Abstract: Under very general conditions the hitting time of a set by a stochastic process is a stopping time. We give a new simple proof of this fact. The section theorems foroptional and predictable sets are easy corollaries of the proof. http://arxiv.org/abs/1001.3619 9834. Schur dynamics of the Schur processes Author(s): Alexei Borodin Abstract: We construct discrete time Markov chains that preserve the class of Schur processes on partitions and signatures. One application is a simple exact sampling algorithm for q^{volume}-distributed skew plane partitions with an arbitrary back wall. Another application is a construction of Markov chains on infinite Gelfand-Tsetlin schemes that represent deterministic flows on the space of extreme characters of the infinite-dimensional unitary group. http://arxiv.org/abs/1001.3442 9835. Stochastic Switching Games and Duopolistic Competition in Emissions Markets Author(s): Michael Ludkovski Abstract: We study optimal behavior of energy producers under a CO_2 emission abatement program. We focus on a two-player discrete-time model where each producer is sequentially optimizing her emission and production schedules. The game-theoretic aspect is captured through a reduced-form price-impact model for the CO_2 allowance price. Such duopolistic competition results in a new type of a non-zero-sum stochastic switching game on finite horizon. Existence of game Nash equilibria is established through generalization to randomized switching strategies. No uniqueness is possible and we therefore consider a variety of correlated equilibrium mechanisms. We prove existence of correlated equilibrium points in switching games and give a recursive description of equilibrium game values. A simulation-based algorithm to solve for the game values is constructed and a numerical example is presented. http://arxiv.org/abs/1001.3455 9836. Optimal relaxed control of dissipative stochastic partial differential equations in Banach spaces Author(s): Zdzislaw Brzezniak and Rafael Serrano Abstract: We study an optimal relaxed control problem for a class of semilinear stochastic PDEs on Banach spaces perturbed by multiplicative noise and driven by a cylindrical Wiener process. The state equation is controlled through the nonlinear part of the drift coefficient which satisfies a dissipative-type condition with respect to the state variable. The main tools of our study are the factorization method for stochastic convolutions in UMD type-2 Banach spaces and certain compactness properties of the factorization operator and of the class of Young measures on Suslin metrisable control sets. http://arxiv.org/abs/1001.3165 9837. Loop-Erasure of Plane Brownian Motion Author(s): Dapeng Zhan Abstract: We use the coupling technique to prove that there exists a loop-erasure of a plane Brownian motion stopped on exiting a simply connected domain, and the loop-erased curve is the reversal of a radial SLE$_2$ curve. http://arxiv.org/abs/1001.3189 9838. Understanding heavy tails in a bounded world or, is a truncated heavy tail heavy or not? Author(s): Arijit Chakrabarty and Gennady Samorodnitsky Abstract: We address the important question of the extent to which random variables and vectors with truncated power tails retain the characteristic features of random variables and vectors with power tails. We define two truncation regimes, soft truncation regime and hard truncation regime, and show that, in the soft truncation regime, truncated power tails behave, in important respects, as if no truncation took place. On the other hand, in the hard truncation regime much of "heavy tailedness" is lost. We show how to estimate consistently the tail exponent when the tails are truncated, and suggest statistical tests to decide on whether the truncation is soft or hard. Finally, we apply our methods to two recent data sets arising from computer networks. http://arxiv.org/abs/1001.3218 9839. Infinitely many Brownian globules with Brownian radii Author(s): Myriam Fradon and Sylvie Roelly Abstract: We consider an infinite system of non overlapping globules undergoing Brownian motions in R^3. The term globules means that the objects we are dealing with are spherical, but with a radius which is random and time-dependent. The dynamics is modelized by an infinite-dimensional Stochastic Differential Equation with local time. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also find a class of reversible measures. http://arxiv.org/abs/1001.3252 9840. Regularly varying time series in Banach spaces Author(s): Thomas Meinguet and Johan Segers Abstract: When a spatial process is recorded over time and the observation at a given time instant is viewed as a point in a function space, the result is a time series taking values in a Banach space. To study the spatio-temporal extremal dynamics of such a time series, the latter is assumed to be jointly regularly varying. This assumption is shown to be equivalent to convergence in distribution of the rescaled time series conditionally on the event that at a given moment in time it is far away from the origin. The limit is called the tail process or the spectral process depending on the way of rescaling. These processes provide convenient starting points to study, for instance, joint survival functions, tail dependence coefficients, extremograms, extremal indices, and point processes of extremes. The theory applies to linear processes composed of infinite sums of linearly transformed independent random elements whose common distribution is regularly varying. http://arxiv.org/abs/1001.3262 9841. Optimal stopping of expected profit and cost yields in an investment under uncertainty Author(s): Boualem Djehiche (KTH Stockolm) and Said Hamad\`ene (LMM) and Marie Am\'elie Morlais (LMM) Abstract: We consider a finite horizon optimal stopping problem related to trade-off strategies between expected profit and cost cash-flows of an investment under uncertainty. The optimal problem is first formulated in terms of a system of Snell envelopes for the profit and cost yields which act as obstacles to each other. We then construct both a minimal and a maximal solutions using an approximation scheme of the associated system of reflected backward SDEs. When the dependence of the cash-flows on the sources of uncertainty, such as fluctuation market prices, assumed to evolve according to a diffusion process, is made explicit, we also obtain a connection between these solutions and viscosity solutions of a system of variational inequalities (VI) with interconnected obstacles. We also provide two counter-examples showing that uniqueness of solutions of (VI) does not hold in general. http://arxiv.org/abs/1001.3289 9842. A Milstein-type scheme without Levy area terms for SDEs driven by fractional Brownian motion Author(s): Aur\'elien Deya (IECN) and Andreas Neuenkirch and Samy Tindel (IECN) Abstract: In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second order Taylor expansion, where the usual Levy area terms are replaced by products of increments of the driving fBm. The convergence of our scheme is shown by means of a combination of rough paths techniques and error bounds for the discretisation of the Levy area terms. http://arxiv.org/abs/1001.3344 9843. On a forward-backward stochastic system associated to the Burgers equation Author(s): Ana Bela Cruzeiro and Evelina Shamarova Abstract: We describe a probabilistic construction of $H^s$-regular solutions for the spatially periodic Burgers equation by using a characterization of this solution through a forward-backward stochastic system. http://arxiv.org/abs/1001.3367 9844. The approach to criticality in sandpiles Author(s): Anne Fey and Lionel Levine and and David B. Wilson Abstract: A popular theory of self-organized criticality relates the critical behavior of driven dissipative systems to that of systems with conservation. In particular, this theory predicts that the stationary density of the abelian sandpile model should be equal to the threshold density of the corresponding fixed-energy sandpile. This "density conjecture" has been proved for the underlying graph Z. We show (by simulation or by proof) that the density conjecture is false when the underlying graph is any of Z^2, the complete graph K_n, the Cayley tree, the ladder graph, the bracelet graph, or the flower graph. Driven dissipative sandpiles continue to evolve even after a constant fraction of the sand has been lost at the sink. These results cast doubt on the validity of using fixed-energy sandpiles to explore the critical behavior of the abelian sandpile model at stationarity. http://arxiv.org/abs/1001.3401 9845. Erasure entropies and Gibbs measures Author(s): Aernout van Enter and Evgeny Verbitskiy Abstract: Recently Verdu and Weissman introduced erasure entropies, which are meant to measure the information carried by one or more symbols given all of the remaining symbols in the realization of the random process or field. A natural relation to Gibbs measures has also been observed. In his short note we study this relation further, review a few earlier contributions from statistical mechanics, and provide the formula for the erasure entropy of a Gibbs measure in terms of the corresponding potentia. For some 2-dimensonal Ising models, for which Verdu and Weissman suggested a numerical procedure, we show how to obtain an exact formula for the erasure entropy. l http://arxiv.org/abs/1001.3122 9846. Tight Bounds for Algebraic Gossip on Graphs Author(s): Michael Borokhovich and Chen Avin and Zvi Lotker Abstract: We study the stopping times of gossip algorithms for network coding. We analyze algebraic gossip (i.e., random linear coding) and consider three gossip algorithms for information spreading Pull, Push, and Exchange. The stopping time of algebraic gossip is known to be linear for the complete graph, but the question of determining a tight upper bound or lower bounds for general graphs is still open. We take a major step in solving this question, and prove that algebraic gossip on any graph of size n is O(D*n) where D is the maximum degree of the graph. This leads to a tight bound of Theta(n) for bounded degree graphs and an upper bound of O(n^2) for general graphs. We show that the latter bound is tight by providing an example of a graph with a stopping time of Omega(n^2). Our proofs use a novel method that relies on Jackson's queuing theorem to analyze the stopping time of network coding; this technique is likely to become useful for future research. http://arxiv.org/abs/1001.3265 9847. Chernoff's theorem for backward propagators and applications to diffusions on manifolds Author(s): Evelina Shamarova Abstract: The classical Chernoff's theorem is a statement about discrete-time approximations of semigroups, where the approximations are consturcted as products of time-dependent contraction operators strongly differentiable at zero. We generalize the version of Chernoff's theorem for semigroups proved in a paper by Smolyanov et al., and obtain a theorem about descrete-time approximations of backward propagators. http://arxiv.org/abs/1001.3373 9848. R-positivity of matrices and Hamiltonians on nearest neighbors trajectories Author(s): Jorge Littin and Servet Martinez Abstract: We revisit the $R-$positivity of nearest neighbors matrices on ${\ZZ_+}$ and the Gibbs measures on the set of nearest neighbors trajectories on ${\ZZ_+}$ whose Hamiltonians award either visits to sites a or visits to edges. We give conditions that guarantee the $R-$positivity or equivalently the existence of the infinite volume Gibbs measure, and we show geometrical recurrence of the associated Markov chain. In this work we generalize and sharpen results obtained in [3] and [6]. http://arxiv.org/abs/1001.2782 9849. Some properties on $G$-evaluation and its applications to $G$-martingale decomposition Author(s): Yongsheng Song Abstract: In this article, a sublinear expectation induced by $G$-expectation is introduced, which is called $G$-evaluation for convenience. As an application, we prove that any $\xi\in L^\beta_G(\Omega_T)$ with some $\beta>1$ the decomposition theorem holds and any $\beta>1$ integrable symmetric $G$-martingale can be represented as an It$\hat{o}'s$ integral w.r.t $G$-Brownian motion. As a byproduct, we prove a regular property for $G$-martingale: Any $G$-martingale $\{M_t\}$ has a quasi-continuous version http://arxiv.org/abs/1001.2802 9850. The optimal control related to Riemannian manifolds and the viscosity solutions to H-J-B equations Author(s): Xuehong Zhu Abstract: This paper is concerned with the Dynamic Programming Principle (DPP in short) with SDEs on Riemannian manifolds. Moreover, through the DPP, we conclude that the cost function is the unique viscosity solution to the related PDEs on manifolds. http://arxiv.org/abs/1001.2820 9851. Quantum stochastic differential equations and continuous measurements: unbounded coefficients Author(s): Ricardo Castro Santis and Alberto Barchielli Abstract: A natural formulation of the theory of quantum measurements in continuous time is based on quantum stochastic differential equations (Hudson-Parthasarathy equations). However, such a theory was developed only in the case of Hudson-Parthasarathy equations with bounded coefficients. By using some results on Hudson-Parthasarathy equations with unbounded coefficients, we are able to extend the theory of quantum continuous measurements to cases in which unbounded operators on the system space are involved. A significant example of a quantum optical system (the degenerate parametric oscillator) is shown to fulfill the hypotheses introduced in the general theory. http://arxiv.org/abs/1001.2826 9852. Exact lower bounds on the exponential moments of Winsorized and truncated random variables Author(s): Iosif Pinelis Abstract: Exact lower bounds on the exponential moments of min(y,X) and XI{X http://arxiv.org/abs/1001.2901 9853. Quenched effective population size Author(s): Serik Sagitov and Peter Jagers and Vladimir Vatutin Abstract: We study the genealogy of a geographically - or otherwise - structured version of the Wright-Fisher population model with fast migration. The new feature is that migration probabilities may change in a random fashion. Applying Takahashi's results on Markov chains with random transition matrices, we establish convergence to the Kingman coalescent, as the population size goes to infinity. This brings a novel formula for the coalescent effective population size (EPS). We call it a quenched EPS to emphasize the key feature of our model - random environment. The quenched EPS is compared with an annealed (mean-field) EPS which describes the case of constant migration probabilities obtained by averaging the random migration probabilities over possible environments. http://arxiv.org/abs/1001.2907 9854. On refined volatility smile expansion in the Heston model Author(s): P. Friz and S. Gerhold and A. Gulisashvili and S. Sturm Abstract: It is known that Heston's stochastic volatility model exhibits moment explosion, and that the critical moment $s^{*}$ can be obtained by solving (numerically) a simple equation. This yields a leading order expansion for the implied volatility at large strikes: $\sigma_{BS}(k,T)^{2}T\sim \Psi (s^*-1) \times k$ (Roger Lee's moment formula). Motivated by recent "tail-wing" refinements of this moment formula, we first derive a novel tail expansion for the Heston density, sharpening previous work of Dr{\u{a}}gulescu and Yakovenko [Quant. Finance 2, 6 (2002), 443--453], and then show the validity of a refined expansion of the type $% \sigma_{BS}(k,T) ^{2}T=(\beta _{1}k^{1/2}+\beta_{2}+...)^{2}$, where all constants are explicitly known as functions of $s^*$, the Heston model parameters, spot vol and maturity $T$. In the case of the "zero-correlation" Heston model such an expansion was derived by Gulisashvili and Stein [Appl. Math. Opt., DOI: 10.1007/s002450099085]. Our methods and results may prove useful beyond the Heston model: the entire quantitative analysis is based on affine principles; at no point do we need knowledge of the (explicit, but cumbersome) closed form expression of the Fourier transform of $\log S_{T}$ (equivalently: Mellin transform of $S_{T}$% ). Secondly, our analysis reveals a new parameter ("\textit{critical slope}"% ), defined in a model free manner, which drives the second and higher order terms in tail- and implied volatility expansions. http://arxiv.org/abs/1001.3003 9855. Stochastic differential equations with coefficients in Sobolev spaces Author(s): Shizan Fang and Dejun Luo and Anto Thalmaier Abstract: We consider It\^o SDE $\d X_t=\sum_{j=1}^m A_j(X_t) \d w_t^j + A_0(X_t) \d t$ on $\R^d$. The diffusion coefficients $A_1,..., A_m$ are supposed to be in the Sobolev space $W_\text{loc}^{1,p} (\R^d)$ with $p>d$, and to have linear growth; for the drift coefficient $A_0$, we consider two cases: (i) $A_0$ is continuous whose distributional divergence $\delta(A_0)$ w.r.t. the Gaussian measure $\gamma_d$ exists, (ii) $A_0$ has the Sobolev regularity $W_\text{loc}^{1,p'}$ for some $p'>1$. Assume $\int_{\R^d} \exp\big[\lambda_0\bigl(|\delta(A_0)| + \sum_{j=1}^m (|\delta(A_j)|^2 +|\nabla A_j|^2)\bigr)\big] \d\gamma_d<+\infty$ for some $\lambda_0>0$, in the case (i), if the pathwise uniqueness of solutions holds, then the push-forward $(X_t)_# \gamma_d$ admits a density with respect to $\gamma_d$. In particular, if the coefficients are bounded Lipschitz continuous, then $X_t$ leaves the Lebesgue measure $\Leb_d$ quasi-invariant. In the case (ii), we develop a method used by G. Crippa and C. De Lellis for ODE and implemented by X. Zhang for SDE, to establish the existence and uniqueness of stochastic flow of maps. http://arxiv.org/abs/1001.3007 9856. Discrete Time and Finite State Reflected Backward Stochastic Difference Equations Author(s): Lifen An and Shaolin Ji Abstract: In this paper, we firstly establish the discrete time and finite state reflected backward stochastic difference equations(DF-RBSDE for short); then we explore the corresponding basic properties and theorems including the Existence and Uniqueness Theorem as well as the Comparison Theorem in our framework by "one step" method; afterwards, we show the connections between DF-RBSDE and optimal stopping time problems. For applications, we study the connection between DF-RBSDE and the general theory of g-martingales and multiple prior martingale including Doob-Mayer Decomposition Theorem and Optional Sampling Theorem in our framework; and then we apply the theory of DF-RBSDEs to multiple prior martingale and optimal stopping problems under Knightian uncertainty; finally, applying the above theories, we consider the pricing models of American Option in complete and incomplete markets. http://arxiv.org/abs/1001.3054 9857. Incremental moments and H\"older exponents of multifractional multistable processes Author(s): Ronan Le Gu\'evel (LMJL) and Jacques L\'evy-V\'ehel (INRIA Saclay - Ile de France) Abstract: Multistable processes, that is, processes which are, at each ``time'', tangent to a stable process, but where the index of stability varies along the path, have been recently introduced as models for phenomena where the intensity of jumps is non constant. In this work, we give further results on (multifractional) multistable processes related to their local structure. We show that, under certain conditions, the incremental moments display a scaling behaviour, and that the pointwise exponent is, as expected, equal to the localisability index. http://arxiv.org/abs/1001.3130 9858. Asymptotic equivalence and sufficiency for volatility estimation under microstructure noise Author(s): Markus Rei\ss Abstract: The basic model for high-frequency data in finance is considered, where an efficient price process is observed under microstructure noise. It is shown that this nonparametric model is in Le Cam's sense asymptotically equivalent to a Gaussian shift experiment in terms of the square root of the volatility function $\sigma$. As an application, simple rate-optimal estimators of the volatility and efficient estimators of the integrated volatility are constructed. http://arxiv.org/abs/1001.3006 9859. Square-mean almost automorphic solutions for some stochastic differential equations Author(s): Miaomiao Fu and Zhenxin Liu Abstract: The concept of square-mean almost automorphy for stochastic processes is introduced. The existence and uniqueness of square-mean almost automorphic solutions to some linear and non-linear stochastic differential equations are established provided the coefficients satisfy some conditions. The asymptotic stability of the unique square-mean almost automorphic solution in square-mean sense is discussed. http://arxiv.org/abs/1001.3049 9860. Stochastic perturbation of sweeping process and a convergence result for an associated numerical scheme Author(s): Frederic Bernicot (LPP) and Juliette Venel (LAMAV) Abstract: Here we present well-posedness results for first order stochastic differential inclusions, more precisely for sweeping process with a stochastic perturbation. These results are provided in combining both deterministic sweeping process theory and methods concerning the reflection of a Brownian motion. In addition, we prove convergence results for a Euler scheme, discretizing theses stochastic differential inclusions. http://arxiv.org/abs/1001.3128 9861. A Causal Construction of Diffusion Processes Author(s): Tadeusz Banek Abstract: A simple nonlinear integral equation for Ito's map is obtained. Although, it does not include stochastic integrals, it does give causal construction of diffusion processes which can be easily implemented by iteration systems. Applications in financial modelling and extension to fBm are discussed. http://arxiv.org/abs/1001.2715 9862. A Break of the Complexity of the Numerical Approximation of Nonlinear SPDEs with Multiplicative Noise Author(s): Arnulf Jentzen and Michael Roeckner Abstract: A new numerical method for stochastic partial differential equations (SPDEs) of evolutionary type, which is in some sense the infinite dimensional analog of Milstein's scheme for finite dimensional stochastic ordinary differential equations (SODEs), is introduced and analyzed in this article. The Milstein scheme is known to be impressively efficient for scalar one-dimensional SODEs but only for some special multidimensional SODEs due to difficult simulations of iterated stochastic integrals in the general multidimensional SODE case. It is a key observation of this article that, in contrast to what one may expect, its infinite dimensional counterpart introduced here is very easy to simulate and this, therefore, leads to a break of the complexity (number of computational operations and random variables needed to compute the scheme) in comparison to previously considered algorithms for simulating nonlinear SPDEs with multiplicative trace class noise. http://arxiv.org/abs/1001.2751 9863. The genealogy of branching Brownian motion with absorption Author(s): Julien Berestycki and Nathanael Berestycki and Jason Schweinsberg Abstract: We consider a system of particles which perform branching Brownian motion with negative drift and are killed upon reaching zero, in the near-critical regime where the total population stays roughly constant with approximately N particles. We show that the characteristic time scale for the evolution of this population is of order (log N)^3, in the sense that when time is measured in these units, the scaled number of particles converges to a variant of Neveu's continuous-state branching process. Furthermore, the genealogy of the particles is then governed by a coalescent process known as the Bolthausen-Sznitman coalescent. This validates the non-rigorous predictions by Brunet, Derrida, Muller, and Munier for a closely related model. http://arxiv.org/abs/1001.2337 9864. Continuity of a queueing integral representation in the ${M}_{\mathbf{1}}$ topology Author(s): Guodong Pang and Ward Whitt Abstract: We establish continuity of the integral representation $y(t)=x(t)+\int_0^th(y(s)) ds$, $t\ge0$, mapping a function $x$ into a function $y$ when the underlying function space $D$ is endowed with the Skorohod $M_1$ topology. We apply this integral representation with the continuous mapping theorem to establish heavy-traffic stochastic-process limits for many-server queueing models when the limit process has jumps unmatched in the converging processes as can occur with bursty arrival processes or service interruptions. The proof of $M_1$-continuity is based on a new characterization of the $M_1$ convergence, in which the time portions of the parametric representations are absolutely continuous with respect to Lebesgue measure, and the derivatives are uniformly bounded and converge in $L_1$. http://arxiv.org/abs/1001.2381 9865. Branching processes in random environment which extinct at a given moment Author(s): C. Boeinghoff and E.E. Dyakonova and G. Kersting and V.A. Vatutin Abstract: Let ${Z_{n},n\geq 0} $ be a critical branching process in random environment and let $T$ be its moment of extinction. Under the annealed approach we prove, as $n\to \infty ,$ a limit theorem for the number of particles in the process at moment $n$ given $T=n+1$ and a functional limit theorem for the properly scaled process ${Z_{nt},\delta \leq t\leq 1-\delta} $ given $T=n+1$ and $\delta \in (0,1/2)$. http://arxiv.org/abs/1001.2413 9866. Scaling limit of the random walk among random traps on Z^d Author(s): Jean-Christophe Mourrat Abstract: Attributing a positive value \tau_x to each x in Z^d, we investigate a nearest-neighbour random walk which is reversible for the measure with weights (\tau_x), often known as "Bouchaud's trap model". We assume that these weights are independent, identically distributed and non-integrable random variables (with polynomial tail), and that d > 4. We obtain the quenched subdiffusive scaling limit of the model, the limit being the fractional kinetics process. We begin our proof by expressing the random walk as a time change of a random walk among random conductances. We then focus on proving that the time change converges, under the annealed measure, to a stable subordinator. This is achieved using previous results concerning the mixing properties of the environment viewed by the time-changed random walk. http://arxiv.org/abs/1001.2459 9867. The unscaled paths of branching Brownian motion Author(s): Simon C. Harris and Matthew I. Roberts Abstract: For a set $A\subset C[0,\infty)$, we give new results on the growth of the number of particles in a dyadic branching Brownian motion whose paths fall within A. We show that it is possible to work without rescaling the paths. We give large deviations probabilities as well as a more sophisticated proof of a result on growth in the number of particles along certain sets of paths. Our results reveal that the number of particles can oscillate dramatically. As a byproduct of our methods we also obtain new results on the number of particles near the frontier of the model. The methods used are entirely probabilistic. http://arxiv.org/abs/1001.2471 9868. Fluid limit theorems for stochastic hybrid systems with application to neuron models Author(s): K. Pakdaman and M. Thieullen and G. Wainrib Abstract: This paper establishes limit theorems for a class of stochastic hybrid systems (continuous deterministic dynamic coupled with jump Markov processes) in the fluid limit (small jumps at high frequency), thus extending known results for jump Markov processes. We prove a functional law of large numbers with exponential convergence speed, derive a diffusion approximation and establish a functional central limit theorem. We apply these results to neuron models with stochastic ion channels, as the number of channels goes to infinity, estimating the convergence to the deterministic model. In terms of neural coding, we apply our central limit theorems to estimate numerically impact of channel noise both on frequency and spike timing coding. http://arxiv.org/abs/1001.2474 9869. Spiders in random environment Author(s): Christophe Gallesco and Sebastian Muller and Serguei Popov and Marina Vachkovskaia Abstract: A spider consists of several, say $N$, particles. Particles can jump independently according to a random walk if the movement does not violate some given restriction rules. If the movement violates a rule it is not carried out. We consider random walk in random environment (RWRE) on $\Z$ as underlying random walk. We suppose the environment $\omega=(\omega_x)_{x \in \Z}$ to be elliptic, with positive drift and nestling, so that there exists a unique positive constant $\kappa$ such that $\E[((1-\omega_0)/\omega_0)^{\kappa}]=1$. The restriction rules are kept very general; we only assume transitivity and irreducibility of the spider. The main result is that the speed of a spider is positive if $\kappa/N>1$ and null if $\kappa/N<1$. In particular, if $\kappa/N <1$ a spider has null speed but the speed of a (single) RWRE is positive. http://arxiv.org/abs/1001.2533 9870. Information Theoretic Bounds for Low-Rank Matrix Completion Author(s): Sriram Vishwanath Abstract: This paper studies the low-rank matrix completion problem from an information theoretic perspective. The completion problem is rephrased as a communication problem of an (uncoded) low-rank matrix source over an erasure channel. The paper then uses achievability and converse arguments to present order-wise optimal bounds for the completion problem. http://arxiv.org/abs/1001.2331 9871. On physical diffusion and stochastic diffusion Author(s): T. N. Narasimhan Abstract: Although the same mathematical expression is used to describe physical diffusion and stochastic diffusion, there are intrinsic similarities and differences in their nature. A comparative study shows that characteristic terms of physical and stochastic diffusion cannot be placed exactly in one-to-one correspondence. Therefore, judgment needs to be exercised in transferring ideas between physical and stochastic diffusion. http://arxiv.org/abs/1001.2357 9872. Asymptotics of the probability minimizing a "down-side" risk Author(s): Hiroaki Hata and Hideo Nagai and Shuenn-Jyi Sheu Abstract: We consider a long-term optimal investment problem where an investor tries to minimize the probability of falling below a target growth rate. From a mathematical viewpoint, this is a large deviation control problem. This problem will be shown to relate to a risk-sensitive stochastic control problem for a sufficiently large time horizon. Indeed, in our theorem we state a duality in the relation between the above two problems. Furthermore, under a multidimensional linear Gaussian model we obtain explicit solutions for the primal problem. http://arxiv.org/abs/1001.2131 9873. Stochastic equations with boundary noise Author(s): Roland Schnaubelt and Mark Veraar Abstract: We study the wellposedness and pathwise regularity of semilinear non-autonomous parabolic evolution equations with boundary and interior noise in an $L^p$ setting. We obtain existence and uniqueness of mild and weak solutions. The boundary noise term is reformulated as a perturbation of a stochastic evolution equation with values in extrapolation spaces. http://arxiv.org/abs/1001.2137 9874. Remarks on restricted Nevanlinna transforms Author(s): Lech Jankowski and Zbigniew J. Jurek Abstract: The Nevanlinna transform K(z), of a measure and a real constant, plays an important role in the complex analysis and more recently in the free probability theory (boolean convolution). It is shown that its restriction k(it) (the restricted Nevanlinna transform) to the imaginary axis can be expressed as the Laplace transform of the Fourier transform (characteristic function) of the corresponding measure. Finally, a relation between the Voiculescu and the boolean convolution is indicated. http://arxiv.org/abs/1001.2154 9875. On many-server queues in heavy traffic Author(s): Anatolii A. Puhalskii and Josh E. Reed Abstract: We establish a heavy-traffic limit theorem on convergence in distribution for the number of customers in a many-server queue when the number of servers tends to infinity. No critical loading condition is assumed. Generally, the limit process does not have trajectories in the Skorohod space. We give conditions for the convergence to hold in the topology of compact convergence. Some new results for an infinite server are also provided. http://arxiv.org/abs/1001.2163 9876. Consistency properties of a simulation-based estimator for dynamic processes Author(s): Manuel S. Santos Abstract: This paper considers a simulation-based estimator for a general class of Markovian processes and explores some strong consistency properties of the estimator. The estimation problem is defined over a continuum of invariant distributions indexed by a vector of parameters. A key step in the method of proof is to show the uniform convergence (a.s.) of a family of sample distributions over the domain of parameters. This uniform convergence holds under mild continuity and monotonicity conditions on the dynamic process. The estimator is applied to an asset pricing model with technology adoption. A challenge for this model is to generate the observed high volatility of stock markets along with the much lower volatility of other real economic aggregates. http://arxiv.org/abs/1001.2173 9877. Convergence of some random functionals of discretized semimartingales Author(s): Assane Diop (PMA) Abstract: In this paper, we study the asymptotic behavior of sums of functions of the increments of a given semimartingale, taken along a regular grid whose mesh goes to 0. The function of the $i$th increment may depend on the current time, and also on the past of the semimartingale before this time. We study the convergence in probability of two types of such sums, and we also give associated central limit theorems. This extends known results when the summands are a function depending only on the increments, and this is motivated mainly by statistical applications. http://arxiv.org/abs/1001.2182 9878. Asymptotics of q-Plancherel measures Author(s): Valentin Feray (LaBRI) and Pierre-Lo\"ic M\'eliot (IGM-LabInfo) Abstract: In this paper, we are interested in the asymptotic size of rows and columns of a random Young diagram under a natural deformation of the Plancherel measure coming from Hecke algebras. The first lines of such diagrams are typically of order $n$, so it does not fit in the context studied by P. Biane and P. \'Sniady. Using the theory of polynomial functions on Young diagrams of Kerov and Olshanski, we are able to compute explicitly the first- and second-order asymptotics of the length of the first rows. Our method works also for other measures, for instance those coming from Schur-Weyl representations. http://arxiv.org/abs/1001.2180 9879. On a general many-dimensional excited random walk Author(s): Mikhail Menshikov and Serguei Popov and Alejandro Ramirez and Marina Vachkovskaia Abstract: The generalized excited random walk is a generalization of the excited random walk, introduced in 2003 by Benjamini and Wilson, which is a discrete-time stochastic process $(X_n,n=0,1,2,...)$ taking values on $\Z^d$, $d\geq 2$, described as follows: when the particle visits a site for the first time, it has a uniformly positive drift in a given direction $\ell$; when the particle is at a site which was already visited before, it has zero drift. Assuming uniform ellipticity and that the jumps of the process are uniformly bounded, we prove that the process is ballistic in the direction $\ell$ so that $\liminf_{n\to\infty}\frac{X_n\cdot \ell}{n}>0$. A key ingredient in the proof of this result is an estimate on the probability that the process visits less than $n^{{1/2}+\alpha}$ distinct sites by time $n$, where $\alpha$ is some positive number depending on the parameters of the model. This approach completely avoids the use of tan points and coupling methods specific to the excited random walk. Furthermore, we apply this technique to prove that the excited random walk in an i.i.d. random environment satisfies a ballistic law of large numbers and a central limit theorem. http://arxiv.org/abs/1001.1741 9880. Functional Inequalities via Lyapunov conditions Author(s): Patrick Cattiaux (IMT) and Arnaud Guillin Abstract: We review here some recent results by the authors, and various coauthors, on (weak,super) Poincar\'e inequalities, transportation-information inequalities or logarithmic Sobolev inequality via a quite simple and efficient technique: Lyapunov conditions. http://arxiv.org/abs/1001.1822 9881. Weighted Dickey-Fuller Processes for Detecting Stationarity Author(s): Ansgar Steland Abstract: Aiming at monitoring a time series to detect stationarity as soon as possible, we introduce monitoring procedures based on kernel-weighted sequential Dickey-Fuller (DF) processes, and related stopping times, which may be called weighted Dickey-Fuller control charts. Under rather weak assumptions, (functional) central limit theorems are established under the unit root null hypothesis and local-to-unity alternatives. For gen- eral dependent and heterogeneous innovation sequences the limit processes depend on a nuisance parameter. In this case of practical interest, one can use estimated control limits obtained from the estimated asymptotic law. Another easy-to-use approach is to transform the DF processes to obtain limit laws which are invariant with respect to the nuisance pa- rameter. We provide asymptotic theory for both approaches and compare their statistical behavior in finite samples by simulation. http://arxiv.org/abs/1001.1833 9882. Sequentially Updated Residuals and Detection of Stationary Errors in Polynomial Regression Models Author(s): Ansgar Steland Abstract: The question whether a time series behaves as a random walk or as a station- ary process is an important and delicate problem, particularly arising in financial statistics, econometrics, and engineering. This paper studies the problem to detect sequentially that the error terms in a polynomial regression model no longer behave as a random walk but as a stationary process. We provide the asymptotic distribution theory for a monitoring procedure given by a control chart, i.e., a stopping time, which is related to a well known unit root test statistic calculated from sequentially updated residuals. We provide a functional central limit theorem for the corresponding stochastic process which implies a central limit theorem for the control chart. The finite sample properties are investigated by a simulation study. http://arxiv.org/abs/1001.1845 9883. Mixing times for random k-cycles and coalescence-fragmentation chains Author(s): Nathanael Berestycki and Oded Schramm and Ofer Zeitouni Abstract: Let S_n be the permutation group on n elements, and consider a random walk on S_n whose step distribution is uniform on k-cycles. We prove a well-known conjecture that the mixing time of this process is (1/k) n \log n, with threshold of width linear in n. Our proofs are elementary and purely probabilistic, and do not appeal to the representation theory of S_n. http://arxiv.org/abs/1001.1894 9884. Another observation about operator compressions Author(s): Elizabeth S. Meckes and Mark W. Meckes Abstract: Let $T$ be a self-adjoint operator on a finite dimensional Hilbert space. It is shown that the distribution of the eigenvalues of a compression of $T$ to a subspace of a given dimension is almost the same for almost all subspaces. This is a coordinate-free analogue of a recent result of Chatterjee and Ledoux on principal submatrices. The proof is based on measure concentration and entropy techniques, and the result improves on some aspects of the result of Chatterjee and Ledoux. http://arxiv.org/abs/1001.1954 9885. Centering problems for probability measures on finite dimensional vector spaces Author(s): Andrzej {\L}uczak Abstract: The paper deals with various centering problems for probability measures on finite dimensional vector spaces. We show that for every such measure there exists a vector $h$ satisfying $\mu*\delta(h)=S(\mu*\delta (h))$ for each symmetry $S$ of $\mu$, generalizing thus Jurek's result obtained for full measures. An explicit form of the $h$ is given for infinitely divisible $\mu$. The main result of the paper consists in the analysis of quasi-decomposable (operator-semistable and operator-stable) measures and finding conditions for the existence of a `universal centering' of such a measure to a strictly quasi-decomposable one. http://arxiv.org/abs/1001.1963 9886. Improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation Author(s): Kathrin Bringmann and Karl Mahlburg Abstract: We generalize and improve results of Andrews, Gravner, Holroyd, Liggett, and Romik on metastability thresholds for generalized two-dimensional bootstrap percolation models, and answer several of their open problems and conjectures. Specifically, we prove slow convergence and localization bounds for Holroyd, Liggett, and Romik's k-percolation models, and in the process provide a unified and improved treatment of existing results for bootstrap, modified bootstrap, and Frobose percolation. Furthermore, we prove improved asymptotic bounds for the generating functions of partitions without k-gaps, which are also related to certain infinite probability processes relevant to these percolation models. One of our key technical probability results is also of independent interest. We prove new upper and lower bounds for the probability that a sequence of independent events with monotonically increasing probabilities contains no ``k-gap'' patterns, which interpolates the general Markov chain solution that arises in the case that all of the probabilities are equal. http://arxiv.org/abs/1001.1977 9887. Random walks - a sequential approach Author(s): Ansgar Steland Abstract: In this paper sequential monitoring schemes to detect nonparametric drifts are studied for the random walk case. The procedure is based on a kernel smoother. As a by-product we obtain the asymptotics of the Nadaraya-Watson estimator and its as- sociated sequential partial sum process under non-standard sampling. The asymptotic behavior differs substantially from the stationary situation, if there is a unit root (random walk component). To obtain meaningful asymptotic results we consider local nonpara- metric alternatives for the drift component. It turns out that the rate of convergence at which the drift vanishes determines whether the asymptotic properties of the monitoring procedure are determined by a deterministic or random function. Further, we provide a theoretical result about the optimal kernel for a given alternative. http://arxiv.org/abs/1001.1828 9888. Fractional order Taylor's series and the neo-classical inequality Author(s): Keisuke Hara and Masanori Hino Abstract: We prove the neo-classical inequality with the optimal constant, which was conjectured by T. J. Lyons [Rev. Mat. Iberoamericana 14 (1998) 215-310]. For the proof, we introduce the fractional order Taylor's series with residual terms. Their application to a particular function provides an identity that deduces the optimal neo-classical inequality. http://arxiv.org/abs/1001.1775 9889. On Ergodicity, Infinite Flow and Consensus in Random Models Author(s): Behrouz Touri and Angelia Nedi'c Abstract: We consider the ergodicity and consensus problem for a discrete-time linear dynamic model driven by random matrices, which is equivalent to studying these concepts for the product of random matrices. Our focus is on the model where the matrices are "stochastic". We introduce a new phenomena, the infinite flow, and we study its fundamental properties and relations with the ergodicity and consensus. We establish several new and important results. The central result of this work is the infinite flow theorem establishing the role of infinite flow in the ergodicity of a general independent random model. Through the use of infinite flow, we show that the ergodicity of the model is equivalent to the ergodicity of the expected model when the matrices in the model have a common steady state in expectation and a feedback property. This result demonstrates that for such models, the expected infinite flow is both necessary and sufficient for the ergodicity. The result is providing us with a powerful deterministic characterization of the ergodicity, which renders a new elegant tool that can be used for studying the consensus and average consensus over random graphs, as well as random consensus algorithms. http://arxiv.org/abs/1001.1890 9890. Quantum stochastic integrals as operators Author(s): Andrzej {\L}uczak Abstract: We construct quantum stochastic integrals for the integrator being a martingale in a von Neumann algebra, and the integrand -- a suitable process with values in the same algebra, as densely defined operators affiliated with the algebra. In the case of a finite algebra we allow the integrator to be an $L^2$--martingale in which case the integrals are $L^2$--martingales too. http://arxiv.org/abs/1001.1959 9891. Multivariate concentration of measure type results using exchangeable pairs and size biasing Author(s): Subhankar Ghosh Abstract: Let $(\mathbf{W,W'})$ be an exchangeable pair of vectors in $\mathbb{R}^k$. Suppose this pair satisfies \beas E(\mathbf{W}'|\mathbf{W})=(I_k-\Lambda)\mathbf{W}+\mathbf{R(W)}. \enas If $||\mathbf{W-W'}||_2\le K$ and $\mathbf{R(W)}=0$, then concentration of measure results of following form is proved for all $\mathbf{w}\succeq 0$ when the moment generating function of $\mathbf{W}$ is finite. \beas P(\mathbf{W}\succeq\mathbf{w}),P(\mathbf{W}\preceq -\mathbf{w})\le \exp(-\frac{||\mathbf{w}||_2^2}{2K^2\nu_1}), \enas for an explicit constant $\nu_1$, where $\succeq$ stands for coordinate wise $\ge$ ordering. This result is applied to examples like complete non degenerate U-statistics. Also, we deal with the example of doubly indexed permutation statistics where $\mathbf{R(W)}\neq 0$ and obtain similar concentration of measure inequalities. Practical examples from doubly indexed permutation statistics include Mann-Whitney-Wilcoxon statistic and random intersection of two graphs. Both these two examples are used in nonparametric statistical testing. We conclude the paper with a multivariate generalization of a recent concentration result due to Ghosh and Goldstein \cite{cnm} involving bounded size bias couplings. http://arxiv.org/abs/1001.1396 9892. Phase separation in random cluster models I: uniform upper bounds on local deviation Author(s): Alan Hammond Abstract: This is the first in a series of three papers that addresses the behaviour of the droplet that results, in the percolating phase, from conditioning the Fortuin-Kasteleyn random cluster model on the presence of an open dual circuit Gamma_0 encircling the origin and enclosing an area of at least (or exactly) n^2. (By the Fortuin-Kasteleyn representation, the model is a close relative of the droplet formed by conditioning the Potts model on an excess of spins of a given type.) We consider local deviation of the droplet boundary, measured in a radial sense by the maximum local roughness, MLR(Gamma_0), this being the maximum distance from a point in the circuit Gamma_0 to the boundary of the circuit's convex hull; and in a longitudinal sense by what we term maximum facet length, MFL(Gamma_0), namely, the length of the longest line segment of which the boundary of the convex hull is formed. The principal conclusion of the series of papers is the following uniform control on local deviation: that there are positive constants c and C such that the conditional probability that the normalised quantity n^{-1/3}\big(\log n \big)^{-2/3} MLR(Gamma_0) lies in the interval [c,C] tends to 1 in the high n-limit; and that the same statement holds for n^{-2/3}\big(\log n \big)^{-1/3} MFL(Gamma_0). In this way, we confirm the anticipated n^{1/3} scaling of maximum local roughness, and provide a sharp logarithmic power-law correction. This local deviation behaviour occurs by means of locally Gaussian effects constrained globally by curvature, and we believe that it arises in a range of radially defined stochastic interface models, including several in the Kardar-Parisi-Zhang universality class. This paper is devoted to proving the upper bounds in these assertions, and includes a heuristic overview of the surgical technique used in the three papers. http://arxiv.org/abs/1001.1527 9893. Phase separation in random cluster models II: the droplet at equilibrium, and local deviation lower bounds Author(s): Alan Hammond Abstract: We study the droplet that results from conditioning the subcritical Fortuin-Kasteleyn random cluster model on the presence of an open circuit Gamma_0 encircling the origin and enclosing an area of at least (or exactly) n^2. We consider local deviation of the droplet boundary, measured in a radial sense by the maximum local roughness, MLR(Gamma_0), this being the maximum distance from a point in the circuit Gamma_0 to the boundary of the circuit's convex hull; and in a longitudinal sense by what we term maximum facet length, MFL(Gamma_0), namely, the length of the longest line segment of which the boundary of the convex hull is formed. We prove that that there exists a constant c > 0 such that the conditional probability that the normalised quantity n^{-1/3}\big(\log n \big)^{-2/3} MLR(Gamma_0) exceeds c tends to 1 in the high n-limit; and that the same statement holds for n^{-2/3}\big(\log n \big)^{-1/3} MFL(Gamma_0). To obtain these bounds, we exhibit the random cluster measure conditional on the presence of an open circuit trapping high area as the invariant measure of a Markov chain that resamples sections of the circuit boundary. We analyse the chain at equilibrium to prove the local roughness lower bounds. Alongside complementary upper bounds provided in arXiv:1001.1527, the fluctuations MLR(Gamma_0) and MFL(Gamma_0) are determined up to a constant factor. http://arxiv.org/abs/1001.1528 9894. Phase separation in random cluster models III: circuit regularity Author(s): Alan Hammond Abstract: We study the droplet that results from conditioning the subcritical Fortuin-Kasteleyn random cluster model on the presence of an open circuit Gamma_0 encircling the origin and enclosing an area of at least (or exactly) n^2. In this paper, we prove that the resulting circuit is highly regular: we define a notion of a regeneration site in such a way that, for any such element v of Gamma_0, the circuit Gamma_0 cuts through the radial line segment through v only at v. We show that, provided that the conditioned circuit is centred at the origin in a natural sense, the set of regeneration sites reaches into all parts of the circuit, with maximal distance from one such site to the next being at most logarithmic in n with high probability. The result provides a flexible control on the conditioned circuit that permits the use of surgical techniques to bound its fluctuations, and, as such, it plays a crucial role in the derivation of bounds on the local fluctuation of the circuit carried out in arXiv:1001.1527 and arXiv:1001.1528. http://arxiv.org/abs/1001.1529 9895. Stability of parallel queueing systems with coupled service rates Author(s): Sem Borst and Matthieu Jonckheere and Lasse Leskel\"a Abstract: This paper considers a parallel system of queues fed by independent arrival streams, where the service rate of each queue depends on the number of customers in all of the queues. Necessary and sufficient conditions for the stability of the system are derived, based on stochastic monotonicity and marginal drift properties of multiclass birth and death processes. These conditions yield a sharp characterization of stability for systems, where the service rate of each queue is decreasing in the number of customers in other queues, and has uniform limits as the queue lengths tend to infinity. The results are illustrated with applications where the stability region may be nonconvex. http://arxiv.org/abs/1001.1560 9896. Critical Ising on the square lattice mixes in polynomial time Author(s): Eyal Lubetzky and Allan Sly Abstract: The Ising model is widely regarded as the most studied model of spin-systems in statistical physics. The focus of this paper is its dynamic (stochastic) version, the Glauber dynamics, introduced in 1963 and by now the most popular means of sampling the Ising measure. Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the dynamics on $\Z^2$ everywhere except at criticality. While the critical behavior of the Ising model has long been the focus for physicists, mathematicians have only recently developed an understanding of its critical geometry with the advent of SLE, CLE and new tools to study conformally invariant systems. A rich interplay exists between the static and dynamic models. At the static phase-transition for Ising, the dynamics is conjectured to undergo a critical slowdown: At high temperature the inverse-gap is O(1), at the critical $\beta_c$ it is polynomial in the side-length and at low temperature it is exponential in it. A seminal series of papers verified this on $\Z^2$ except at $\beta=\beta_c$ where the behavior remained a challenging open problem. Here we establish the first rigorous polynomial upper bound for the critical mixing, thus confirming the critical slowdown for the Ising model in $\Z^2$. Namely, we show that on a finite box with arbitrary (e.g. fixed, free, periodic) boundary conditions, the inverse-gap at $\beta=\beta_c$ is polynomial in the side-length. The proof harnesses recent understanding of the scaling limit of critical Fortuin-Kasteleyn representation of the Ising model together with classical tools from the analysis of Markov chains. http://arxiv.org/abs/1001.1613 9897. Testability of minimum balanced multiway cut densities Author(s): Marianna Bolla and Tamas Koi and Andras Kramli Abstract: Testable weighted graph parameters and equivalent notions of testability are investigated based on papers of Laszlo Lovasz and coauthors. We prove that certain balanced minimum multiway cut densities are testable. Using this fact, quadratic programming techniques are applied to approximate some of these quantities. The problem is related to cluster analysis and statistical physics. Convergence of special noisy graph sequences is also discussed. http://arxiv.org/abs/1001.1623 9898. Limit theorems for weakly subcritical branching processes in random environment Author(s): V.I. Afanasyev and C. Boeinghoff and G. Kersting and V.A. Vatutin Abstract: For a branching process in random environment it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. Interestingly there is the possibility that the process may at the same time be subcritical and, conditioned on nonextinction, 'supercritical'. This so-called weakly subcritical case is considered in this paper. We study the asymptotic survival probability and the size of the population conditioned on non-extinction. Also a functional limit theorem is proven, which makes the conditional supercriticality manifest. A main tool is a new type of functional limit theorems for conditional random walks. http://arxiv.org/abs/1001.1672 9899. Embeddable Markov Matrices Author(s): E B Davies Abstract: We give an account of some results, both old and new, about any $n\times n$ Markov matrix that is embeddable in a one-parameter Markov semigroup. These include the fact that its eigenvalues must lie in a certain region in the unit ball. We prove that a well-known procedure for approximating a non-embeddable Markov matrix by an embeddable one is optimal in a certain sense. http://arxiv.org/abs/1001.1693 9900. Relative Complexity of random walks in random sceneries Author(s): Jon. Aaronson Abstract: Relative complexity measures the complexity of a probability preserving transformation relative to a factor being a sequence of random variables whose exponential growth rate is the relative entropy of the extension. We prove distributional limit theorems for the relative complexity of certain zero entropy extensions: RWRSs whose associated random walks satisfy the alpha-stable CLT (alpha>1). The results give invariants for relative isomorphism of these. http://arxiv.org/abs/1001.1433 9901. On Wiener-Hopf factors for stable processes Author(s): Piotr Graczyk and Tomasz Jakubowski Abstract: We give a series representation of the logarithm of the bivariate Laplace exponent $\kappa$ of $\alpha$-stable processes for almost all $\alpha \in (0,2]$. http://arxiv.org/abs/1001.1230 9902. Fluctuations of the Longest Common Subsequence for Sequences of Independent Blocks Author(s): Heinrich Matzinger and Felipe Torres Abstract: The problem of the fluctuation of the Longest Common Subsequence (LCS) of two i.i.d. sequences of length $n>0$ has been open for decades. There exist contradicting conjectures on the topic. Chvatal and Sankoff conjectured in 1975 that asymptotically the order should be $n^{2/3}$, while Waterman conjectured in 1994 that asymptotically the order should be $n$. A contiguous substring consisting only of one type of symbol is called a block. In the present work, we determine the order of the fluctuation of the LCS for a special model of sequences consisting of i.i.d. blocks whose lengths are uniformly distributed on the set $\{l-1,l,l+1\}$, with $l$ a given positive integer. We showed that the fluctuation in this model is asymptotically of order $n$, which confirm Waterman's conjecture. For achieving this goal, we developed a new method which allows us to reformulate the problem of the order of the variance as a (relatively) low dimensional optimization problem. http://arxiv.org/abs/1001.1273 9903. A functional limit theorem for partial sums of dependent random variables with infinite variance Author(s): Bojan Basrak and Danijel Krizmani\'c and and Johan Segers Abstract: Under an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-Gaussian stable random variable. A functional version of this is known to be true as well, the limit process being a stable L\'evy process. The main result in the paper is that for a stationary, regularly varying sequence for which clusters of high-threshold excesses can be broken down into asymptotically independent blocks, the properly centered partial sum process still converges to a stable L\'evy process. Due to clustering, the L\'evy triple of the limit process can be different from the one in the independent case. The convergence takes place in the space of c\`adl\`ag functions endowed with Skorohod's $M_1$ topology, the more usual $J_1$ topology being inappropriate as the partial sum processes may exhibit rapid successions of jumps within temporal clusters of large values, collapsing in the limit to a single jump. The result rests on a new limit theorem for point processes which is of independent interest. The theory is applied to moving average processes, squared GARCH(1,1) processes, and stochastic volatility models. http://arxiv.org/abs/1001.1345 9904. On some Non Asymptotic Bounds for the Euler Scheme Author(s): Vincent Lemaire (PMA) and Stephane Menozzi (PMA) Abstract: We obtain non asymptotic bounds for the Monte Carlo algorithm associated to the Euler discretization of some diffusion processes. The key tool is the Gaussian concentration satisfied by the density of the discretization scheme. This Gaussian concentration is derived from a Gaussian upper bound of the density of the scheme and a modification of the so-called ``Herbst argument'' used to prove Logarithmic Sobolev inequalities. We eventually establish a Gaussian lower bound for the density of the scheme that emphasizes the concentration is sharp. http://arxiv.org/abs/1001.1347 9905. A simple reduction from a biased measure on the discrete cube to the uniform measure Author(s): Nathan Keller Abstract: We show that certain statements related to the Fourier-Walsh expansion of functions with respect to a biased measure on the discrete cube can be deduced from the respective results for the uniform measure by a simple reduction. In particular, we present simple generalizations to the biased measure $\mu_p$ of the Bonami-Beckner hypercontractive inequality, and of Talagrand's lower bound on the size of the boundary of subsets of the discrete cube. Our generalizations are tight up to constant factors. http://arxiv.org/abs/1001.1167 9906. Typical Geometry, Second-Order Properties and Central Limit Theory for Iteration Stable Tessellations Author(s): Tomasz Schreiber and Christoph Thaele Abstract: Since the seminal work of Mecke, Nagel and Weiss, the iteration stable (STIT) tessellations have attracted considerable interest in stochastic geometry as a natural and flexible yet analytically tractable model for hierarchical spatial cell-splitting and crack-formation processes. The purpose of this paper is to describe large scale asymptotic geometry of STIT tessellations in $\mathbb{R}^d$ and more generally that of non-stationary iteration infinitely divisible tessellations. We study several aspects of the typical first-order geometry of such tessellations resorting to martingale techniques as providing a direct link between the typical characteristics of STIT tessellations and those of suitable mixtures of Poisson hyperplane tessellations. Further, we also consider second-order properties of STIT and iteration infinitely divisible tessellations, such as the variance of the total surface area of cell boundaries inside a convex observation window. Our techniques, relying on martingale theory and tools from integral geometry, allow us to give explicit and asymptotic formulae. Based on these results, we establish a functional central limit theorem for the length/surface increment processes induced by STIT tessellations. We conclude a central limit theorem for total edge length/facet surface, with normal limit distribution in the planar case and non-normal ones in all higher dimensions. http://arxiv.org/abs/1001.0990 9907. On extrema of stable processes Author(s): Alexey Kuznetsov Abstract: We study Wiener-Hopf factorization and distribution of extrema for general stable processes. By connecting Wiener-Hopf factors with a certain elliptic-like function we are able to obtain many explicit and general results, such as expressions for Wiener-Hopf factors and Mellin transform of supremum in terms of double gamma functions, quasi-periodicity and functional identities for these functions, finite product representations in some special cases and identities in distribution satisfied by the supremum functional. http://arxiv.org/abs/1001.0991 9908. Emergence of a Giant Component in Random Site Subgraphs of a d-Dimensional Hamming Torus Author(s): David Sivakoff Abstract: The d-dimensional Hamming torus is the graph whose vertices are all of the integer points inside an a_1 n X a_2 n X ... X a_d n box in R^d (for constants a_1, ..., a_d > 0), and whose edges connect all vertices within Hamming distance one. We study the size of the largest connected component of the subgraph generated by independently removing each vertex of the Hamming torus with probability 1-p. We show that if p=\lambda / n, then there exists \lambda_c > 0, which is the positive root of a degree d polynomial whose coefficients depend on a_1, ..., a_d, such that for \lambda < \lambda_c the largest component has O(log n) vertices (a.a.s. as n \to \infty), and for \lambda > \lambda_c the largest component has (1-q) \lambda (\prod_i a_i) n^{d-1} + o(n^{d-1}) vertices and the second largest component has O(log n) vertices (a.a.s.). An implicit formula for q < 1 is also given. Additionally, we show that if p = c log n / n, then when c < (d-1) / (\sum a_i) the site subgraph of the Hamming torus is not connected, and when c > (d-1) / (\sum a_i) the subgraph is connected (a.a.s.). We also show that the subgraph is connected precisely when it contains no isolated vertices. http://arxiv.org/abs/1001.1007 9909. Directed polymers in random environment with heavy tails Author(s): Antonio Auffinger and Oren Louidor Abstract: We study the model of Directed Polymers in Random Environment in 1+1 dimensions, where the distribution at a site has a tail which decays regularly polynomially with power \alpha, where \alpha \in (0,2). After proper scaling of temperature \beta^{-1}, we show strong localization of the polymer to a favorable region in the environment where energy and entropy are best balanced. We prove that this region has a weak limit under linear scaling and identify the limiting distribution as an (\alpha, \beta)-indexed family of measures on Lipschitz curves lying inside the 45-degrees-rotated square with unit diagonal. In particular, this shows order n transversal fluctuations of the polymer. If, and only if, \alpha is small enough, we find that there exists a random critical temperature below which, but not above, the effect of the environment is macroscopic. The results carry over to d+1 dimensions for d>1 with minor modifications. http://arxiv.org/abs/1001.1028 9910. Two-parameter Levy processes along decreasing paths Author(s): Shai Covo (Bar Ilan University) Abstract: Let {X_{t_1,t_2}: t_1,t_2 >= 0} be a two-parameter L\'evy process on R^d. We study basic properties of the one-parameter process {X_{x(t),y(t)}: t \in T} where x and y are, respectively, nondecreasing and nonincreasing nonnegative continuous functions on the interval T. We focus on and characterize the case where the process has stationary increments. http://arxiv.org/abs/1001.1134 9911. Equilibrium solution to the lowest unique positive integer game Author(s): Seung Ki Baek and Sebastian Bernhardsson Abstract: We address the equilibrium concept of a reverse auction game so that no one can enhance the individual payoff by a unilateral change when all the others follow a certain strategy. In this approach the combinatorial possibilities to consider become very much involved even for a small number of players, which has hindered a precise analysis in previous works. We here present a systematic way to reach the solution for a general number of players, and show that this game is an example of conflict between the group and the individual interests. http://arxiv.org/abs/1001.1065 9912. Stochastic integrals for spde's: a comparison Author(s): Robert C. Dalang and Lluis Quer-Sardanyons Abstract: We present the Walsh theory of stochastic integrals with respect to martingale measures, alongside of the Da Prato and Zabczyk theory of stochastic integrals with respect to Hilbert-space-valued Wiener processes and some other approaches to stochastic integration, and we explore the links between these theories. We then show how each theory can be used to study stochastic partial differential equations, with an emphasis on the stochastic heat and wave equations driven by spatially homogeneous Gaussian noise that is white in time. We compare the solutions produced by the different theories. http://arxiv.org/abs/1001.0856 9913. Averaging over fast variables in the fluid limit for Markov chains: application to the supermarket model with memory Author(s): M.J. Luczak and J.R. Norris Abstract: We set out a general procedure which allows the approximation of certain Markov chains by the solutions of differential equations. The chains considered have some components which oscillate rapidly and randomly, while others are close to deterministic. The limiting dynamics are obtained by averaging the drift of the latter with respect to a local equilibrium distribution of the former. Some general estimates are proved under a uniform mixing condition on the fast variable which give explicit error probabilities for the fluid approximation. Mitzenmacher, Prabhakar and Shah \cite{MPS} introduced a variant with memory of the `join the shortest queue' or `supermarket' model, and obtained a limit picture for the case of a stable system in which the number of queues and the total arrival rate are large. In this limit, the empirical distribution of queue sizes satisfies a differential equation, while the memory of the system oscillates rapidly and randomly. We illustrate our general fluid limit estimate in giving a proof of this limit picture. http://arxiv.org/abs/1001.0895 9914. The evolution of the cover time Author(s): Martin T. Barlow and Jian Ding and Asaf Nachmias and Yuval Peres Abstract: The cover time of a graph is a celebrated example of a parameter that is easy to approximate using a randomized algorithm, but for which no constant factor deterministic polynomial time approximation is known. A breakthrough due to Kahn, Kim, Lovasz and Vu yielded a (log log n)^2 polynomial time approximation. We refine this upper bound, and show that the resulting bound is sharp and explicitly computable in random graphs. Cooper and Frieze showed that the cover time of the largest component of the Erdos-Renyi random graph G(n,c/n) in the supercritical regime with c>1 fixed, is asymptotic to f(c) n \log^2 n, where f(c) tends to 1 as c tends to 1. However, our new bound implies that the cover time for the critical Erdos-Renyi random graph G(n,1/n) has order n, and shows how the cover time evolves from the critical window to the supercritical phase. Our general estimate also yields the order of the cover time for a variety of other concrete graphs, including critical percolation clusters on the Hamming hypercube {0,1}^n, on high-girth expanders, and on tori Z_n^d for fixed large d. For the graphs we consider, our results show that the blanket time, introduced by Winkler and Zuckerman, is within a constant factor of the cover time. Finally, we prove that for any connected graph, adding an edge can increase the cover time by at most a factor of 4. http://arxiv.org/abs/1001.0609 9915. Conditional negative association for competing urns Author(s): Jeff Kahn and Michael Neiman Abstract: Competing urns refers to the random experiment where m balls are dropped, randomly and independently, into urns 1,...,n. Formally, we have a random map $\sigma$ from {1,...,m} to {1,...,n} with the $\sigma(i)$'s i.i.d. With $x_j$ the indicator of the event that at least $t_j$ balls land in urn j (for some threshold $t_j$), we prove conditional negative association for the random variables $x_1,...,x_n$. We mostly deal with the more general situation in which the $\sigma(i)$'s need not be identically distributed, proving results which imply conditional negative association in the i.i.d. case. Some of the results--particularly Lemma 8 on graph orientations--are thought to be of independent interest. We also give a counterexample to a negative correlation conjecture of D. Welsh, a strong version of a (still open) conjecture of G. Farr. http://arxiv.org/abs/1001.0610 9916. A Berry Esseen Theorem for the Lightbulb Process Author(s): Larry Goldstein and Haimeng Zhang Abstract: In the so called lightbulb process, on days $r=1,...,n$, out of $n$ lightbulbs, all initially off, exactly $r$ bulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice versa. With $X$ the number of bulbs on at the terminal time $n$, an even integer, and $\mu=n/2, \sigma^2={Var}(X)$, we have $$ \sup_{z \in \mathbb{R}} |P(\frac{X-\mu}{\sigma})-P(Z \le z)| \le \frac{n}{2\sigma^2}\Delta_0 + 1.64 \frac{n}{\sigma^3}+ \frac{2}{\sigma} $$ where $$ \Delta_0 \le \frac{1}{2\sqrt{n}} + \frac{1}{2n} + e^{-n/2} \qmq {for $n \ge 4$,} $$ yielding a bound of order $O(n^{-1/2})$ as $n \to \infty$. A similar, though slightly larger bound holds for $n$ odd. The results are shown using a version of Stein's method for bounded, monotone size bias couplings. The argument for even $n$ depends on the construction of a variable $X^s$ on the same space as $X$ which has the $X$ size bias distribution, that is, that satisfies E X g(X)=\mu Eg(X^s) \quad for all bounded continuous $g$, and for which there exists a $B \ge 0$, in this case B=2, such that $X \le X^s \le X+B$ almost surely. The argument for $n$ odd is similar to that for $n$ even, but one first couples $X$ closely to $V$, a symmetrized version of $X$, for which a size bias coupling of $V$ to $V^s$ can proceed as in the even case. http://arxiv.org/abs/1001.0612 9917. On the distribution of the Brownian motion process on its way to hitting zero Author(s): Konstantin Borovkov Abstract: We present functional versions of recent results on the univariate distributions of the process $V_{x,u} = x + W_{u\tau(x)},$ $0\le u\le 1$, where $W_\bullet$ is the standard Brownian motion process, $x>0$ and $\tau (x) =\inf\{t>0 : W_{t}=-x\}$. http://arxiv.org/abs/1001.0628 9918. Maharam extension and spectral representation of stable processes Author(s): Emmanuel Roy (LAGA) Abstract: We give a second look at stable processes (especially stationary) by interpreting the self-similar property at the level of the L\'evy measure as characteristic of a Maharam system. This allows us to derive structural results and their ergodic consequences. As a byproduct, we obtain a ?stable processes? proof of Banach-Lamperti Theorem for \alpha<2. http://arxiv.org/abs/1001.0638 9919. Asymptotic properties of random matrices and pseudomatrices Author(s): Romuald Lenczewski Abstract: We study the asymptotics of sums of matricially free random variables called random pseudomatrices, and we compare it with that of random matrices with block-identical variances. For objects of both types we find the limit joint distributions of blocks and give their Hilbert space realizations, using operators called `matricially free Gaussian operators'. In particular, if the variance matrices are symmetric, the asymptotics of symmetric blocks of random pseudomatrices agrees with that of symmetric random blocks. We also show that blocks of random pseudomatrices are `asymptotically matricially free' whereas the corresponding symmetric random blocks are `asymptotically symmetrically matricially free', where symmetric matricial freeness is obtained from matricial freeness by an operation of symmetrization. Finally, we show that row blocks of square, lower-block-triangular and block-diagonal pseudomatrices are asymptotically free, monotone independent and boolean independent, respectively. http://arxiv.org/abs/1001.0667 9920. On the so-called Boy or Girl Paradox Author(s): G. D'Agostini Abstract: A quite old problem has been recently revitalized by Leonard Mlodinow's book The Drunkard's Walk, where it is presented in a way that has definitely confused several people, that wonder why the prevalence of the name of one daughter among the population should change the probability that the other child is a girl too. I try here to discuss the problem from scratch, showing that the rarity of the name plays no role, unless the strange assumption of two identical names in the same family is taken into account. But also the name itself does not matter. What is really important is `identification', meant in an acceptation broader than usual, in the sense that a child is characterized by a set of attributes that make him/her uniquely identifiable (`that one') inside a family. The important point of how the information is acquired is also commented, suggesting an explanation of why several people tend to consider the informations "at least one boy" and "a well defined boy" (elder/youngest or of a given name) equivalent. http://arxiv.org/abs/1001.0708 9921. A law of large numbers approximation for Markov population processes with countably many types Author(s): A.D. Barbour and M.J. Luczak Abstract: When modelling metapopulation dynamics, the influence of a single patch on the metapopulation depends on the number of individuals in the patch. Since the population size has no natural upper limit, this leads to systems in which there are countably infinitely many possible types of individual. Analogous considerations apply in the transmission of parasitic diseases. In this paper, we prove a law of large numbers for rather general systems of this kind, together with a rather sharp bound on the rate of convergence in an appropriately chosen weighted $\ell_1$ norm. http://arxiv.org/abs/1001.0044 9922. Stochastic Monge-Kantorovich Problem and its Duality Author(s): Xicheng Zhang Abstract: In this article we prove the existence of a stochastic optimal transference plan for a stochastic Monge-Kantorovich problem by measurable selection theorem. A stochastic version of Kantorovich duality and the characterization of stochastic optimal transference plan are also established. Moreover, Wasserstein distance between two probability kernels are discussed too. http://arxiv.org/abs/1001.0094 9923. On the generalized Feynman-Kac transformation for nearly symmetric Markov processes Author(s): Li Ma and Wei Sun Abstract: Suppose $X$ is a right process which is associated with a non-symmetric Dirichlet form $(\mathcal{E},D(\mathcal{E}))$ on $L^{2}(E;m)$. For $u\in D(\mathcal{E})_{e}$, we have Fukushima's decomposition: $\tilde{u}(X_{t})-\tilde{u}(X_{0})=M^{u}_{t}+N^{u}_{t}$. In this paper, we investigate the strong continuity of the generalized Feynman-Kac semigroup defined by $P^{u}_{t}f(x)=E_{x}[e^{N^{u}_{t}}f(X_{t})]$. Let $Q^{u}(f,g)=\mathcal{E}(f,g)+\mathcal{E}(u,fg)$ for $f,g\in D(\mathcal{E})_{b}$. Denote by $J_1$ the dissymmetric part of the jumping measure $J$ of $(\mathcal{E},D(\mathcal{E}))$. Under the assumption that $J_1$ is finite, we show that $(Q^{u},D(\mathcal{E})_{b})$ is lower semi-bounded if and only if there exists a constant $\alpha_0\ge 0$ such that $\|P^{u}_{t}\|_2\leq e^{\alpha_0 t}$ for every $t>0$. If one of these conditions holds, then $(P^{u}_{t})_{t\geq0}$ is strongly continuous on $L^{2}(E;m)$. If $X$ is equipped with a differential structure, then this result also holds without assuming that $J_1$ is finite. http://arxiv.org/abs/1001.0203 9924. Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management Author(s): Huyen Pham (PMA and Crest) Abstract: We formulate and investigate a general stochastic control problem under a progressive enlargement of filtration. The global information is enlarged from a reference filtration and the knowledge of multiple random times together with associated marks when they occur. By working under a density hypothesis on the conditional joint distribution of the random times and marks, we prove a decomposition of the original stochastic control problem under the global filtration into classical stochastic control problems under the reference filtration, which are determined in a finite backward induction. Our method revisits and extends in particular stochastic control of diffusion processes with finite number of jumps. This study is motivated by optimization problems arising in default risk management, and we provide applications of our decomposition result for the indifference pricing of defaultable claims, and the optimal investment under bilateral counterparty risk. The solutions are expressed in terms of BSDEs involving only Brownian filtration, and remarkably without jump terms coming from the default times and marks in the global filtration. http://arxiv.org/abs/1001.0206 9925. Bayesian nonparametric analysis for a species sampling model with finitely many types Author(s): Annalisa Cerquetti Abstract: We derive explicit Bayesian nonparametric analysis for a species sampling model with finitely many types of Gibbs form of type $\alpha= -1$ recently introduced in Gnedin (2009). Our results complement existing analysis under Gibbs priors of type $\alpha \in [0, 1)$ proposed in Lijoi et al. (2008). Calculations rely on a groups sequential construction of Gibbs partitions introduced in Cerquetti (2008). http://arxiv.org/abs/1001.0245 9926. Periodically Correlated-Locally Stationary Processes Author(s): N. Modarresi and S. Rezakhah Abstract: In this paper we introduce a new class of non-stationary processes called, Periodically correlated-locally stationary (PC-LS) processes. It is concerned with spectral analysis of the harmonizable representation of the processes. Let $X(t)=X^s(t)+X^p(t)$ represents a stochastic process, where $X^s(t)$ is a continuous time stationary process and $X^p(t)$ is a discrete time periodically correlated (PC) process, then $X(t)$ is PC-LS. We also show that $X(t)$ is linearly correlated, which is include of periodically correlated and locally stationary (LS) processes. http://arxiv.org/abs/1001.0296 9927. Entropy of random walk range on uniformly transient and on uniformly recurrent graphs Author(s): David Windisch Abstract: We study the entropy of the distribution of the set R_n of vertices visited by a simple random walk on a graph with bounded degrees in its first n steps. It is shown that this quantity grows linearly in the expected size of R_n if the graph is uniformly transient, and sublinearly in the expected size if the graph is uniformly recurrent with subexponential volume growth. This in particular answers a question asked by Benjamini, Kozma, Yadin and Yehudayoff (arXiv:0903.3179v1). In the recurrent setting, our proof shows that R_n can be compressed into a string of 0-1-bits of length sublinear in its expected size with low probability of error. http://arxiv.org/abs/1001.0355 9928. Numerical simulation of BSDEs with drivers of quadratic growth Author(s): Adrien Richou (IRMAR) Abstract: We consider Markovian backward stochastic differential equations (BSDEs) with drivers of quadratic growth and bounded terminal conditions. We first show some bound estimations on the process $Z$. Then we give a new time discretization scheme for such BSDEs and we obtain an explicit convergence rate for this scheme. http://arxiv.org/abs/1001.0401 9929. Existence and Comparisons for BSDEs in general spaces Author(s): Samuel N. Cohen and Robert J. Elliott Abstract: We present a theory of Backward Stochastic Differential Equations in continuous time with an arbitrary filtered probability space. No assumptions are made regarding the continuity of the filtration, or of the predictable quadratic variations of martingales in this space. We present conditions for existence and uniqueness of square-integrable solutions, using Lipschitz continuity of the driver. These conditions unite the requirements for existence in continuous and discrete time, and allow discrete processes to be embedded with continuous ones. We also present conditions for a comparison theorem, and hence construct time consistent nonlinear expectations in these general spaces. http://arxiv.org/abs/1001.0439 9930. Intersection local times of independent fractional Brownian motions as generalized white noise functionals Author(s): Maria Joao Oliveira and Jose Luis da Silva and and Ludwig Streit Abstract: In this work we present expansions of intersection local times of fractional Brownian motions in $\R^d$, for any dimension $d\geq 1$, with arbitrary Hurst coefficients in $(0,1)^d$. The expansions are in terms of Wick powers of white noises (corresponding to multiple Wiener integrals), being well-defined in the sense of generalized white noise functionals. As an application of our approach, a sufficient condition on $d$ for the existence of intersection local times in $L^2$ is derived, extending the results of D. Nualart and S. Ortiz-Latorre in "Intersection Local Time for Two Independent Fractional Brownian Motions" (J. Theoret. Probab.,20(4)(2007), 759-767) to different and more general Hurst coefficients. http://arxiv.org/abs/1001.0513 9931. Post-L1-Penalized Estimators in High-Dimensional Linear Regression Models Author(s): Alexandre Belloni and Victor Chernozhukov Abstract: In this paper we study the post-penalized estimator which applies ordinary, unpenalized linear regression to the model selected by the first step penalized estimators, typically the LASSO. We show that post-LASSO can perform as well or nearly as well as the LASSO in terms of the rate of convergence. We show that this performance occurs even if the LASSO-based model selection "fails", in the sense of missing some components of the "true" regression model. Furthermore, post-LASSO can perform strictly better than LASSO, in the sense of a strictly faster rate of convergence, if the LASSO-based model selection correctly includes all components of the "true" model as a subset and enough sparsity is obtained. Of course, in the extreme case, when LASSO perfectly selects the true model, the past-LASSO estimator becomes the oracle estimator. We show that the results hold in both parametric and non-parametric models; and by the "true" model we mean the best $s$-dimensional approximation to the true regression model, where the dimension $s$ is can be chosen to maximize the rate of convergence of LASSO or post-LASSO estimators. Moreover, our analysis is not limited to the LASSO estimator in the first step, and also applies to other estimators, for example, the trimmed LASSO or Dantzig selector estimator. Our analysis also highlights the importance of sparsity induced by the first estimators. That motivated us to also study the impact of trimming small components of the initial estimator to achieve a sparser support for the post-LASSO. Our analysis covers both traditional trimming, as well as a new practical completely data-driven trimming scheme that induces maximal sparsity subject to maintaining a certain goodness-of-fit. http://arxiv.org/abs/1001.0188 9932. Asymptotic variance of random digital search trees Author(s): Hsien-Kuei Hwang and Michael Fuchs and Vytas Zacharovas Abstract: Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotically equivalent approximation to the variance in question. Several new ingredients are also introduced such as a combined use of Laplace and Mellin transforms and a simple, mechanical technique for justifying the analytic de-Poissonization procedures involved. The methodology we develop can be easily adapted to many other problems with an underlying binomial distribution. In particular, the less expected and somewhat surprising $n(\log n)^2$-variance for certain notions of total path-length is also clarified. http://arxiv.org/abs/1001.0095 ----------------------------------- Stefano M. Iacus Department of Economics, Business and Statistics University of Milan Via Conservatorio, 7 I-20123 Milan - Italy Ph.: +39 02 50321 461 Fax: +39 02 50321 505 http://www.economia.unimi.it/iacus ------------------------------------------------------------------------------------ Please don't send me Word or PowerPoint attachments if not absolutely necessary. See: http://www.gnu.org/philosophy/no-word-attachments.html