Showing papers in "ALEA-Latin American Journal of Probability and Mathematical Statistics in 2013"

Gamma limits and U-statistics on the Poisson space

Abstract: Using Stein's method and the Malliavin calculus of variations, we de- rive explicit estimates for the Gamma approximation of functionals of a Poisson measure. In particular, conditions are presented under which the distribution of a sequence of multiple Wiener-Ito stochastic integrals with respect to a compen- sated Poisson measure converges to a Gamma distribution. As an illustration, we present a quantitative version and a non-central extension of a classical theorem by de Jong in the case of degenerate U-statistics of order two. Several multidimen- sional extensions, in particular allowing for mixed or hybrid limit theorems, are also provided.

On the Law of Free Subordinators

Abstract: We study the freely infinitely divisible distributions that appear as the laws of free subordinators. This is the free analog of classically infinitely divisible distributions supported on (0,∞), called the free regular measures. We prove that the class of free regular measures is closed under the free multiplicative convolution, t th boolean power for 0 ≤ t ≤ 1, t th free multiplicative power for t ≥ 1 and weak convergence. In addition, we show that a symmetric distribution is freely infinitely divisible if and only if its square can be represented as the free multiplicative con- volution of a free Poisson and a free regular measure. This gives two new explicit examples of distributions which are infinitely divisible with respect to both classi- cal and free convolutions: � 2 (1) and F(1,1). Another consequence is that the free

Transient random walk in symmetric exclusion: limit theorems and an Einstein relation

Abstract: We consider a one-dimensional simple symmetric exclusion process in equilibrium as a dynamic random environment for a nearest-neighbor random walk that on occupied/vacant sites has two different local drifts to the right. We obtain a LLN, a functional CLT and large deviation bounds for the random walk under the annealed measure by means of a renewal argument. We also obtain an Einstein relation under a suitable perturbation. A brief discussion on the topic of random walks in slowly mixing dynamic random environments is presented.

On the easiest way to connect k points in the Random Interlacements process

Abstract: We consider the random interlacements process with intensity $u$ on ${\mathbb Z}^d$, $d\ge 5$ (call it $I^u$), built from a Poisson point process on the space of doubly infinite nearest neighbor trajectories on ${\mathbb Z}^d$. For $k\ge 3$ we want to determine the minimal number of trajectories from the point process that is needed to link together $k$ points in $\mathcal I^u$. Let $$n(k,d):=\lceil \frac d 2 (k-1) \rceil - (k-2).$$ We prove that almost surely given any $k$ points $x_1,...,x_k\in \mathcal I^u$, there is a sequence ofof $n(k,d)$ trajectories $\gamma^1,...,\gamma^{n(k,d)}$ from the underlying Poisson point process such that the union of their traces $\bigcup_{i=1}^{n(k,d)}\tr(\gamma^{i})$ is a connected set containing $x_1,...,x_k$. Moreover we show that this result is sharp, i.e. that a.s. one can find $x_1,...,x_k in I^u$ that cannot be linked together by $n(k,d)-1$ trajectories.

On the mixing time and spectral gap for birth and death chains

Abstract: For birth and death chains, we derive bounds on the spectral gap and mixing time in terms of birth and death rates. Together with the results of Ding et al. (2010), this provides a criterion for the existence of a cutoff in terms of the birth and death rates. A variety of illustrative examples are treated.

On the length of one-dimensional reactive paths

Abstract: Motivated by some numerical observations on molecular dynamics sim- ulations, we analyze metastable trajectories in a very simple setting, namely paths generated by a one-dimensional overdamped Langevin equation for a double well potential. Specifically, we are interested in so-called reactive paths, namely tra- jectories which leave definitely one well and reach the other one. The aim of this paper is to precisely analyze the distribution of the lengths of reactive paths in the limit of small temperature, and to compare the theoretical results to numerical re- sults obtained by a Monte Carlo method, namely the multi-level splitting approach (see Cerou et al. (2011)).

Strongly Vertex-Reinforced-Random-Walk on the complete graph

Abstract: We study Vertex-Reinforced-Random-Walk (VRRW) on a complete graph with weights of the form w(n) = n α , with α > 1. Unlike for the Edge-Reinforced-Random-Walk, which in this case localizes a.s. on 2 sites, here we observe various phase transitions, and in particular localization on arbitrary large sets is possible, provided α is close enough to 1. Our proof relies on stochastic approximation techniques. At the end of the paper, we also prove a general result ensuring that any strongly reinforced VRRW on any bounded degree graph localizes a.s. on a finite subgraph.

LAN property for some fractional type Brownian motion

Abstract: We study asymptotic expansion of the likelihood of a certain class of Gaussian processes characterized by their spectral density $f_\theta$. We consider the case where $f_\theta\PAR{x} \sim_{x\to 0} \ABS{x}^{-\al(\theta)}L_\theta(x)$ with $L_\theta$ a slowly varying function and $\al\PAR{\theta}\in (-\infty,1)$. We prove LAN property for these models which include in particular fractional Brownian motion %$B^\alpha_t,\: \alpha \geq 1/2$ or ARFIMA processes.

On the critical value function in the divide and color model

Abstract: The divide and color model on a graph G arises by first deleting each edge of G with probability 1-p independently of each other, then coloring the resulting connected components (i.e., every vertex in the component) black or white with respective probabilities r and 1-r, independently for different components. Viewing it as a (dependent) site percolation model, one can denote the critical point r^G_c(p). In this paper, we mainly study the continuity properties of the function r^G_c, which is an instance of the question of locality for percolation. Our main result is the fact that in the case G=Z^2, r^G_c is continuous on the interval [0,1/2); we also prove continuity at p=0 for the more general class of graphs with bounded degree. We then investigate the sharpness of the bounded degree condition and the monotonicity of r^G_c(p) as a function of p.

Optimal Convergence Rates and One-Term Edgeworth Expansions for Multidimensional Functionals of Gaussian Fields

Abstract: We develop techniques for determining the exact asymptotic speed of convergence in the multidimensional normal approximation of smooth functions of Gaussian fields. As a by-product, our findings yield exact limits and often give rise to one-term generalized Edgeworth expansions increasing the speed of convergence. Our main mathematical tools are Malliavin calculus, Stein's method and the Fourth Moment Theorem. This work can be seen as an extension of the results of Nourdin and Peccati (2009a) to the multi-dimensional case, with the notable difference that in our framework covariances are allowed to fluctuate. We apply our findings to exploding functionals of Brownian sheets, vectors of Toeplitz quadratic functionals and the Breuer-Major Theorem.

Convergence to the Tracy-Widom distribution for longest paths in a directed random graph

Abstract: We consider a directed graph on the 2-dimensional integer lattice, plac- ing a directed edge from vertex (i1;i2) to (j1;j2), whenever i1 j1, i2 j2, with probability p, independently for each such pair of vertices. Let Ln;m denote the maximum length of all paths contained in an n m rectangle. We show that there is a positive exponent a, such that, if m=n a ! 1, as n ! 1, then a properly cen- tered/rescaled version of Ln;m converges weakly to the Tracy-Widom distribution. A generalization to graphs with non-constant probabilities is also discussed.

A two-scale approach to the hydrodynamic limit Part II: local Gibbs behavior

Abstract: This work is a follow-up on Grunewald et al. (2009). In that previous work a two-scale approach was used to prove the logarithmic Sobolev inequality for a system of spins with xed mean whose potential is a bounded perturbation of a Gaussian, and to derive an abstract theorem for the convergence to the hydro- dynamic limit. This strategy was then successfully applied to Kawasaki dynamics. Here we shall use again this two-scale approach to show that the microscopic vari- able in such a model behaves according to a local Gibbs state. As a consequence, we shall prove the convergence of the microscopic entropy to the hydrodynamic entropy.

Empirical central limit theorems for ergodic automorphisms of the torus

Abstract: Let T be an ergodic automorphism of the d-dimensional torus T^d , and f be a continuous function from T^d to R . On the probability space T^d equipped with the Lebesgue-Haar measure, we prove the weak convergence of the sequential empirical process of the sequence (f ° T^ i )i≥1 under some condition on the modulus of continuity of f . The proofs are based on new limit theorems and new inequalities for non-adapted sequences, and on new estimates of the conditional expectations of f with respect to a natural ltration.

Confidence intervals for the critical value in the divide and color model

Abstract: We obtain condence intervals for the location of the percolation phase transition in Haggstrom's divide and color model on the square lattice Z 2 and the hexagonal lattice H. The resulting probabilistic bounds are much tighter than the best deterministic bounds up to date; they give a clear picture of the behavior of the DaC models on Z 2 and H and enable a comparison with the triangular lattice T. In particular, our numerical results suggest similarities between DaC model on these three lattices that are in line with universality considerations, but with a remarkable dierence: while the critical value function rc(p) is known to be constant in the parameter p for p < pc on T and appears to be linear on Z 2 , it is almost certainly non-linear on H.

Comments on the Influence of Disorder for Pinning Model in Correlated Gaussian Environment

Abstract: We study the random pinning model, in the case of a Gaussian environment presenting power-law decaying correlations, of exponent decay a > 0. A similar study was done in a hierachical version of the model [5], and we extend here the results to the non-hierarchical (and more natural) case. We comment on the annealed (i.e. averaged over disorder) model, which is far from being trivial, and we discuss the influence of disorder on the critical properties of the system. We show that the annealed critical exponent ν a is the same as the homogeneous one ν pur , provided that correlations are decaying fast enough (a > 2). If correlations are summable (a > 1), we also show that the disordered phase transition is at least of order 2, showing disorder relevance if ν pur < 2. If correlations are not summable (a < 1), we show that the phase transition disappears.

The Isolation Time of Poisson Brownian Motions

Abstract: Let the nodes of a Poisson point process move independently in R^d according to Brownian motions. We study the isolation time for a target particle that is placed at the origin, namely how long it takes until there is no node of the Poisson point process within distance r of it. In the case when the target particle does not move, we obtain asymptotics for the tail probability which are tight up to constants in the exponent in dimension d ≥ 3 and tight up to logarithmic factors in the exponent for dimensions d = 1, 2. In the case when the target particle is allowed to move independently of the Poisson point process, we show that the best strategy for the target to avoid isolation is to stay put.

Comparison-type theorems for It\^o processes and differentially subordinated semimartingales

Abstract: The purpose of this paper is to provide a wide family of sharp estimates for certain class of Ito processes and, more generally, for the class of semimartingales satisfying the so-called α-subordination relation. To describe our motivation, it is convenient to start with the setting of Ito processes. Suppose that (Ω,F ,P) is a complete probability space, filtered by a nondecreasing right-continuous family (Ft)t≥0 of sub-σ-fields of F . As usual, we assume that F0 contains all the sets A satisfying P(A) = 0. Let B = (Bt)t≥0 be an adapted Brownian motion starting from 0 and let X = (Xt)t≥0, Y = (Yt)t≥0 be Ito processes with respect to B (cf.

Some aspects of fluctuations of random walks on R and applications to random walks on R + with non-elastic reflection at 0

Abstract: In this article we refine well-known results concerning the fluctuations of one-dimensional random walks. More precisely, if $(S_n)_{n \geq 0}$ is a random walk starting from $0$ and $r\geq 0$, we obtain the precise asymptotic behavior as $n\to\infty$ of $\mathbb P[\tau^{>r}=n, S_n\in K]$ and $\mathbb P[\tau^{>r}>n, S_n\in K]$, where $\tau^{>r}$ is the first time that the random walk reaches the set $]r,\infty[$, and $K$ is a compact set. Our assumptions on the jumps of the random walks are optimal. Our results give an answer to a question of Lalley stated in \cite{L}, and are applied to obtain the asymptotic behavior of the return probabilities for random walks on $\mathbb R^+$ with non-elastic reflection at $0$.

Tanaka's equation on the circle and stochastic flows

Abstract: We define a Tanaka's equation on an oriented graph with two edges and two vertices. This graph will be embedded in the unit circle. Extending this equation to flows of kernels, we show that the laws of the flows of kernels K solutions of Tanaka's equation can be classified by pairs of probability measures (m + ,m − ) on (0,1), with mean 1/2. What happens at the first vertex is governed by m + , and at the second by m − . For each vertex P, we construct a sequence of stopping times along which the image of the whole circle by K is reduced to P. We also prove that the supports of these flows contain a finite number of points, and that except for some particular cases this number of points can be arbitrarily large.