Walk In Random Scenery Research Articles

Entropy dimension for deterministic walks in random sceneries

AbstractEntropy dimension is an entropy-type quantity which takes values in $[0,1]$ and classifies different levels of intermediate growth rate of complexity for dynamical systems. In this paper, we consider the complexity of skew products of irrational rotations with Bernoulli systems, which can be viewed as deterministic walks in random sceneries, and show that this class of models can have any given entropy dimension by choosing suitable rotations for the base system.

Global observables for RW: Law of large numbers

We consider the sums TN=∑n=1NF(Sn) where Sn is a random walk on Zd and F:Zd→R is a global observable, that is, a bounded function which admits an average value when averaged over large cubes. We show that TN always satisfies the weak Law of Large Numbers but the strong law fails in general except for one dimensional walks with drift. Under additional regularity assumptions on F, we obtain the Strong Law of Large Numbers and estimate the rate of convergence. The growth exponents which we obtain turn out to be optimal in two special cases: for quasiperiodic observables and for random walks in random scenery.

Moderate deviations for the self-normalized random walk in random scenery

Let G be an infinite connected graph with vertex set V. Let {Sn:n∈N0} be the simple random walk on G and let {ξ(v):v∈V} be a collection of i.i.d. random variables which are independent of the random walk. Define the random walk in random scenery as Tn= ∑k=0nξ(Sk), and the normalization variables Vn=(∑k=0nξ2(Sk))1∕2 and Ln,2=(∑v∈Vln2(v))1∕2. For G=Zd and G=Td, the d-ary tree, we provide large deviations results for the self-normalized process Tnn∕(L n,2Vn) under only finite moment assumptions on the scenery.

Limit theorems for additive functionals of random walks in random scenery

We study the asymptotic behaviour of additive functionals of random walks in random scenery. We establish bounds for the moments of the local time of the Kesten and Spitzer process. These bounds combined with a previous moment convergence result (and an ergodicity result) imply the convergence in distribution of additive observables (with a normalization in n1 4). When the sum of the observable is null, the previous limit vanishes and we prove the convergence in the sense of moments (with a normalization in n1 8).

Quenched tail estimate for the random walk in random scenery and in random layered conductance II

This is a continuation of our earlier work [Stochastic Processes and their Applications, 129(1), pp.102--128, 2019] on the random walk in random scenery and in random layered conductance. We complete the picture of upper deviation of the random walk in random scenery, and also prove a bound on lower deviation probability. Based on these results, we determine asymptotics of the return probability, a certain moderate deviation probability, and the Green function of the random walk in random layered conductance.

Random walks in random sceneries and related models

We present random walks in random sceneries as well as three related models: U-statistics indexed by random walks, a model of stratified media with inhomogeneous layers (random one-way streets) and the one-dimensional Lévy-Lorentz gas (random roundabouts on a line). We present in particular results obtained in collaboration with Castell, Guillotin-Plantard, Br. Schapira, Franke, Wendler, Aurzada, Bianchi and Lenci.

Empirical processes for recurrent and transient random walks in random scenery

In this paper, we are interested in the asymptotic behaviour of the sequence of processes (Wn(s,t))s,t∈[0,1] with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(\mathds{1}_{\{\xi_{S_k}\leq s\}}-s\big) \end{equation*} where (ξx, x ∈ ℤd) is a sequence of independent random variables uniformly distributed on [0, 1] and (Sn)n ∈ ℕ is a random walk evolving in ℤd, independent of the ξ’s. In M. Wendler [Stoch. Process. Appl. 126 (2016) 2787–2799], the case where (Sn)n ∈ ℕ is a recurrent random walk in ℤ such that (n−1/αSn)n≥1 converges in distribution to a stable distribution of index α, with α ∈ (1, 2], has been investigated. Here, we consider the cases where (Sn)n ∈ ℕ is either: (a) a transient random walk in ℤd, (b) a recurrent random walk in ℤd such that (n−1/dSn)n≥1 converges in distribution to a stable distribution of index d ∈{1, 2}.

Self-normalized Moderate Deviations for Random Walk in Random Scenery

Let $$\{S_k:k\ge 0\}$$ be a symmetric and aperiodic random walk on $$\mathbb {Z}^d$$, $$d\ge 3$$, and $$\{\xi (z),z\in \mathbb {Z}^d\}$$ a collection of independent and identically distributed random variables. Consider a random walk in random scenery defined by $$T_n=\sum _{k=0}^n\xi (S_k)=\sum _{z\in \mathbb {Z}^d}l_n(z)\xi (z)$$, where $$l_n(z)=\sum _{k=0}^nI{\{S_k=z\}}$$ is the local time of the random walk at the site z. Using $$(\sum _{z\in \mathbb {Z}^d}l_n(z)|\xi (z)|^p)^{1/p}$$, $$p\ge 2$$, as the normalizing constants, we establish self-normalized moderate deviations for random walk in random scenery under a much weaker condition than a finite moment-generating function of the scenery variables.

Extremes for transient random walks in random sceneries under weak independence conditions

Let {ξ(k),k∈Z} be a stationary sequence of random variables with conditions of type D(un) and D′(un). Let {Sn,n∈N} be a transient random walk in the domain of attraction of a stable law. We provide a limit theorem for the maximum of the first n terms of the sequence {ξ(Sn),n∈N} as n goes to infinity. This paper extends a result due to Franke and Saigo who dealt with the case where the sequence {ξ(k),k∈Z} is i.i.d.

Continuous-time random walk between Lévy-spaced targets in the real line

We consider a continuous-time random walk which is defined as an interpolation of a random walk on a point process on the real line. The distances between neighboring points of the point process are i.i.d. random variables in the normal domain of attraction of an α-stable distribution with 0<α<1. This is therefore an example of a random walk in a Lévy random medium. Specifically, it is a generalization of a process known in the physical literature as Lévy–Lorentz gas. We prove that the annealed version of the process is superdiffusive with scaling exponent 1∕(α+1) and identify the limiting process, which is not càdlàg. The proofs are based on the technique of Kesten and Spitzer for random walks in random scenery.

Generalized Divide and Color Models

In this paper, we initiate the study of and Color Models. A very interesting special case of this is the Divide and Color (which motivates the name we use) introduced and studied by Olle Haggstrom. In this generalized model, one starts with a finite or countable set V, a random partition of V and a parameter p ∈ [0; 1]. The corresponding Generalized and Color Model is the [0; 1]-valued process indexed by V obtained by independently, for each partition element in the random partition chosen, with probability p, assigning all the elements of the partition element the value 1, and with probability 1 - p, assigning all the elements of the partition element the value 0. Some of the questions which we study here are the following. Under what situations can different random partitions give rise to the same color process? What can one say concerning exchangeable random partitions? What is the set of product measures that a color process stochastically dominates? For random partitions which are translation invariant, what ergodic properties do the resulting color processes have? The motivation for studying these processes is twofold; on the one hand, we believe that this is a very natural and interesting class of processes that deserves investigation and on the other hand, a number of quite varied well-studied processes actually fall into this class such as (1) the Ising model, (2) the fuzzy Potts model, (3) the stationary distributions for the Voter Model, (4) random walk in random scenery and of course (5) the original and Color Model.

Chover-Type Laws of the Iterated Logarithm for Kesten-Spitzer Random Walks in Random Sceneries Belonging to the Domain of Stable Attraction

Let X={Xi,i≥1} be a sequence of real valued random variables, S0=0 and Sk=∑i=1kXi (k≥1). Let σ={σ(x),x∈Z} be a sequence of real valued random variables which are independent of X’s. Denote by Kn=∑k=0nσ(⌊Sk⌋) (n≥0) Kesten-Spitzer random walk in random scenery, where ⌊a⌋ means the unique integer satisfying ⌊a⌋≤a&lt;⌊a⌋+1. It is assumed that σ’s belong to the domain of attraction of a stable law with index 0&lt;β&lt;2. In this paper, by employing conditional argument, we investigate large deviation inequalities, some sufficient conditions for Chover-type laws of the iterated logarithm and the cluster set for random walk in random scenery Kn. The obtained results supplement to some corresponding results in the literature.

Quenched tail estimate for the random walk in random scenery and in random layered conductance

We discuss the quenched tail estimates for the random walk in random scenery. The random walk is the symmetric nearest neighbor walk and the random scenery is assumed to be independent and identically distributed, non-negative, and has a power law tail. We identify the long time asymptotics of the upper deviation probability of the random walk in quenched random scenery, depending on the tail of scenery distribution and the amount of the deviation. The result is in turn applied to the tail estimates for a random walk in random conductance which has a layered structure.

Approximate local limit theorems with effective rate and application to random walks in random scenery

We show that the Bernoulli part extraction method can be used to obtain approxi- mate forms of the local limit theorem for sums of independent lattice valued random variables, with effective error term, that is with explicit parameters and universal constants. We also show that our estimates allow to recover Gnedenko and Gamkrelidze local limit theorems. We further establish by this method a local limit theorem with effective remainder for random walks in random scenery. 1. Introduction and Main Result. The extraction method of the Bernoulli part of a (lattice valued) random variable was devel- oped by McDonald in (13),(14),(4) for proving local limit theorems in presence of the central limit theorem. Twenty years before McDonald's work, Kolmogorov (11) initiated a similar approach in the study of Levy's concentration function, and is the first having explored this direction. For details and clarifications, we refer to the recent paper by Aizenmann, Germinet, Klein and Warzel (1), where this idea is also developed for general random variables and applications are given. That method allows to transfer results which are available for systems of Bernoulli random variables to systems of arbitrary random variables. It is based on a probabilistic device, and is proved to be an efficient alternative to the characteristic functions method. Kolmogorov wrotes to this effect in his 1958's paper (11) p.29: ...Il semble cependant que nous restons toujours dans une periode ou la competition de ces deux directions (characteristic functions or direct methods from the calculus of probability) conduit aux resultats les plus feconds .... We believe that Kolmogorov's comment is still topical. The main object of this article is to show that this approach can be used to obtain, in a rather simple way, approximate forms of the local limit theorem for sums of independent lattice valued random variables, with effective error term, that is with explicit parameters and universal constants. The approximate form we obtain expresses quite simply, and is thereby very handable. Further, it is precise enough to contain Gnedenko and Gamkrelidze local limit theorems (1.2). Before stating the main results and in view of comparing results, it is necessary to recall and discuss some classical facts and briefly describe the background of this problem. Let ˜

Persistence probabilities for stationary increment processes

We study the persistence probability for processes with stationary increments. Our results apply to a number of examples: sums of stationary correlated random variables whose scaling limit is fractional Brownian motion; random walks in random sceneries; random processes in Brownian scenery; and the Matheron–de Marsily model in Z2 with random orientations of the horizontal layers. Using a new approach, strongly related to the study of the range, we obtain an upper bound of the optimal order in general and improved lower bounds (compared to previous literature) for many specific processes.

Relative complexity of random walks in random scenery in the absence of a weak invariance principle for the local times

We answer a question of Aaronson about the relative complexity of Random Walks in Random Sceneries driven by either aperiodic two-dimensional random walks, two-dimensional Simple Random walk, or by aperiodic random walks in the domain of attraction of the Cauchy distribution. A key step is proving that the range of the random walk satisfies the Fölner property almost surely.

The sequential empirical process of a random walk in random scenery

A random walk in random scenery (Yn)n∈N is given by Yn=ξSn for a random walk (Sn)n∈N and i.i.d. random variables (ξn)n∈Z. In this paper, we will show the weak convergence of the sequential empirical process, i.e. the centered and rescaled empirical distribution function. The limit process shows a new type of behavior, combining properties of the limit in the independent case (roughness of the paths) and in the long range dependent case (self-similarity).

A quenched functional central limit theorem for planar random walks in random sceneries

Random walks in random sceneries (RWRS) are simple examples of stochastic processes in disordered media. They were introduced at the end of the 70's by Kesten-Spitzer and Borodin, motivated by the construction of new self-similar processes with stationary increments. Two sources of randomness enter in their definition: a random field $\xi = (\xi(x))_{x \in \mathbb{Z}^d}$ of i.i.d. random variables, which is called the random scenery, and a random walk $S = (S_n)_{n \in \mathbb{N}}$ evolving in $\mathbb{Z}^d$, independent of the scenery. The RWRS $Z = (Z_n)_{n \in \mathbb{N}}$ is then defined as the accumulated scenery along the trajectory of the random walk, i.e., $Z_n := \sum_{k=1}^n \xi(S_k)$. The law of $Z$ under the joint law of $\xi$ and $S$ is called annealed'', and the conditional law given $\xi$ is called quenched''. Recently, functional central limit theorems under the quenched law were proved for $Z$ by the first two authors for a class of transient random walks including walks with finite variance in dimension $d \ge 3$. In this paper we extend their results to dimension $d=2$.

The quenched limiting distributions of a one-dimensional random walk in random scenery

For a one-dimensional random walk in random scenery (RWRS) on Z, we determine its quenched weak limits by applying Strassen's functional law of iterated logarithm. As a consequence, conditioned on the random scenery, the one dimensional RWRS does not converge in law, in contrast with the multi-dimensional case.

Quenched central limit theorems for random walks in random scenery

Random walks in random scenery are processes defined by Zn:=∑k=1nωSk where S:=(Sk,k≥0) is a random walk evolving in Zd and ω:=(ωx,x∈Zd) is a sequence of i.i.d. real random variables. Under suitable assumptions on the random walk S and the random scenery ω, almost surely with respect to ω, the correctly renormalized sequence (Zn)n≥1 is proved to converge in distribution to a centered Gaussian law with explicit variance.