Random Scenery Research Articles

A local limit theorem for random walks in random scenery and on randomly oriented lattices

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.

On some functional of the hybrid process in random scenery

In this work we wish to investigate an example based on the so-called Kesten–Spitzer random walk in random scenery. Namely, replacing the one-dimensional random walk in a general i.i.d. scenery by the hybrids of empirical and partial sums process (see, for instance, [L. Horváth, Approximations for hybrids of empirical and partial sums process, J. Statist. Plann. Inference 88 (2000) 1–18]), we establish an upper bound in the strong approximation for the corresponding functional.

Planar Lorentz process in a random scenery

Nous considérons le processus de Lorentz dans le plan avec des obstacles convexes disposés de manière périodique (nous supposons de plus que l’horizon est fini). Dans ce modèle, une particule ponctuelle se déplace à vitesse unitaire et sa vitesse obéit à la loi de la réflexion de Descartes à l’instant d’un choc contre un obstacle. La scène aléatoire est donnée par une suite de variables aléatoires indépendantes de même loi, centrées, de variance finie non nulle. Chacune de ces variables aléatoires est associée à un obstacle. Nous associons à la particule une somme qui évolue avec le temps. Cette somme est nulle au départ. A chaque fois que la particule touche un obstacle, elle gagne la valeur de la variable aléatoire associée à cet obstacle. Nous montrons que la somme totale gagnée au temps n (normalisée par $\sqrt{n\log(n)}$) converge en loi vers un processus de Wiener lorsque n tend vers l’infini. Un tel résultat a été établi par Bolthausen [Ann. Probab. 17 (1989) 108–115)] dans le cas de marches aléatoires sur ℤ2 avec des pas indépendants et de même loi. Nous nous inspirons de son travail. Nous remplaçons l’hypothèse d’indépendance de [Ann. Probab. 17 (1989) 108–115)] par des extensions du théorème limite local établi par Szász and Varjú in [Ergodic Theory Dynam. Systems 24 (2004) 257–278]. Ce travail apporte une réponse à une question de Szász concernant le comportement asymptotique de ${∑_{k=0}^{n−1}}ζS_{k}$ où (ζℓ)ℓ est une suite de variables aléatoires indépendantes identiquement distribuées, centrées, de variance finie et non nulle et où Sk désigne le numéro de la cellule dans laquelle se trouve la particule à l’instant de la kème reflexion.

The extremes of random walks in random sceneries

In this article we analyse the behaviour of the extremes of a random walk in a random scenery. The random walk is assumed to be in the domain of attraction of a stable law, and the scenery is assumed to be in the domain of attraction of an extreme value distribution. The resulting random sequence is stationary and strongly dependent if the underlying random walk is recurrent. We prove a limit theorem for the extremes of the resulting stationary process. However, if the underlying random walk is recurrent, the limit distribution is not in the class of classical extreme value distributions.

The extremes of random walks in random sceneries

In this article we analyse the behaviour of the extremes of a random walk in a random scenery. The random walk is assumed to be in the domain of attraction of a stable law, and the scenery is assumed to be in the domain of attraction of an extreme value distribution. The resulting random sequence is stationary and strongly dependent if the underlying random walk is recurrent. We prove a limit theorem for the extremes of the resulting stationary process. However, if the underlying random walk is recurrent, the limit distribution is not in the class of classical extreme value distributions.

A functional approach for random walks in random sceneries

A functional approach for the study of the random walks in random sceneries (RWRS) is proposed. Under fairly general assumptions on the random walk and on the random scenery, functional limit theorems are proved. The method allows to study separately the convergence of the walk and of the scenery: on the one hand, a general criterion for the convergence of the local time of the walk is provided, on the other hand, the convergence of the random measures associated with the scenery is studied. This functional approach is robust enough to recover many of the known results on RWRS as well as new ones, including the case of many walkers evolving in the same scenery.

Convergence of dependent walks in a random scenery to fBm-local time fractional stable motions

It is classical to approximate the distribution of fractional Brownian motion by a renormalized sum $ S_n $ of dependent Gaussian random variables. In this paper we consider such a walk $ Z_n $ that collects random rewards $ \xi_j $ for $ j \in \mathbb Z$, when the ceiling of the walk $ S_n $ is located at $ j$. The random reward (or scenery) $ \xi_j $ is independent of the walk and with heavy tail. We show the convergence of the sum of independent copies of $ Z_n$ suitably renormalized to a stable motion with integral representation, whose kernel is the local time of a fractional Brownian motion (fBm). This work extends a previous work where the random walk $ S_n$ had independent increments limits.

The extremes of a random scenery as seen by a random walk in a random environment

A random walk performs a motion in an iid-environment and observes an iid-scenery along its path. Assuming the scenery is in the domain of attraction of an extreme-value distribution, we prove a limit-theorem for the observed extremes.

Limit theorem for random walk in weakly dependent random scenery

Let $S=(S_k)_{k\geq 0}$ be a random walk on $\mathbb{Z}$ and $\xi=(\xi_{i})_{i\in\mathbb{Z}}$ a stationary random sequence of centered random variables, independent of $S$. We consider a random walk in random scenery that is the sequence of random variables $(\Sigma_n)_{n\geq 0}$ where $$\Sigma_n=\sum_{k=0}^n \xi_{S_k}, n\in\mathbb{N}.$$ Under a weak dependence assumption on the scenery $\xi$ we prove a functional limit theorem generalizing Kesten and Spitzer's theorem (1979).

Discrete approximation of a stable self-similar stationary increments process

The aim of this paper is to present a result of discrete approximation of some class of stable self-similar stationary increments processes. The properties of such processes were intensively investigated, but little is known about the context in which such processes can arise. To our knowledge, discretization and convergence theorems are available only in the case of stable Lévy motions and fractional Brownian motions. This paper yields new results in this direction. Our main result is the convergence of the random rewards schema first introduced by Cohen and Samorodnitsky, which we consider in a more general setting. Strong relationships with Kesten and Spitzer’s random walk in random sceneries are evidenced. Finally, we study some path properties of the limit process.

Weighted power variations of iterated Brownian motion

We characterize the asymptotic behaviour of the weighted power variation processes associated with iterated Brownian motion. We prove weak convergence results in the sense of finite dimensional distributions, and show that the laws of the limiting objects can always be expressed in terms of three independent Brownian motions $X, Y$ and $B$, as well as of the local times of $Y$. In particular, our results involve ''weighted'' versions of Kesten and Spitzer's Brownian motion in random scenery. Our findings extend the theory initiated by Khoshnevisan and Lewis (1999), and should be compared with the recent result by Nourdin and Réveillac (2008), concerning the weighted power variations of fractional Brownian motion with Hurst index $H=1/4$.

Moderate deviations for a random walk in random scenery

We investigate the cumulative scenery process associated with random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cramér’s condition. We prove moderate deviation principles in dimensions d ≥ 2 , covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. For the case d ≥ 4 we even obtain precise asymptotics for the probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. For d ≥ 3 , an important ingredient in the proofs are new concentration inequalities for self-intersection local times of random walks, which are of independent interest, whilst for d = 2 we use a recent moderate deviation result for self-intersection local times, which is due to Bass, Chen and Rosen.

Upper limits of Sinai’s walk in random scenery

We consider Sinai’s walk in i.i.d. random scenery and focus our attention on a conjecture of Révész concerning the upper limits of Sinai’s walk in random scenery when the scenery is bounded from above. A close study of the competition between the concentration property for Sinai’s walk and negative values for the scenery enables us to prove that the conjecture is true if the scenery has “thin” negative tails and is false otherwise.

Strong Laws of Large Numbers for Random Walks in Random Sceneries

In this paper, we study strong laws of large numbers for random walks in random sceneries. Some mild sufficient conditions for the validity of strong laws of large numbers are obtained.

Large deviations estimates for self-intersection local times for simple random walk in $${\mathbb{Z}}^3$$

We obtain large deviations estimates for the self-intersection local times for a simple random walk in dimension 3. Also, we show that the main contribution to making the self-intersection large, in a time period of length n, comes from sites visited less than some power of log(n). This is opposite to the situation in dimensions larger or equal to 5. Finally, we present an application of our estimates to moderate deviations for random walk in random sceneries.

Annealed deviations of random walk in random scenery

Let ( Z n ) n ∈ N be a d-dimensional random walk in random scenery, i.e., Z n = ∑ k = 0 n − 1 Y ( S k ) with ( S k ) k ∈ N 0 a random walk in Z d and ( Y ( z ) ) z ∈ Z d an i.i.d. scenery, independent of the walk. The walker's steps have mean zero and some finite exponential moments. We identify the speed and the rate of the logarithmic decay of P ( 1 n Z n > b n ) for various choices of sequences ( b n ) n in [ 1 , ∞ ) . Depending on ( b n ) n and the upper tails of the scenery, we identify different regimes for the speed of decay and different variational formulas for the rate functions. In contrast to recent work [A. Asselah, F. Castell, Large deviations for Brownian motion in a random scenery, Probab. Theory Related Fields 126 (2003) 497–527] by A. Asselah and F. Castell, we consider sceneries unbounded to infinity. It turns out that there are interesting connections to large deviation properties of self-intersections of the walk, which have been studied recently by X. Chen [X. Chen, Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks, Ann. Probab. 32 (4) 2004].

A note on random walk in random scenery

We consider a random walk in random scenery { X n = η ( S 0 ) + ⋯ + η ( S n ) , n ∈ N } , where a centered walk { S n , n ∈ N } is independent of the scenery { η ( x ) , x ∈ Z d } , consisting of symmetric i.i.d. with tail distribution P ( η ( x ) > t ) ∼ exp ( − c α t α ) , with 1 ⩽ α < d / 2 . We study the probability, when averaged over both randomness, that { X n > n y } for y > 0 , and n large. In this note, we show that the large deviation estimate is of order exp ( − c ( n y ) a ) , with a = α / ( α + 1 ) .

Random walk in random scenery and self-intersection local times in dimensions d ≥ 5

Let {S k , k ≥ 0} be a symmetric random walk on $${\mathbb Z}^d$$ , and $$\{\eta(x), x\in {\mathbb Z^d\}}$$ an independent random field of centered i.i.d. random variables with tail decay $$P(\eta(x)> t)\approx\exp(-t^{\alpha})$$ . We consider a random walk in random scenery, that is $$X_n=\eta(S_0)+\dots+\eta(S_n)$$ . We present asymptotics for the probability, over both randomness, that {X n > n β} for β > 1/2 and α > 1. To obtain such asymptotics, we establish large deviations estimates for the self-intersection local times process $$\sum_x l_n^2(x)$$ , where l n (x) is the number of visits of site x up to time n.

Reconstruction of periodic sceneries seen along a random walk

This article was motivated by a question of Kesten. Kesten asked for which random walks periodic sceneries can be reconstructed. Among others, he asked the question for random walks which at each step can move by one or two units to the right. Previously, Howard [C.D. Howard, Distinguishing certain random sceneries on Z via random walks, Statist. Probab. Lett. 34 (2) (1997) 123–132] proved that all periodic sceneries can be reconstructed provided they are observed along the path of a simple random walk. We prove that for a large class of random walks it is possible to reconstruct periodic sceneries. Among others, we consider the random walk that can only move by one or two units to the right at each step. We show that in this case, the scenery can be reconstructed, provided that the two unit step is less likely than the one unit step.

Deviations of a random walk in a random scenery with stretched exponential tails

Let ( Z n ) n ∈ N be a d-dimensional random walk in random scenery, i.e., Z n = ∑ k = 0 n - 1 Y S k with ( S k ) k ∈ N 0 a random walk in Z d and ( Y z ) z ∈ Z d an i.i.d. scenery, independent of the walk. We assume that the random variables Y z have a stretched exponential tail. In particular, they do not possess exponential moments. We identify the speed and the rate of the logarithmic decay of P ( Z n > nt n ) for all sequences ( t n ) n ∈ N satisfying a certain lower bound. This complements results of Gantert et al. [Annealed deviations of random walk in random scenery, preprint, 2005], where it was assumed that Y z has exponential moments of all orders. In contrast to the situation (Gantert et al., 2005), the event { Z n > nt n } is not realized by a homogeneous behavior of the walk's local times and the scenery, but by many visits of the walker to a particular site and a large value of the scenery at that site. This reflects a well-known extreme behavior typical for random variables having no exponential moments.