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

TL;DR

This paper establishes a local limit theorem for random walks in random scenery and on randomly oriented lattices, detailing the asymptotic behavior of their probability distributions under stable law domains.

Contribution

It extends previous convergence results by providing a local limit theorem for these processes, including the case of randomly oriented lattices.

Findings

Proves a local limit theorem for random walks in random scenery.

Derives similar results for random walks on randomly oriented lattices.

Provides asymptotic probabilities for the position of the walks.

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.