Limit theorems for one and two-dimensional random walks in random scenery

TL;DR

This paper establishes limit theorems for one- and two-dimensional random walks in random scenery, focusing on cases where the stability index matches the dimension, extending previous results to new parameter regimes.

Contribution

It proves convergence of finite-dimensional distributions and local limit theorems for cases where the stability index equals the dimension, filling gaps in existing literature.

Findings

Convergence of finite-dimensional distributions for the specified cases.

Local limit theorem established for the cases where α=d.

Extends previous functional limit theorems to new parameter regimes.

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 $(\xi_y,y\in{\mathbb Z}^d)$ are two independent sequences of i.i.d. random variables with values in ${\mathbb Z}^d$ and $\mathbb R$ respectively. We suppose that the distributions of $X_1$ and $\xi_0$ belong to the normal basin of attraction of stable distribution of index $\alpha\in(0,2]$ and $\beta\in(0,2]$. When $d=1$ and $\alpha\ne 1$, a functional limit theorem has been established in \cite{KestenSpitzer} and a local limit theorem in \cite{BFFN}. In this paper, we establish the convergence of the finite-dimensional distributions and a local limit theorem when $\alpha=d$ (i.e. $\alpha = d=1$ or $\alpha=d=2$) and $\beta \in (0,2]$. Let us mention that functional limit theorems have been established in \cite{bolthausen} and recently in \cite{DU} in the particular case where $\beta=2$ (respectively for $\alpha=d=2$ and $\alpha=d=1$).