Limit theorem for random walk in weakly dependent random scenery

TL;DR

This paper establishes a functional limit theorem for a random walk in a weakly dependent random scenery, extending the classical results to broader dependence structures.

Contribution

It generalizes Kesten and Spitzer's theorem to include weakly dependent scenery sequences, broadening the scope of limit theorems for random walks in random environments.

Findings

Proves a functional limit theorem under weak dependence conditions.

Extends classical results to more general dependence structures.

Provides a framework for analyzing random walks in complex random sceneries.

Abstract

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).