Limit theorem for random walk in weakly dependent random scenery

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).