Asymptotics of a dynamic random walk in a random scenery: I. Law of large numbers

Abstract

In this paper, we consider a Z -random walk (S n) n∈ N on nearest neighbours with dynamical quasiperiodic transition probabilities in a random scenery ξ( α), α∈ Z , a family of i.i.d. random variables, independent of the random walk. We prove, at first, that (S n) n∈ N verifies a local limit theorem and is recurrent on its moving average. Then, we show, explicitly, that Z n= ∑ i=0 n ξ(S i) satisfies a law of large numbers.