Dynamic ℤ d-random walks in a random scenery: a strong law of large numbers

Abstract

In this paper, we study a ℤ d -random walk ( S n ) n∈ℕ on nearest neighbours with transition probabilities generated by a dynamical system S=( E, A, μ, T). We prove, at first, that under some hypotheses, ( S n ) n∈ℕ verifies a local limit theorem. Then, we study these walks in a random scenery ( ξ x ) x ∈ℤ d, a sequence of i.i.d. centered random variables and show that for certain dynamic random walks, ( ξ s κ ) κ ≥0 satisfies a strong law of large numbers.