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 $$(S_n )_{n{\text{ }} \in {\text{ }}\mathbb{N}}$$ on nearest neighbours with transition probabilities generated by a dynamical system $$S = (E,\mathcal{A},\mu ,T)$$ . We prove, at first, that under some hypotheses, $$(S_n )_{n{\text{ }} \in {\text{ }}\mathbb{N}}$$ verifies a local limit theorem. Then, we study these walks in a random scenery $$(\xi _x )_{x \in \mathbb{Z}^d }$$ , a sequence of independent, identically distributed and centred random variables and show that for certain dynamic random walks, $$(\xi _{S_k } )_{k \geqslant 0}$$ satisfies a strong law of large numbers.