Dynamic Z^d -Random Walks in a Random Scenery: A Strong Law of Large Numbers

In this paper, we study a Z^d -random walk (S n)n(element of)N on nearest neighbours with transition probabilities generated by a dynamical system S=(E,A,(mu),T). We prove, at first, that under some hypotheses, (S n)n(element of)N verifies a local limit theorem. Then, we study these walks in a random scenery ((zeta) x)x(element of)Z^d , a sequence of independent, identically distributed and centred random variables and show that for certain dynamic random walks, ((zeta)S k)k>=0 satisfies a strong law of large numbers.