Deviations of a random walk in a random scenery with stretched exponential tails

Abstract

Let ( Z n ) n ∈ N be a d-dimensional random walk in random scenery, i.e., Z n = ∑ k = 0 n - 1 Y S k with ( S k ) k ∈ N 0 a random walk in Z d and ( Y z ) z ∈ Z d an i.i.d. scenery, independent of the walk. We assume that the random variables Y z have a stretched exponential tail. In particular, they do not possess exponential moments. We identify the speed and the rate of the logarithmic decay of P ( Z n > nt n ) for all sequences ( t n ) n ∈ N satisfying a certain lower bound. This complements results of Gantert et al. [Annealed deviations of random walk in random scenery, preprint, 2005], where it was assumed that Y z has exponential moments of all orders. In contrast to the situation (Gantert et al., 2005), the event { Z n > nt n } is not realized by a homogeneous behavior of the walk's local times and the scenery, but by many visits of the walker to a particular site and a large value of the scenery at that site. This reflects a well-known extreme behavior typical for random variables having no exponential moments.