Deviations of a Random Walk in a Random Scenery with Stretched Exponential Tails

Let (Zn)n∈N0 be a d-dimensional random walk in random scenery, i.e., Zn = ∑n−1 k=0 YSk with (Sk)k∈N0 a random walk in Z d and (Yz)z∈Zd an i.i.d. scenery, independent of the walk. We assume that the random variables Yz 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( 1 nZn > tn) for all sequences (tn)n∈N satisfying a certain lower bound. This complements results of [GKS05], where it was assumed that Yz has exponential moments of all orders. In contrast to the situation [GKS05], the event { 1 nZn > tn} 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. MSC 2000. 60K37, 60F10, 60J55.