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

Let (Z(n))(n is an element of N) be a d-dimensional random walk in random scenery, i.e., Z(n) = Sigma(n-1)(k=0) Y-Sk with (S-k)(k is an element of N0) a random walk in Z(d) and (Y-z)(z is an element of Zd) 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 is an element of 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. (C) 2005 Elsevier B.V. All rights reserved.