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

Research paper by **Remco van der Hofstad, Nina Gantert, Wolfgang K{ö}nig**

Indexed on: **09 Aug '05**Published on: **09 Aug '05**Published in: **Mathematics - Probability**

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#### Abstract

Let (Z_n)_{n\in\N_0} be a d-dimensional random walk in random scenery, i.e.,
Z_n=\sum_{k=0}^{n-1}Y_{S_k} with (S_k)_{k\in\N_0} a random walk in Z^d and
(Y_z)_{z\in 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 Pr(Z_n>t_n n) for all sequences
(t_n)_{n\in\N} satisfying a certain lower bound. This complements previous
results, where it was assumed that Y_z has exponential moments of all orders.
In contrast to the previous situation,the event \{Z_n>t_nn\} 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.