Moderate deviations for a random walk in random scenery

We investigate the cumulative scenery process associated with random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cramér’s condition. We prove moderate deviation principles in dimensions d≥2$d\ge 2$, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. For the case d≥4$d\ge 4$ we even obtain precise asymptotics for the probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. For d≥3$d\ge 3$, an important ingredient in the proofs are new concentration inequalities for self-intersection local times of random walks, which are of independent interest, whilst for d=2$d=2$ we use a recent moderate deviation result for self-intersection local times, which is due to Bass, Chen and Rosen.