WIAS Preprint No. 1167, (2006)

Moderate deviations for random walk in random scenery


  • Fleischmann, Klaus
  • Mörters, Peter
  • Wachtel, Vitali

2010 Mathematics Subject Classification

  • 60F10 60K37


  • Moderate deviation principles, self-intersection local times, concentration inequalities, large deviations, moderate deviation regimes, maximum of local times, precise asymptotics, annealed probabilities, Cramér's condition




We investigate 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 $dge 2$, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. In the case $dge 4$ we even obtain precise asymptotics for the annealed probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. In $dge 3$, an important ingredient in the proofs are new concentration inequalities for self-intersection local times of random walks, which are of independent interest, whilst in $d=2$ we use a recent moderate deviation result for self-intersection local times, which is due to Bass, Chen and Rosen.

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