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Moderate deviations for a random walk in random scenery

Author

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

Abstract

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, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. For the case d>=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, 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 we use a recent moderate deviation result for self-intersection local times, which is due to Bass, Chen and Rosen.

Suggested Citation

  • Fleischmann, Klaus & Mörters, Peter & Wachtel, Vitali, 2008. "Moderate deviations for a random walk in random scenery," Stochastic Processes and their Applications, Elsevier, vol. 118(10), pages 1768-1802, October.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:10:p:1768-1802
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    References listed on IDEAS

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    1. Gantert, Nina & van der Hofstad, Remco & König, Wolfgang, 2006. "Deviations of a random walk in a random scenery with stretched exponential tails," Stochastic Processes and their Applications, Elsevier, vol. 116(3), pages 480-492, March.
    2. Castell, F. & Pradeilles, F., 2001. "Annealed large deviations for diffusions in a random Gaussian shear flow drift," Stochastic Processes and their Applications, Elsevier, vol. 94(2), pages 171-197, August.
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    Cited by:

    1. Xinwei Feng & Qi-Man Shao & Ofer Zeitouni, 2021. "Self-normalized Moderate Deviations for Random Walk in Random Scenery," Journal of Theoretical Probability, Springer, vol. 34(1), pages 103-124, March.
    2. Deuschel, Jean-Dominique & Fukushima, Ryoki, 2019. "Quenched tail estimate for the random walk in random scenery and in random layered conductance," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 102-128.

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    2. Xinwei Feng & Qi-Man Shao & Ofer Zeitouni, 2021. "Self-normalized Moderate Deviations for Random Walk in Random Scenery," Journal of Theoretical Probability, Springer, vol. 34(1), pages 103-124, March.
    3. Gantert, Nina & van der Hofstad, Remco & König, Wolfgang, 2006. "Deviations of a random walk in a random scenery with stretched exponential tails," Stochastic Processes and their Applications, Elsevier, vol. 116(3), pages 480-492, March.
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    5. Guillotin-Plantard, Nadine & Poisat, Julien, 2013. "Quenched central limit theorems for random walks in random scenery," Stochastic Processes and their Applications, Elsevier, vol. 123(4), pages 1348-1367.

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