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

Author

Listed:
  • Gantert, Nina
  • van der Hofstad, Remco
  • König, Wolfgang

Abstract

Let be a d-dimensional random walk in random scenery, i.e., with a random walk in and an i.i.d. scenery, independent of the walk. We assume that the random variables Yz 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 for all sequences 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 Yz has exponential moments of all orders. In contrast to the situation (Gantert et al., 2005), the event {Zn>ntn} 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.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:3:p:480-492
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    References listed on IDEAS

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    1. 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.
    2. Chen, Xia, 2001. "Moderate deviations for Markovian occupation times," Stochastic Processes and their Applications, Elsevier, vol. 94(1), pages 51-70, July.
    3. Asselah, A. & Castell, F., 2003. "Quenched large deviations for diffusions in a random Gaussian shear flow drift," Stochastic Processes and their Applications, Elsevier, vol. 103(1), pages 1-29, January.
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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.
    3. 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.

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