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Self-normalized Moderate Deviations for Random Walk in Random Scenery

Author

Listed:
  • Xinwei Feng

    (Shandong University)

  • Qi-Man Shao

    (Southern University of Science and Technology
    The Chinese University of Hong Kong)

  • Ofer Zeitouni

    (Weizmann Institute of Science
    New York University)

Abstract

Let $$\{S_k:k\ge 0\}$$ { S k : k ≥ 0 } be a symmetric and aperiodic random walk on $$\mathbb {Z}^d$$ Z d , $$d\ge 3$$ d ≥ 3 , and $$\{\xi (z),z\in \mathbb {Z}^d\}$$ { ξ ( z ) , z ∈ Z d } a collection of independent and identically distributed random variables. Consider a random walk in random scenery defined by $$T_n=\sum _{k=0}^n\xi (S_k)=\sum _{z\in \mathbb {Z}^d}l_n(z)\xi (z)$$ T n = ∑ k = 0 n ξ ( S k ) = ∑ z ∈ Z d l n ( z ) ξ ( z ) , where $$l_n(z)=\sum _{k=0}^nI{\{S_k=z\}}$$ l n ( z ) = ∑ k = 0 n I { S k = z } is the local time of the random walk at the site z. Using $$(\sum _{z\in \mathbb {Z}^d}l_n(z)|\xi (z)|^p)^{1/p}$$ ( ∑ z ∈ Z d l n ( z ) | ξ ( z ) | p ) 1 / p , $$p\ge 2$$ p ≥ 2 , as the normalizing constants, we establish self-normalized moderate deviations for random walk in random scenery under a much weaker condition than a finite moment-generating function of the scenery variables.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:jotpro:v:34:y:2021:i:1:d:10.1007_s10959-019-00965-2
    DOI: 10.1007/s10959-019-00965-2
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    References listed on IDEAS

    as
    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. Qi-Man Shao, 1999. "A Cramér Type Large Deviation Result for Student's t-Statistic," Journal of Theoretical Probability, Springer, vol. 12(2), pages 385-398, April.
    3. Mathias Becker & Wolfgang König, 2009. "Moments and Distribution of the Local Times of a Transient Random Walk on ℤ d," Journal of Theoretical Probability, Springer, vol. 22(2), pages 365-374, June.
    4. 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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