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Moments and Distribution of the Local Times of a Transient Random Walk on ℤ d

Author

Listed:
  • Mathias Becker

    (Universität Leipzig)

  • Wolfgang König

    (Universität Leipzig)

Abstract

Consider an arbitrary transient random walk on ℤ d with d∈ℕ. Pick α∈[0,∞), and let L n (α) be the spatial sum of the αth power of the n-step local times of the walk. Hence, L n (0) is the range, L n (1)=n+1, and for integers α, L n (α) is the number of the α-fold self-intersections of the walk. We prove a strong law of large numbers for L n (α) as n→∞. Furthermore, we identify the asymptotic law of the local time in a random site uniformly distributed over the range. These results complement and contrast analogous results for recurrent walks in two dimensions recently derived by Černý (Stoch. Proc. Appl. 117:262–270, 2007). Although these assertions are certainly known to experts, we could find no proof in the literature in this generality.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:22:y:2009:i:2:d:10.1007_s10959-008-0168-4
    DOI: 10.1007/s10959-008-0168-4
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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. Inna M. Asymont & Dmitry Korshunov, 2020. "Strong Law of Large Numbers for a Function of the Local Times of a Transient Random Walk in $${\mathbb {Z}}^d$$ Z d," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2315-2336, December.

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