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Dynamic ℤ d -Random Walks in a Random Scenery: A Strong Law of Large Numbers

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  • N. Guillotin-Plantard

    (Université Claude Bernard, Lyon I)

Abstract

In this paper, we study a ℤ d -random walk $$(S_n )_{n{\text{ }} \in {\text{ }}\mathbb{N}}$$ on nearest neighbours with transition probabilities generated by a dynamical system $$S = (E,\mathcal{A},\mu ,T)$$ . We prove, at first, that under some hypotheses, $$(S_n )_{n{\text{ }} \in {\text{ }}\mathbb{N}}$$ verifies a local limit theorem. Then, we study these walks in a random scenery $$(\xi _x )_{x \in \mathbb{Z}^d }$$ , a sequence of independent, identically distributed and centred random variables and show that for certain dynamic random walks, $$(\xi _{S_k } )_{k \geqslant 0}$$ satisfies a strong law of large numbers.

Suggested Citation

  • N. Guillotin-Plantard, 2001. "Dynamic ℤ d -Random Walks in a Random Scenery: A Strong Law of Large Numbers," Journal of Theoretical Probability, Springer, vol. 14(1), pages 241-260, January.
    • Handle: RePEc:spr:jotpro:v:14:y:2001:i:1:d:10.1023_a:1007885418401
    DOI: 10.1023/A:1007885418401
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    References listed on IDEAS

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    1. Csáki, Endre & König, Wolfgang & Shi, Zhan, 1999. "An embedding for the Kesten-Spitzer random walk in random scenery," Stochastic Processes and their Applications, Elsevier, vol. 82(2), pages 283-292, August.
    2. Khoshnevisan, Davar & Lewis, Thomas M., 1998. "A law of the iterated logarithm for stable processes in random scenery," Stochastic Processes and their Applications, Elsevier, vol. 74(1), pages 89-121, May.
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