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

N. Guillotin-Plantard ()
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N. Guillotin-Plantard: Université Claude Bernard, Lyon I

Journal of Theoretical Probability, 2001, vol. 14, issue 1, 241-260

Abstract: 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.

Keywords: random walk; random scenery; ergodic theory; Denjoy–Koksma's inequality; low discrepancy sequences (search for similar items in EconPapers)
Date: 2001
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DOI: 10.1023/A:1007885418401

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