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October, 1988 Mixing Properties for Random Walk in Random Scenery
W. Th. F. Den Hollander
Ann. Probab. 16(4): 1788-1802 (October, 1988). DOI: 10.1214/aop/1176991597

Abstract

Consider the lattice $Z^d, d \geq 1$, together with a stochastic black-white coloring of its points and on it a random walk that is independent of the coloring. A local scenery perceived at a given time is a pattern of colors seen by the walker in a finite box around his current position. Under weak assumptions on the probability distributions governing walk and coloring, we prove asymptotic independence of local sceneries perceived at times 0 and $n$, in the limit as $n\rightarrow\infty$, and at times 0 and $T_k$, in the limit as $k \rightarrow \infty$, where $T_k$ is the random $k$th hitting time of a black point. An immediate corollary of the latter result is the convergence in distribution of the interarrival times between successive black hits, i.e., of $T_{k+1} - T_k$ as $k\rightarrow\infty$. The limit distribution is expressed in terms of the distribution of the first hitting time $T_1$. The proof uses coupling arguments and ergodic theory.

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W. Th. F. Den Hollander. "Mixing Properties for Random Walk in Random Scenery." Ann. Probab. 16 (4) 1788 - 1802, October, 1988. https://doi.org/10.1214/aop/1176991597

Information

Published: October, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0651.60108
MathSciNet: MR958216
Digital Object Identifier: 10.1214/aop/1176991597

Subjects:
Primary: 60K99
Secondary: 60F05 , 60G99 , 60J15

Keywords: coupling , induced dynamical system , interarrival times , local scenery , Random walk , stochastically colored lattice , Strong mixing

Rights: Copyright © 1988 Institute of Mathematical Statistics

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Vol.16 • No. 4 • October, 1988
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