The Annals of Probability

Mixing Properties for Random Walk in Random Scenery

W. Th. F. Den Hollander

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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.

Article information

Ann. Probab., Volume 16, Number 4 (1988), 1788-1802.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60K99: None of the above, but in this section
Secondary: 60F05: Central limit and other weak theorems 60G99: None of the above, but in this section 60J15

Random walk stochastically colored lattice local scenery strong mixing interarrival times coupling induced dynamical system


Hollander, W. Th. F. Den. Mixing Properties for Random Walk in Random Scenery. Ann. Probab. 16 (1988), no. 4, 1788--1802. doi:10.1214/aop/1176991597.

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