The Annals of Probability

Mixing Properties for Random Walk in Random Scenery

W. Th. F. Den Hollander

Full-text: Open access

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.

Article information

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

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991597

Digital Object Identifier
doi:10.1214/aop/1176991597

Mathematical Reviews number (MathSciNet)
MR958216

Zentralblatt MATH identifier
0651.60108

JSTOR
links.jstor.org

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.aop/1176991597


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