## The Annals of Probability

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

### Mixing Properties for Random Walk in Random Scenery

#### 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