Illinois Journal of Mathematics

Random walks with occasionally modified transition probabilities

Olivier Raimond and Bruno Schapira

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We study recurrence properties and the validity of the (weak) law of large numbers for (discrete time) processes which, in the simplest case, are obtained from simple symmetric random walk on $\mathbb{Z}$ by modifying the distribution of a step from a fresh point. If the process is denoted as $\{S_n\}_{n \ge0}$, then the conditional distribution of $S_{n+1} - S_n$ given the past through time $n$ is the distribution of a simple random walk step, provided $S_n$ is at a point which has been visited already at least once during $[0,n-1]$. Thus, in this case, $P\{S_{n+1}-S_n = \pm1|S_\ell, \ell\le n\} = 1/2$. We denote this distribution by $P_1$. However, if $S_n$ is at a point which has not been visited before time $n$, then we take for the conditional distribution of $S_{n+1}-S_n$, given the past, some other distribution $P_2$. We want to decide in specific cases whether $S_n$ returns infinitely often to the origin and whether $(1/n)S_n \to0$ in probability. Generalizations or variants of the $P_i$ and the rules for switching between the $P_i$ are also considered.

Article information

Illinois J. Math., Volume 54, Number 4 (2010), 1213-1238.

First available in Project Euclid: 24 September 2012

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]


Raimond, Olivier; Schapira, Bruno. Random walks with occasionally modified transition probabilities. Illinois J. Math. 54 (2010), no. 4, 1213--1238. doi:10.1215/ijm/1348505527.

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