Illinois Journal of Mathematics

Random walks with occasionally modified transition probabilities

Olivier Raimond and Bruno Schapira

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Abstract

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

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

Dates
First available in Project Euclid: 24 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1348505527

Digital Object Identifier
doi:10.1215/ijm/1348505527

Mathematical Reviews number (MathSciNet)
MR2981846

Zentralblatt MATH identifier
1259.60110

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

Citation

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. https://projecteuclid.org/euclid.ijm/1348505527


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