# Random walks with occasionally modified transition probabilities

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 $\Z$ by modifying the distribution of a step from a fresh point. If the process is denoted as $\{S_n\}_{n \ge 0}$, 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 $P\{S_{n+1}-S_n = \pm 1|S_\ell, \ell \le n\} = 1/2$. 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 $(1/n)S_n \to 0$ in probability. Generalizations or variants of the $P_i$ and the rules for switching between the $P_i$ are also considered.
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Preprints, Working Papers, ...
previous Section 2 removed, to appear in Illinois J. Math. 2012
Domain :

https://hal.archives-ouvertes.fr/hal-00433533
Contributor : Bruno Schapira <>
Submitted on : Wednesday, April 11, 2012 - 11:09:29 AM
Last modification on : Wednesday, April 11, 2012 - 12:28:56 PM

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### Identifiers

• HAL Id : hal-00433533, version 2
• ARXIV : 0911.3886

### Citation

Olivier Raimond, Bruno Schapira. Random walks with occasionally modified transition probabilities. previous Section 2 removed, to appear in Illinois J. Math. 2012. <hal-00433533v2>

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