Random walks with occasionally modified transition probabilities

TL;DR

This paper investigates the recurrence and law of large numbers for a modified random walk on integers, where the transition probabilities depend on whether the current position has been visited before, exploring conditions for recurrence and convergence.

Contribution

It introduces a model of random walk with position-dependent transition probabilities based on visitation history, analyzing recurrence and convergence properties.

Findings

Conditions for infinite recurrence to the origin.

Criteria for the law of large numbers to hold.

Impact of transition probability modifications on walk behavior.

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.