Random walks with occasionally modified transition probabilities
Illinois Journal of Mathematics(2009)
摘要
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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关键词
transition probability,random walk,conditional distribution,law of large numbers,discrete time
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