On the Time of Reaching a High Level by a Transient Random Walk in a Random Environment

Abstract

Let a sequence of independent identically distributed pairs of random variables $(p_{i},q_{i}) $, $i\in {\bf Z}$, be given, with ${p_{0}+q_{0}=1}$ and $p_{0}>0$, $q_{0}>0$ a.s. We consider a random walk in the random environment $(p_{i},q_{i}) $, $i\in {\bf Z}$. This means that under a fixed environment a walking particle located at some moment in a state $i$ jumps either to the state $(i+1) $ with probability $p_{i}$ or to the state $(i-1) $ with probability $q_{i}$. It is assumed that ${\bf E}\,\log (p_{0}/q_{0}) <0$, i.e., that the random walk tends with time to $-\infty$. The set of such random walks may be divided into three types according to the value of the quantity ${\bf E}\,((p_{0}/q_{0}) \log (p_{0}/q_{0}))$. In the case when the expectation above is zero we prove a limit theorem as $n\to \infty $ for the of time distribution of reaching the level $n$ by the mentioned random walk.