Conditional Limit Theorem for the Maximum of a Random Walk in Random Environment

Abstract

Let $(p_{i},q_{i}) $, $i\in {\bf Z}$, be a sequence of independent identically distributed pairs of random variables, where $p_{0}+q_{0}=1$ and, in addition, $p_{0}>0$ and $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, given the environment, a walking particle located at some moment $n$ at a state $i$ jumps at the moment $n+1$ either to the state $(i+1)$ with probability $p_{i}$ or to the state $(i-1)$ with probability $q_{i}$. It is assumed that the random variable $\log ( q_{0}/p_{0}) $ has zero mean and finite positive variance $\sigma^{2} $. Let $X_{n}$ be the state of working particle at the moment $n$. It is shown that given $X_{1}\ge 0,\ldots ,X_{n}\ge 0,$ the random variable $\sigma^{2}\max_{0\le i\le n}X_{i}/\log^{2}n$ converges in distribution, as $n\to \infty $ to a positive random variable whose distribution function is known.