Random Walks in a Random Environment

Abstract

Let $\{\alpha_n\}$ be a sequence of independent, identically distributed random variables with $0 \leqq \alpha_n \leqq 1$ for all $n$. The random walk in a random environment on the integers is the sequence $\{X_n\}$ where $X_0 = 0$ and inductively $X_{n+1} = X_n + 1, (X_n - 1)$, with probability $\alpha_{X_n}, (1 - \alpha_{X_n})$. In this paper we consider limit theorems for the random walk in a random environment. We show that "randomizing the environment" in some sense "slows down" the random walk in Section One. The remaining sections are concerned with features of this "slowing down" in some simple models.