On the Duration of Random Walks

Abstract

In a recent paper [1] the author investigated the mean number of steps in random walks in $n$-dimensional domains. The purpose of the present article is to generalize those results by applying similar methods to the study of the moment generating function for the number of steps and of its distribution function. As an application explicit asymptotic expressions for the variance in special cases and estimates for the likelihood of very long walks are obtained. The author wishes to express his thanks to Professor R. Fortet for many helpful discussions. The walks take place in an open bounded domain $B$ of $n$-dimensional Euclidean space $E$ with boundary $C$. A point moves in $E$ according to a given transition probability law $F(y, x)$. Here $x$ and $y$ are points of $E$ with coordinates $x_i, i = 1, 2, \cdots, n$, and $y_i, i = 1, 2, \cdots, n$, and $F(y, x)$ is the probability that a jump known to start at $x$ end at a point all of whose coordinates are less than the corresponding ones of $y$. The function $F(y, x)$ is a distribution function with respect to $y$, and it is assumed to be Borel measurable with respect to all variables. Let $N = N_x$ be the number of steps in a random walk that begins at a point $x$ of $B$ and ends with the step on which the moving point leaves $B$ for the first time. If the probability of the moving point eventually leaving $B$ is equal to one, then $N$ is a random variable. It is called the duration of the walk. It is useful to extend the definition of $N$ by setting $N_x = 0,\quad x \varepsilon E - B$.