A Ratio Limit Theorem for Subterminal Times

Abstract

Consider a recurrent random walk $\{X_n\}$ with state space $S \subseteqq R^d (d \leqq 2)$. A stopping time $T$ is called subterminal if it satisfies a technical condition which essentially states that it is the first time a path possesses some property which does not depend on how long the process has been running. Suppose $T$ is a subterminal time which can occur only when $\{X_n\}$ is in a bounded set; then under an additional assumption a ratio limit theorem (as $n \rightarrow \infty$) is obtained for $P(T > n \mid X_0 = x) (x\in S)$. The theorem applies in particular to the hitting time of a bounded set with nonempty interior in the general case, and to the hitting time of a bounded set with nonzero Haar measure in the nonsingular case.