A Method for Computing the Asymptotic Limit of a Class of Expected First Passage Times

Abstract

The precise asymptotic behavior of certain expected first passage times plays an important role in C. Stone's theory of weak convergence of Markov processes. For a special class of random walks studied by Harris (1952), Lamperti (1962) and Karlin-McGregor (1959) we present a new method, using a maximum principle for a linear second order difference operator, that yields these asymptotic estimates. As a corollary we obtain an alternative proof of Lamperti's (1962) invariance principle.