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.
"A Method for Computing the Asymptotic Limit of a Class of Expected First Passage Times." Ann. Probab. 1 (6) 1035 - 1043, December, 1973. https://doi.org/10.1214/aop/1176996809