Passage times of random walks and Lévy processes across power law boundaries

Abstract

We establish an integral test involving only the distribution of the increments of a random walk S which determines whether lim supn→∞(Sn/n κ ) is almost surely zero, finite or infinite when 1/2 <κ <1 and a typical step in the random walk has zero mean. This completes the results of Kesten and Maller (9) concerning finiteness of one-sided passage times over power law boundaries, so that we now have quite explicit criteria for all values of κ ≥ 0. The results, and those of (9), are also extended to Levy processes. 1. Random walks In (9) an almost complete solution was given to the problem of finding analytic conditions, expressed directly in terms of the step distribution F of the random walk S = (Sn ,n ≥ 0), for first-passage times over one-sided and two-sided power law boundaries of the random walk to be almost surely (a.s.) finite. The exception was for the one-sided passage time T ∗