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

We establish an integral test involving only the distribution of the increments of a random walk S which determines whether lim sup _n→∞(S_n/n^k) is almost surely zero, finite or infinite when 1/2 < k < 1 and a typical step in the random walk has zero mean. This completes the results of Kesten and Mailer [9] concerning finiteness of one-sided passage times over power law boundaries, so that we now have quite explicit criteria for all values of k ≥ 0. The results, and those of [9], are also extended to Lévy processes.