Moment Convergence of Reciprocals of some First Passage Times

Synoptic Abstract

Let T_c be the first passage time above the level c in a random walk. When the random walk has a positive drift, there are many studies about the limiting behavior of ET_C as c → ∞. However, for driftless random walks, ET_c is infinite for every positive c and ET_c yields little information. To remedy this situation, this paper is devoted to the studies of the quantity E(l/T_c) as c → ∞ for random walks with nonnegative drifts. An example is given as an application.

Key Words and Phrases:

• Uniformly integrable
• first passage times
• asymptotic efficiency