ASYMPTOTIC EXPECTED NUMBER OF PASSAGES OF A RANDOM WALK THROUGH AN INTERVAL

Abstract

In this note we find a new result concerning the asymptotic expected number of passages of a finite or infinite interval (x, x h] as x -> infinity for a random walk with increments having a positive expected value. If the increments are distributed like X then the limit for 0 < h < infinity turns out to have the form E min(vertical bar X vertical bar, h)/EX, which unexpectedly is independent of h for the special case where vertical bar X vertical bar <= b < infinity almost surely and h > b. When h = infinity, the limit is E max(X, 0)/EX. For the case of a simple random walk, a more pedestrian derivation of the limit is given.