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→∞ for a random walk with increments having a positive expected value. If the increments are distributed like X then the limit for 0<h<∞ turns out to have the form Emin(|X|,h)/EX, which unexpectedly is independent of h for the special case where |X|≤b<∞ almost surely and h>b. When h=∞, the limit is Emax(X,0)/EX. For the case of a simple random walk, a more pedestrian derivation of the limit is given.