Asymptotic expected number of passages of a random walk through an interval

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.