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 an 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 indpendent 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 and is observed to be consistent with the general results.