TY - JOUR

T1 - Asymptotic expected number of passages of a random walk through an interval

AU - Kella, Offer

AU - Stadje, Wolfgang

PY - 2013/3

Y1 - 2013/3

N2 - 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.

AB - 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.

KW - Generalized renewal theorem

KW - Passage

KW - Random walk

KW - Two-sided renewal theorem

UR - http://www.scopus.com/inward/record.url?scp=84879116286&partnerID=8YFLogxK

U2 - 10.1239/jap/1363784439

DO - 10.1239/jap/1363784439

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AN - SCOPUS:84879116286

SN - 0021-9002

VL - 50

SP - 288

EP - 294

JO - Journal of Applied Probability

JF - Journal of Applied Probability

IS - 1

ER -