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Asymptotic Expected Number of Passages of a Random Walk Through an Interval

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Offer Kella  and
Wolfgang Stadje
Offer Kella*
Affiliation:
The Hebrew University of Jerusalem
Wolfgang Stadje*
Affiliation:
University of Osnabrück
*
∗ Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel. Email address: offer.kella@huji.ac.il
∗∗ Postal address: Institute of Mathematics, University of Osnabrück, 49069 Osnabrück, Germany. Email address: wstadje@uos.de
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Abstract

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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.

Keywords

Random walkpassagegeneralized renewal theoremtwo-sided renewal theorem

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60K05: Renewal theory
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 288 - 294
Copyright
Copyright © Applied Probability Trust 2013 

Footnotes

Supported in part by grant no. 434/09 from the Israel Science Foundation and the Vigevani Chair in Statistics.

Supported by grant no. 306/13-2 of the Deutsche Forschungsgemeinschaft.

References

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Breiman, L. (1992). Probability (Classics Appl. Math. 7). Society for Industrial and Applied Mathematics, Philadelphia, PA.CrossRefGoogle Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar