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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: [email protected]
∗∗ Postal address: Institute of Mathematics, University of Osnabrück, 49069 Osnabrück, Germany. Email address: [email protected]
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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

Asmussen, S. (2003). Applied Probability and Queues (Appl. Math. 51), 2nd edn. Springer, New York.Google Scholar
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