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On overshoots and hitting times for random walks

Published online by Cambridge University Press:  14 July 2016

Jean Bertoin
Jean Bertoin*
Affiliation:
Université Pierre et Marie Curie
*
∗Postal address: Laboratoire de Probabilités, Université Pierre et Marie Curie, 4, Place Jussieu, F-75252 Paris Cedex 05, France. Email address: jbe@ccr.jussieu.fr.
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Abstract

Consider an oscillating integer valued random walk up to the first hitting time of some fixed integer x > 0. Suppose there is a fee to be paid each time the random walk crosses the level x, and that the amount corresponds to the overshoot. We determine the distribution of the sum of these fees in terms of the renewal functions of the ascending and descending ladder heights. The proof is based on the observation that some path transformation of the random walk enables us to translate the problem in terms of the intersection of certain regenerative sets.

Keywords

Random walkovershootladder heightregenerative set
Type
Short Communications
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999 , pp. 593 - 600
Copyright
Copyright © Applied Probability Trust 1999 

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References

Chow, Y. S. (1986). On moments of ladder height variables. Adv. Appl. Math. 7, 46–54.Google Scholar
Doney, R. A. (1987). On the maxima of random walks and stable processes and the arc-sine law. Bull. London Math. Soc. 19, 177–182.Google Scholar
Feller, W. E. (1968). An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Feller, W. E. (1971). An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Fristedt, B. E. (1996). Intersections and limits of regenerative sets. In Random Discrete Structures, eds. Aldous, D. and Pemantle, R. Springer, Berlin, pp. 121–151.Google Scholar
Spitzer, F. (1964). Principles of Random Walks. Van Nostrand, Princeton, NJ.Google Scholar