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On the maximum of a stationary independent increment process

Published online by Cambridge University Press:  14 July 2016

Sheldon M. Ross
Sheldon M. Ross*
Affiliation:
University of California, Berkeley
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Abstract

A stationary independent increment process is the continuous time analogue of the discrete random walk, and, as such, has a wide variety of applications. In this paper we consider M(t), the maximum value that such a process attains by time t. By using renewal theoretic methods we obtain results about M(t). In particular we show that if μ, the mean drift of the process, is positive, then M(t)/t converges to μ, and E[M(t + h) – M(t)] → hμ.

Keywords

STATIONARY INDEPENDENT INCREMENT PROCESSRENEWAL PROCESSLADDER HEIGHTMAXIMUM VALUE
Type
Short Communications
Information
Journal of Applied Probability , Volume 9 , Issue 3 , June 1972 , pp. 677 - 680
Copyright
Copyright © Applied Probability Trust 1972 

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References

[1] Ross, S. (1970) Applied Probability Models with Optimization Applications. Holden-Day, San Francisco.Google Scholar
[2] Heyde, C. C. (1966) Some renewal theorems with application to a first passage problem. Ann. Math. Statist. 37, 699–711.Google Scholar
[3] Rubinovitch, M. (1968) Ladder regenerative events with applications to dam models. Technical Report No. 58, Department of Operations Research, Cornell University.Google Scholar
[4] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. John Wiley, New York.Google Scholar