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The behavior near the origin of the supremum functional in a process with stationary independent increments

Published online by Cambridge University Press:  14 July 2016

Sidney I. Resnick  and
Michael Rubinovitch
Sidney I. Resnick
Affiliation:
Stanford University
Michael Rubinovitch
Affiliation:
Technion-Israel Institute of Technology
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Abstract

Let {X(t),t ≧0} be a process with stationary independent increments which is stochastically continuous with right-continuous paths and normalized so that X(0)=0. Let Z1(t) = X(t), Z2(t) = sup0≦s≦tX(s) and Z3 (t) = largest positive jump of X in (0, t] if there is one; = 0 otherwise. Then for i = 1,2,3 and x > 0: limt↓0t—1P[Zi(t) > x] = M+(x) at all points of continuity of M+, the Lévy measure of X.

Keywords

PROCESS WITH STATIONARY INDEPENDENT INCREMENTSEXTREMAL PROCESSSUPREMUM FUNCTIONAL
Type
Short Communications
Information
Journal of Applied Probability , Volume 12 , Issue 1 , March 1975 , pp. 159 - 160
Copyright
Copyright © Applied Probability Trust 1975 

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References

[1] Breiman, L. (1968) Probability , Addison-Wesley. Reading, Mass. Google Scholar
[2] Feller, W. (1970) An Introduction to Probability Theory and its Applications . Vol. 2, Wiley, New York.Google Scholar
[3] Resnick, S. I. and Rubinovitch, M. (1973) The structure of extremal processes. Adv. Appl. Prob. 5, 287–307.Google Scholar