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Time spent below a random threshold by a Poisson driven sequence of observations

Part of: Special processes Stochastic processes Distribution theory

Published online by Cambridge University Press:  14 July 2016

S. N. U. A. Kirmani  and
Jacek Wesołowski
S. N. U. A. Kirmani*
Affiliation:
University of Northern Iowa
Jacek Wesołowski*
Affiliation:
Politechnika Warszawska
*
∗Postal address: Department of Mathematics, University of Northern Iowa, Cedar Falls, IA 50614-0506, USA
∗∗Postal address: Wydział Matematyki i Nauk Informacyjnych, Politechnika Warszawska, Warszawa 00-661, Poland. Email address: [email protected]
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Abstract

The mean and the variance of the time S(t) spent by a system below a random threshold until t are obtained when the system level is modelled by the current value of a sequence of independent and identically distributed random variables appearing at the epochs of a nonhomogeneous Poisson process. In the case of the homogeneous Poisson process, the asymptotic distribution of S(t)/t as t → ∞ is derived.

Keywords

Exceedance statisticslimit theoremsnonhomogeneous Poisson processPoisson driven sequence of observationsrandom thresholdorder statistics

MSC classification

Primary: 60G55: Point processes 60G70: Extreme value theory; extremal processes 60K10: Applications (reliability, demand theory, etc.)
Secondary: 60K37: Processes in random environments 62E15: Exact distribution theory 62E20: Asymptotic distribution theory
Type
Short Communications
Information
Journal of Applied Probability , Volume 40 , Issue 3 , September 2003 , pp. 807 - 814
Copyright
Copyright © Applied Probability Trust 2003 

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