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

Part of: Stochastic processes Special 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: wesolo@mini.pw.edu.pl
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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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References

Bairamov, I. G., and Eryilmaz, S. N. (2000). Distributional properties of statistics based on minimal spacing and record exceedance statistics. J. Statist. Planning Infer. 90, 21–33.Google Scholar
Bairamov, I. G., and Kotz, S. (2001). On distributions of exceedances associated with order statistics and record values for arbitrary distributions. Statist. Papers 42, 171–185.Google Scholar
Eryilmaz, S. (2003). Random threshold models based on multivariate observations. J. Statist. Planning Infer. 113, 557–568.Google Scholar
Esary, G. D., Marshall, A. W., and Proschan, F. (1973). Shock models and wear processes. Ann. Prob. 1, 627–649.CrossRefGoogle Scholar
Kirmani, S. N. U. A., and Gupta, R. C. (1989). On repair age and residual repair life in the minimal repair process. Prob. Eng. Inf. Sci. 3, 381–391.Google Scholar
Nevzorov, V. B. (2001). Records: Mathematical Theory (Translations Math. Monogr. 194). American Mathematical Society, Providence, RI.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V., and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.Google Scholar
Wesołowski, J., and Ahsanullah, M. (1998). Distributional properties of exceedance statistics. Ann. Inst. Statist. Math. 50, 543–565.CrossRefGoogle Scholar