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High-Level exceedances of regenerative and semi-stationary processes

Published online by Cambridge University Press:  14 July 2016

Richard Serfozo
Richard Serfozo*
Affiliation:
Bell Laboratories
*
∗Postal address: Bell Laboratories, WB–1G 306, Holmdel, N.J. 07733, U.S.A.
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Abstract

The cumulative amount of time that a regenerative or semi-stationary process exceeds a high level and other measures of these exceedances are considered as special cases of a non-decreasing stochastic process of partial sums. We present necessary and sufficient conditions for these exceedance processes to converge in distribution to Poisson processes or processes with stationary independent non-negative increments as the level goes to infinity. We apply our results to random walks, M/M/s queues, and thinnings of point processes.

Keywords

HIGH-LEVEL EXCEEDANCESMARKOV CHAINSREGENERATIVE PROCESSESSEMI-STATIONARY PROCESSESWEAK CONVERGENCEM/M/s QUEUETHINNING OF POINT PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 2 , June 1980 , pp. 423 - 431
Copyright
Copyright © Applied Probability Trust 1980 

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Footnotes

Part of this research was done at Syracuse University under the sponsorship of Air Force Office of Scientific Research Grant #AFOSR–74–2627, and National Science Foundation Grant #ENG75–13653.

References

Berman, S. (1971) Asymptotic independence of the number of high and low level crossings of stationary Gaussian processes. Ann. Math. Statist. 42, 927–945.CrossRefGoogle Scholar
Berman, S. (1978) A compound Poisson limit for stationary sums, and sojourns of Gaussian processes. Technical Report, New York University.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities, Springer-Verlag, New York.Google Scholar
Cramer, H. and Leadbetter, M. (1967) Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. I, 3rd edn. Wiley, New York.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Jagers, P. (1972) On the weak convergence of superpositions of point processes. Z. Wahrscheinlichkeitsth. 22, 1–7.CrossRefGoogle Scholar
Jagers, P. and Lindvall, T. (1974) Thinning and rare events in point processes. Z. Wahrscheinlichkeitsth. 28, 89–98.CrossRefGoogle Scholar
Kallenberg, O. (1975) Random Measures. Akademie-Verlag, Berlin, (Academic Press, New York (1976).) Google Scholar
Kedem, B. (1978) Sufficiency and the number of level crossings by a stationary process. Biometrika 65, 207–210.CrossRefGoogle Scholar
Keilson, J. (1966) A limit theorem for passage times in ergodic processes. Ann. Math. Statist. 37, 866–870.CrossRefGoogle Scholar
Leadbetter, M. (1976) Weak convergence of high-level exceedances of a stationary sequence. Z. Wahrscheinlichkeitsth. 34, 11–15.CrossRefGoogle Scholar
Lindvall, T. (1980) An invariance principle for thinned random measures. Stud. Scient. Math. Hungar. Tomas X, Fasc. 3–4. To appear.Google Scholar
Meyer, R. M. (1973) A Poisson-type limit theorem for mixing sequences of dependent events. Ann. Prob. 1, 480–483.CrossRefGoogle Scholar
Serfozo, R. (1972) Semi-stationary processes. Z. Wahrscheinlichkeitsth. 23, 125–132.CrossRefGoogle Scholar
Serfozo, R. (1975) Functional limit theorems for stochastic processes based on embedded processes. Adv. Appl. Prob. 7, 123–139.CrossRefGoogle Scholar
Serfozo, R. (1976) Compositions, inverses and thinnings of random measures. Z. Wahrscheinlichkeitsth. 37, 253–265.CrossRefGoogle Scholar
Whitt, W. (1980) Some useful functions for functional limit theorems. Math. Operat. Res. To appear.CrossRefGoogle Scholar
Wichura, M. (1974) On the functional form of the law of the iterated logarithm for the partial maxima of independent identically distributed random variables. Ann. Prob. 2, 202–230.CrossRefGoogle Scholar