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Some properties of the crossings process generated by a stationary χ2 process

Published online by Cambridge University Press:  01 July 2016

Ken Sharpe
Ken Sharpe*
Affiliation:
University of Melbourne
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Abstract

The process generated by the crossings of a fixed level, u, by the process Pn(t) is considered, where

and the Xi(t) are identical, independent, separable, stationary, zero mean, Gaussian processes. A simple formula is obtained for the expected number of upcrossings in a given time interval, sufficient conditions are given for the upcrossings process to tend to a Poisson process as u→∞, and it is shown that under suitable scaling the distribution of the length of an excursion of Pn(t) above u tends to a Rayleigh distribution as u→ ∞.

Keywords

χ2 PROCESSCROSSINGSGAUSSIAN PROCESSPOISSON PROCESSLENGTH OF EXCURSIONSRAYLEIGH DISTRIBUTION
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 2 , June 1978 , pp. 373 - 391
Copyright
Copyright © Applied Probability Trust 1978 

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References

Berman, S. M. (1971a) Excursions above high levels for stationary Gaussian processes. Pacific J. Maths 36, 63–80.Google Scholar
Berman, S. M. (1971b) Asymptotic independence of the number of high and low level crossings of stationary Gaussian processes. Ann. Math. Statist. 42, 927–945.Google Scholar
Berman, S. M. (1974) Sojourns and extremes of Gaussian processes. Ann. Prob. 2, 999–1026.Google Scholar
Cramér, H. and Leadbetter, M. R. (1967) Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
Hasofer, A. M. (1974) The upcrossings rate of a class of stochastic processes. In Studies in Probability and Statistics, ed. Williams, E. J., Jerusalem Academic Press.Google Scholar
Kac, M. and Slepian, D. (1959) Large excursions of Gaussian processes. Ann. Math. Statist. 30, 1215–1228.Google Scholar
Leadbetter, M. R. (1966) On crossings of levels and curves by a wide class of stochastic process. Ann. Math. Statist. 37, 260–267.Google Scholar
Leadbetter, M. R. (1972) Point processes generated by level crossings. In Stochastic Point Processes, ed. Lewis, P. A. W., Wiley, New York.Google Scholar
Marcus, M. B. (1977) Level crossings of a stochastic process with absolutely continuous sample paths. Ann. Prob. 5, 52–71.CrossRefGoogle Scholar
Rice, S. O. (1945) Mathematical analysis of random noise. Bell Syst. Tech. J. 19, 46–156.Google Scholar
Volkonski, V. A. and Rozanov, Yu. A. (1959), (1961) Some limit theorems for random functions I and II. Theory Prob. Appl. 4, 178–197; 6, 186–198.CrossRefGoogle Scholar