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Slepian models for X2-processes with dependent components with application to envelope upcrossings

Published online by Cambridge University Press:  14 July 2016

Georg Lindgren
Georg Lindgren*
Affiliation:
University of Lund
*
∗Postal address: Department of Mathematical Statistics, University of Lund, Box 118, S-221 00 Lund, Sweden.
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Abstract

A Slepian model for the local behaviour near the level upcrossings of a x2-process with dependent Gaussian components is presented. In case of independent components, this model is shown to take on a rather simple form, thereby simplifying earlier results by Aronowich and Adler.

The Slepian model is applied to the envelope of a stationary Gaussian process and used to approximate the probability of ‘empty' envelope upcrossings, i.e. the probability that an envelope upcrossing is not followed by a level crossing in the original process.

Keywords

LEVEL CROSSINGRELIABILITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 1 , March 1989 , pp. 36 - 49
Copyright
Copyright © Applied Probability Trust 1989 

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Footnotes

Part of this work was carried out during a visit to the Department of Structural Engineering, the Technical University of Denmark, Lyngby.

References

Adler, R. J. (1981) The Geometry of Random Fields. Wiley, New York.Google Scholar
Albin, P. (1986) On extremal theory for low levels of stationary nondifferentiable x2-processes. University of Lund, Statistical Research Report 1986:8, 1–33.Google Scholar
Albin, P. (1987) On extremal theory for nondifferentiable stationary processes. University of Lund, Dept. of Math. Statist., Ph.D. thesis, October 1987, 1–114.Google Scholar
Aronowich, M. and Adler, R. J. (1985) Behaviour of x2-processes at extrema. Adv. Appl. Prob. 17, 280–297.CrossRefGoogle Scholar
Aronowich, M. and Adler, R. J. (1986) Extrema and level crossing of x2-processes. Adv. Appl. Prob. 18, 901–920.Google Scholar
Cramer, H. and Leadbetter, M.-R. (1967) Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
Ditlevsen, O. and Lindgren, G. (1988) Empty envelope excursions in stationary Gaussian processes. J. Sound and Vibration 122, 571–587.Google Scholar
Kac, M. and Slepian, D. (1959) Large excursions of Gaussian processes, Ann. Math. Statist. 30, 1215–1228.CrossRefGoogle Scholar
Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.Google Scholar
Leadbetter, M. R. and Rootzen, H. (1988) Extremal theory for stochastic proceses. Ann. Prob. 16, 431–478.Google Scholar
Lindgren, G. (1980a) Point process of exits by bivariate Gaussian processes and extremal theory for the x2-process and its concomitants. J. Multivariate Anal. 10, 181–206.CrossRefGoogle Scholar
Lindgren, G. (1980b) Extreme values and crossings for the x2-process and other functions of multidimensional Gaussian processes with reliability applications. Adv. Appl. Prob. 12, 746–774.Google Scholar
Lindgren, G. (1980c) Model processes in nonlinear prediction with application to detection and alarm. Ann. Prob. 8, 775–792.Google Scholar
Sharpe, K. (1978) Some properties of the crossings process generated by a stationary x2-process. Adv. Appl. Prob. 10, 373–391.Google Scholar
Vanmarcke, E. H. (1975) On the distribution of the first-passage time for normal stationary random processes. J. Appl. Mech. ASME 42, 215–220.Google Scholar