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Extrema and level crossings of χ2 processes

Published online by Cambridge University Press:  01 July 2016

Michael Aronowich  and
Robert J. Adler
Michael Aronowich*
Affiliation:
University of the Witwatersrand
Robert J. Adler*
Affiliation:
Technion—Israel Institute of Technology
*
Postal address for both authors: Faculty of Industrial Engineering and Management Technion—Israel Institute of Technology, Technion City, Haifa 32 000, Israel.
Postal address for both authors: Faculty of Industrial Engineering and Management Technion—Israel Institute of Technology, Technion City, Haifa 32 000, Israel.
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Abstract

We study the sample path behaviour of χ2 processes in the neighbourhood of their level crossings and extrema via the development of Slepian model processes. The results, aside from being of particular interest in the study of χ2 processes, have a general interest insofar as they indicate which properties of Gaussian processes (which have been heavily researched in this regard) are mirrored or lost when the assumption of normality is not made. We place particular emphasis on the behaviour of χ2 processes at both high and low levels, these being of considerable practical importance. We also extend previous results on the asymptotic Poisson form of the point process of high maxima to include also low minima (which are in a different domain of attraction) thus closing a gap in the theory of χ2 processes.

Keywords

UPCROSSINGSSLEPIAN MODEL PROCESSPOISSON LIMITDISTRIBUTION OF MAXIMUM
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 4 , December 1986 , pp. 901 - 920
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

∗

Research supported in part by AFOSR 84–0104.

∗∗

Research supported in part by AFOSR Contract No. F49620 82 0009, while visiting Center for Stochastic Processes, Chapel Hill, North Carolina.

References

Abramowitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions. Dover, New York.Google Scholar
Adler, R. J. (1981) The Geometry of Random Fields. Wiley, New York.Google Scholar
Aronowich, M. and Adler, R. J. (1985) Behaviour of ?2 processes at extrema. Adv. Appl. Prob 17, 280–297.CrossRefGoogle Scholar
Berman, S. (1983) Sojourns of stationary processes in rare sets. Ann. Prob. 11, 847–866.CrossRefGoogle Scholar
Berman, S. (1984) Sojourns of vector Gaussian processes inside and outside spheres. Z. Wahrscheinlichkeitsth. 66, 529–542.CrossRefGoogle Scholar
Cramér, H. and Leadbetter, M. R. (1967) Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
Dudley, R. M. (1973) Sample functions of the Gaussian process. Ann. Prob. 1, 66–103.CrossRefGoogle Scholar
Gradshteyn, I. S. and Ryzhik, I. M. (1980) Table of Integrals, Series, and Products. Academic Press, New York.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) Extremal and Related Properties of Stationary Processes. Springer-Verlag, New York.Google Scholar
Lindgren, G. (1970) Some properties of a normal process near a local maximum. Ann. Math. Statist. 41, 1870–1883.Google Scholar
Lindgren, G. (1980a) Point processes of exits by bivariate Gaussian processes and extremal theory for the ?2-process and its concomitants. J. Multivariate Anal. 10, 181–206.CrossRefGoogle Scholar
Lindgren, G. (1980b) Extreme values and crossings for the ?2-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.CrossRefGoogle Scholar
Lindgren, G. (1984) Use and structure of Slepian model processes for prediction and detection in crossing and extreme value theory. In Statistical Extremes and Applications , Ed. Tiago de Oliveira, J. Reidel, Dordrecht.Google Scholar
Lukacs, E. and Laha, R. G. (1964) Applications of Characteristic Functions. Griffin, London.Google Scholar
Miller, K. S. (1980) An Introduction to Vector Stochastic Processes. Krieger, New York.Google Scholar
Sharpe, K. (1978) Some properties of the crossings process generated by a stationary ?2-process. Adv. Appl. Prob. 10, 373–391.Google Scholar