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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.

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