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Sample path behaviour of Χ2 surfaces at extrema

Published online by Cambridge University Press:  01 July 2016

Michael Aronowich  and
Robert J. Adler
Michael Aronowich*
Affiliation:
Technion—Israel Institute of Technology
Robert J. Adler*
Affiliation:
Technion—Israel Institute of Technology
*
∗Postal address: Faculty of Industrial Engineering and Management, Technion—Israel Institute of Technology, Haifa 32 000, Israel.
∗Postal address: Faculty of Industrial Engineering and Management, Technion—Israel Institute of Technology, Haifa 32 000, Israel.
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Abstract

We study the sample path properties of χ2 random surfaces, in particular in the neighbourhood of their extrema. We show that, as is the case for their Gaussian counterparts, χ2 surfaces at high levels follow the form of certain deterministic paraboloids, but that, unlike their Gaussian counterparts, at low levels their form is much more random. This has a number of interesting implications in the modelling of rough surfaces and the study of the ‘robustness' of Gaussian field models. The general approach of the paper is the study of extrema via the ‘Slepian model process', which, for χ2 fields, is tractable only at asymptotically high or low levels.

Keywords

RANDOM SURFACESLEVEL CROSSINGSSAMPLE PATHSMODEL PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 4 , December 1988 , pp. 719 - 738
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

Research supported in part by AFOSR Grants 84–0104 and 85–0384.

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