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Model fields in crossing theory: a weak convergence perspective

Published online by Cambridge University Press:  01 July 2016

Richard J. Wilson
Richard J. Wilson*
Affiliation:
University of Queensland
*
∗Postal address: Department of Mathematics, University of Queensland, St. Lucia, QLD 4067, Australia.
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Abstract

In this paper, the behavior of a Gaussian random field near an ‘upcrossing' of a fixed level is investigated by strengthening the results of Wilson and Adler (1982) to full weak convergence in the space of functions which have continuous derivatives up to order 2. In Section 1, weak convergence and model processes are briefly discussed. The model field of Wilson and Adler (1982) is constructed in Section 2 using full weak convergence. Some of its properties are also investigated. Section 3 contains asymptotic results for the model field, including the asymptotic distributions of the Lebesgue measure of a particular excursion set and the maximum of the model field as the level becomes arbitrarily high.

Keywords

TIGHTNESSSMOOTH FUNCTIONSHOMOGENEOUSGAUSSIAN RANDOM FIELDUPCROSSINGHORIZONTAL WINDOW CONDITIONINGPALM DISTRIBUTIONERGODICEXCURSION SETMAXIMUM
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 4 , December 1988 , pp. 756 - 774
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

Research supported in part by AFOSR Grant No. F49620 85 C 0144 while the author was visiting the Center for Stochastic Processes, University of North Carolina, Chapel Hill.

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