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On the capacity functional of excursion sets of Gaussian random fields on ℝ2

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  19 September 2016

Marie Kratz  and
Werner Nagel
Marie Kratz*
Affiliation:
ESSEC Business School, CREAR
Werner Nagel*
Affiliation:
Friedrich-Schiller-Universität Jena
*
* Postal address: ESSEC Business School, Avenue Bernard Hirsch BP 50105, Cergy-Pontoise 95021 cedex, France.
** Postal address: Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, D-07737 Jena, Germany. Email address: werner.nagel@uni-jena.de
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Abstract

When a random field (Xt,t∈ℝ2) is thresholded on a given level u, the excursion set is given by its indicator 1[u, ∞)(Xt). The purpose of this work is to study functionals (as established in stochastic geometry) of these random excursion sets as, e.g. the capacity functional as well as the second moment measure of the boundary length. It extends results obtained for the one-dimensional case to the two-dimensional case, with tools borrowed from crossings theory, in particular, Rice methods, and from integral and stochastic geometry.

Keywords

Capacity functionalcrossingsexcursion setGaussian fieldgrowing circle methodRice formulasecond moment measuresweeping line methodstereologystochastic geometry

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60D05: Geometric probability and stochastic geometry 60G60: Random fields
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 3 , September 2016 , pp. 712 - 725
Copyright
Copyright © Applied Probability Trust 2016 

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