Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-18T06:08:17.119Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Stochastic processes Geometric probability and stochastic geometry

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Adler, R. J. (1981).The Geometry of Random Fields.John Wiley,Chichester.Google Scholar
[2] Adler, R. J. and Taylor, J. E. (2007).Random Fields and Geometry.Springer,New York.Google Scholar
[3] Adler, R. J.,Samorodnitsky, G. and Taylor, J. E. (2010).Excursion sets of three classes of stable random fields.Adv. Appl. Prob. 42,293–318.Google Scholar
[4] Azaϊs, J.-M. and Wschebor, M. (2009).Level Sets and Extrema of Random Processes and Fields.John Wiley,Hoboken, NJ.Google Scholar
[5] Brodtkorb, P. A. (2006).Evaluating nearly singular multinormal expectations with application to wave distributions.Methodol. Comput. Appl. Prob. 8,65–91.Google Scholar
[6] Chiu, S. N.,Stoyan, D.,Kendall, W. S. and Mecke, J. (2013).Stochastic Geometry and its Applications,3rd edn.John Wiley,Chichester.Google Scholar
[7] Cramér, H. and Leadbetter, M. R. (1967).Stationary and Related Stochastic Processes.John Wiley,New York.Google Scholar
[8] Estrade, A.,Iribarren, I. and Kratz, M. (2011).Chord-length distribution functions and Rice formulae. Application to random media.Extremes 15,333–352.CrossRefGoogle Scholar
[9] Kratz, M. (2006).Level crossings and other level functionals of stationary Gaussian processes.Prob. Surveys 3,230–288.Google Scholar
[10] Kratz, M. and Leόn, J. R. (2001).Central limit theorems for level functionals of stationary Gaussian processes and fields.J. Theoret. Prob. 14,639–672.Google Scholar
[11] Lindgren, G. (1972).Wave-length and amplitude in Gaussian noise.Adv. Appl. Prob. 4,81–108.Google Scholar
[12] Matheron, G. (1975).Random Sets and Integral Geometry.John Wiley,New York.Google Scholar
[13] Mercadier, C. (2006).Numerical bounds for the distributions of the maximum of some one- and two-parameter Gaussian processes.Adv. Appl. Prob. 38,149–170.Google Scholar
[14] Molchanov, I. (2005).Theory of Random Sets.Springer,London.Google Scholar
[15] Piterbarg, V. I. (1996).Asymptotic Methods in the Theory of Gaussian Processes and Fields.American Mathematical Society,Providence, RI.Google Scholar
[16] Preparata, F. P. and Shamos, M. I. (1985).Computational Geometry: An Introduction.Springer,New York.Google Scholar
[17] Rychlik, I. (1987).Joint distribution of successive zero crossing distances for stationary Gaussian processes.\textit{J. Appl. Prob.}24,378–385.Google Scholar
[18] Schneider, R. and Weil, W. (2008).Stochastic and Integral Geometry.Springer,Berlin.Google Scholar
[19] Serra, J. (1984).Image Analysis and Mathematical Morphology.Academic Press,London.Google Scholar
[20] Weiss, V. and Nagel, W. (1994).Second-order stereology for planar fibre processes.Adv. Appl. Prob. 26,906–918.CrossRefGoogle Scholar
[21] Wschebor, M. (1985).Surfaces Aléatoires(Lecture Notes Math. 1147).Springer,Berlin.Google Scholar