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Excursion sets of three classes of stable random fields

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Robert J. Adler ,
Gennady Samorodnitsky  and
Jonathan E. Taylor
Robert J. Adler*
Affiliation:
Technion
Gennady Samorodnitsky*
Affiliation:
Cornell University
Jonathan E. Taylor*
Affiliation:
Stanford University
*
∗ Postal address: Faculty of Electrical Engineering, Technion, Haifa, 32000, Israel. Email address: robert@ee.technion.ac.il
∗∗ Postal address: Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, USA. Email address: gs18@cornell.edu
∗∗∗ Postal address: Department of Statistics, Stanford University, Stanford, CA 94305-4065, USA. Email address: jonathan.taylor@stanford.edu
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Abstract

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Studying the geometry generated by Gaussian and Gaussian-related random fields via their excursion sets is now a well-developed and well-understood subject. The purely non-Gaussian scenario has, however, not been studied at all. In this paper we look at three classes of stable random fields, and obtain asymptotic formulae for the mean values of various geometric characteristics of their excursion sets over high levels. While the formulae are asymptotic, they contain enough information to show that not only do stable random fields exhibit geometric behaviour very different from that of Gaussian fields, but they also differ significantly among themselves.

Keywords

Stable random fieldharmonisable fieldexcursion setEuler characteristicintrinsic volumegeometry

MSC classification

Primary: 60G52: Stable processes
Secondary: 60D05: Geometric probability and stochastic geometry 60G10: Stationary processes 60G60: Random fields

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 42 , Issue 2 , June 2010 , pp. 293 - 318
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Research supported in part by the US-Israel Binational Science Foundation, grant number 2008262, and the NSF, grant number DMS-0852227.

Research supported in part by the US-Israel Binational Science Foundation, grant number 2008262, by an NSA grant MSPF-05G-049, and an ARO grant W911NF-07-1-0078 at Cornell University.

Research supported in part by the US-Israel Binational Science Foundation, grant number 2008262, by NSF grants DMS-0405970, 0852227, 0906801, and the Natural Sciences and Engineering Research Council of Canada.

References

Adler, R. and Samorodnitsky, G. (1997). Level crossings of absolutely continuous stationary symmetric α-stable processes. Ann. Appl. Prob. 7, 460–493.Google Scholar
Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.Google Scholar
Adler, R. J., Samorodnitsky, G. and Gadrich, T. (1993). The expected number of level crossings for stationary, harmonisable, symmetric, stable processes. Ann. Appl. Prob. 3, 553–575.Google Scholar
Adler, R. J., Taylor, J. and Worsley, K. (2010). Random Fields and Geometry: Applications. In preparation. Early chapters available at http://webee.technion.ac.il/people/adler/publications.html.Google Scholar
Breiman, L. (1965). On some limit theorems similar to the arc-sine law. Theory Prob. Appl. 10, 323–331.CrossRefGoogle Scholar
Hadwiger, H. (1957). Vorlesüngen Über Inhalt, Oberfläche und Isoperimetrie. Springer, Berlin.Google Scholar
Klain, D. A. and Rota, G.-C. (1997). Introduction to Geometric Probability. Cambridge University Press.Google Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.Google Scholar
Worsley, K. (1997). The geometry of random images. Chance 9, 27–40.Google Scholar