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Excursion sets of three classes of stable random fields

  • Robert J. Adler (a1), Gennady Samorodnitsky (a2) and Jonathan E. Taylor (a3)

Abstract

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.

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Copyright

Corresponding author

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

Footnotes

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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.

Footnotes

References

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[1] Adler, R. and Samorodnitsky, G. (1997). Level crossings of absolutely continuous stationary symmetric α-stable processes. Ann. Appl. Prob. 7, 460493.
[2] Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.
[3] 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, 553575.
[4] 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.
[5] Breiman, L. (1965). On some limit theorems similar to the arc-sine law. Theory Prob. Appl. 10, 323331.
[6] Hadwiger, H. (1957). Vorlesüngen Über Inhalt, Oberfläche und Isoperimetrie. Springer, Berlin.
[7] Klain, D. A. and Rota, G.-C. (1997). Introduction to Geometric Probability. Cambridge University Press.
[8] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.
[9] Worsley, K. (1997). The geometry of random images. Chance 9, 2740.

Keywords

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Excursion sets of three classes of stable random fields

  • Robert J. Adler (a1), Gennady Samorodnitsky (a2) and Jonathan E. Taylor (a3)

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