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January 2013 High level excursion set geometry for non-Gaussian infinitely divisible random fields
Robert J. Adler, Gennady Samorodnitsky, Jonathan E. Taylor
Ann. Probab. 41(1): 134-169 (January 2013). DOI: 10.1214/11-AOP738

Abstract

We consider smooth, infinitely divisible random fields $(X(t),t\in M)$, $M\subset\mathbb{R}^{d}$, with regularly varying Lévy measure, and are interested in the geometric characteristics of the excursion sets

\[A_{u}=\{t\in M:X;(t)>u\}\]

over high levels $u$.

For a large class of such random fields, we compute the $u\to\infty$ asymptotic joint distribution of the numbers of critical points, of various types, of $X$ in $A_{u}$, conditional on $A_{u}$ being nonempty. This allows us, for example, to obtain the asymptotic conditional distribution of the Euler characteristic of the excursion set.

In a significant departure from the Gaussian situation, the high level excursion sets for these random fields can have quite a complicated geometry. Whereas in the Gaussian case nonempty excursion sets are, with high probability, roughly ellipsoidal, in the more general infinitely divisible setting almost any shape is possible.

Citation

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Robert J. Adler. Gennady Samorodnitsky. Jonathan E. Taylor. "High level excursion set geometry for non-Gaussian infinitely divisible random fields." Ann. Probab. 41 (1) 134 - 169, January 2013. https://doi.org/10.1214/11-AOP738

Information

Published: January 2013
First available in Project Euclid: 23 January 2013

zbMATH: 1269.60051
MathSciNet: MR3059195
Digital Object Identifier: 10.1214/11-AOP738

Subjects:
Primary: 60G52, 60G60
Secondary: 60D05, 60G10, 60G17

Rights: Copyright © 2013 Institute of Mathematical Statistics

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