The Annals of Probability

High level excursion set geometry for non-Gaussian infinitely divisible random fields

Robert J. Adler, Gennady Samorodnitsky, and Jonathan E. Taylor

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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.

Article information

Source
Ann. Probab., Volume 41, Number 1 (2013), 134-169.

Dates
First available in Project Euclid: 23 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1358951983

Digital Object Identifier
doi:10.1214/11-AOP738

Mathematical Reviews number (MathSciNet)
MR3059195

Zentralblatt MATH identifier
1269.60051

Subjects
Primary: 60G52: Stable processes 60G60: Random fields
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G10: Stationary processes 60G17: Sample path properties

Keywords
Infinitely divisible random fields moving average excursion sets extrema critical points Euler characteristic Morse theory geometry

Citation

Adler, Robert J.; Samorodnitsky, Gennady; Taylor, Jonathan E. High level excursion set geometry for non-Gaussian infinitely divisible random fields. Ann. Probab. 41 (2013), no. 1, 134--169. doi:10.1214/11-AOP738. https://projecteuclid.org/euclid.aop/1358951983


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