## The Annals of Probability

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

#### 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

https://projecteuclid.org/euclid.aop/1358951983

Digital Object Identifier
doi:10.1214/11-AOP738

Mathematical Reviews number (MathSciNet)
MR3059195

Zentralblatt MATH identifier
1269.60051

#### 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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