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
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
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