Open Access
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


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.


Download 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.


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

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

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

Keywords: critical points , Euler characteristic , Excursion sets , Extrema , geometry , Infinitely divisible random fields , Morse theory , moving average

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 1 • January 2013
Back to Top