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

作者：Robert J. Adler, Gennady Samorodnitsky, Jonathan Taylor · 发表于：The Annals of Probability · 年份：2013 · DOI：10.1214/11-aop738 · 被引用次数：30 · 研究领域：Stochastic processes and statistical mechanics、Geometry and complex manifolds、Probability and Statistical Research

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.