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

TL;DR

This paper analyzes the geometric properties of high-level excursion sets in non-Gaussian infinitely divisible random fields, revealing complex shapes and asymptotic distributions of critical points and Euler characteristic.

Contribution

It provides the first detailed asymptotic analysis of the geometry of excursion sets for non-Gaussian infinitely divisible fields, including critical point counts and Euler characteristic distributions.

Findings

High-level excursion sets can have complex, non-ellipsoidal shapes.

Asymptotic joint distribution of critical points is derived.

Euler characteristic distribution conditioned on nonempty excursion sets is obtained.

Abstract

We consider smooth, infinitely divisible random fields $(X(t),t\in M)$, $M\subset {\mathbb{R}}^d$, with regularly varying Levy 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.