Excursion sets of stable random fields

TL;DR

This paper investigates the geometric properties of excursion sets in stable non-Gaussian random fields, providing asymptotic formulas that reveal distinct behaviors from Gaussian fields and differences among stable field classes.

Contribution

It introduces the first asymptotic analysis of excursion set geometry for stable non-Gaussian fields, highlighting their unique geometric characteristics.

Findings

Stable fields exhibit different geometric behaviors than Gaussian fields.

Asymptotic formulas quantify mean geometric characteristics of excursion sets.

Distinct stable field classes show significant differences in geometric properties.

Abstract

Studying the geometry generated by Gaussian and Gaussian- related random fields via their excursion sets is now a well developed and well understood subject. The purely non-Gaussian scenario has, however, not been studied at all. In this paper we look at three classes of stable random fields, and obtain asymptotic formulae for the mean values of various geometric characteristics of their excursion sets over high levels. While the formulae are asymptotic, they contain enough information to show that not only do stable random fields exhibit geometric behaviour very different from that of Gaussian fields, but they also differ significantly among themselves.