Limit theorems for excursion sets of subordinated Gaussian random fields with long-range dependence

TL;DR

This paper investigates the asymptotic distribution of the volume of excursion sets of subordinated Gaussian fields with long-range dependence, using Hermite expansions and multiple Wiener-Itô integrals, covering non-stationary cases.

Contribution

It introduces new limit theorems for excursion set volumes of subordinated Gaussian fields, including non-stationary and anisotropic cases, with explicit distributional characterizations.

Findings

Derived limit distributions involving multiple Wiener-Itô integrals

Applicable to non-stationary and anisotropic Gaussian fields

Illustrated results with concrete examples

Abstract

This paper considers the asymptotic behaviour of volumes of excursion sets of subordinated Gaussian random fields with (possibly) infinite variance. Actually, we consider integral functionals of such fields and obtain their limiting distribution using the Hermite expansion of the integrand. We consider the general non-stationary Gaussian random fields, including stationary and anisotropic special cases. The limiting random variables in our limit theorems have the form of multiple Wiener-It\^{o} integrals. We illustrate most results with corresponding examples.