Central limit theorems for the excursion set volumes of weakly dependent random fields

TL;DR

This paper establishes multivariate central limit theorems for the volumes of excursion sets of stationary weakly dependent random fields, including Gaussian and shot noise fields, with formulas for covariance and statistical applications.

Contribution

It provides new multivariate CLTs for excursion set volumes of weakly dependent random fields, including explicit covariance formulas and a statistical version.

Findings

Multivariate CLTs are proved for excursion set volumes.

Explicit covariance matrix formulas are derived.

Numerical results illustrate the theoretical findings.

Abstract

The multivariate central limit theorems (CLT) for the volumes of excursion sets of stationary quasi-associated random fields on $\mathbb{R}^d$ are proved. Special attention is paid to Gaussian and shot noise fields. Formulae for the covariance matrix of the limiting distribution are provided. A statistical version of the CLT is considered as well. Some numerical results are also discussed.