CLT for Lipschitz-Killing curvatures of excursion sets of Gaussian fields

TL;DR

This paper establishes a central limit theorem for Lipschitz-Killing curvatures of excursion sets of isotropic Gaussian fields, extending previous results on geometric functionals like Euler-Poincaré characteristic.

Contribution

It introduces a CLT for Lipschitz-Killing curvatures of Gaussian excursion sets, broadening the understanding of their asymptotic geometric behavior.

Findings

Proves a CLT for Lipschitz-Killing curvatures of Gaussian excursion sets.

Extends previous CLTs from Euler-Poincaré characteristic to more general geometric functionals.

Provides theoretical foundation for statistical analysis of geometric features in Gaussian fields.

Abstract

Our interest in this paper is to explore limit theorems for various geometric functionals of excursion sets of isotropic Gaussian random fields. In the past, limit theorems have been proven for various geometric functionals of excursion sets/sojourn times. Most recently a CLT for Euler-Poincar\'e characteristic of the excursions set of a Gaussian random field has been proven under appropriate conditions. In this paper, we shall obtain a central limit theorem for some global geometric functionals, called the Lipschitz-Killing curvatures of excursion sets of Gaussian random fields in an appropriate setting.