A Central Limit Theorem for Lipschitz-Killing Curvatures of Gaussian Excursions

TL;DR

This paper proves a central limit theorem for the Lipschitz-Killing curvatures of Gaussian excursion sets, demonstrating their convergence to a normal distribution as the observation window expands, with bounds on the variance.

Contribution

It establishes a new CLT for Lipschitz-Killing curvatures of Gaussian excursions, including variance bounds, advancing geometric probability theory.

Findings

Lipschitz-Killing curvatures converge to normal distribution

Asymptotic variance has a proven lower bound

Results apply to large-scale Gaussian excursion sets

Abstract

This paper studies the excursion set of a real stationary isotropic Gaussian random field above a fixed level. We show that the standardized Lipschitz-Killing curvatures of the intersection of the excursion set with a window converges in distribution to a normal distribution as the window grows to the $d$-dimensional Euclidean space. Moreover a lower bound for the asymptotic variance is shown.