CLT for Lipschitz-Killing curvatures of excursion sets of 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.