Central Limit Theorem for Lipschitz-Killing Curvatures of Excursion Sets of Gaussian Random Fields

摘要

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, asymptotics of nonlinear functionals of Gaussian random fields have been studied [see Berman (Sojourns and extremes of stochastic processes, Wadsworth & Brooks, Monterey, 1991), Kratz and Len (Extremes 3(1):57-86, 2000), Kratz and Len (J Theor Probab 14(3):639-672, 2001), Meshenmoser and Shashkin (Stat Probab Lett 81(6):642-646, 2011), Pham (Stoch Proc Appl 123(6):2158-2174, 2013), Spodarev (Chapter in modern stochastics and applications, volume 90 of the series Springer optimization and its applications, pp 221-241, 2013) for a sample of works in such settings], the most recent addition being (Adler and Naitzat in Stoch Proc Appl 2016; Estrade and Len in Ann Probab 2016) where a central limit theorem (CLT) for Euler integral and Euler-Poincar, characteristic, respectively, of the excursions set of a Gaussian random field is proven under some conditions. In this paper, we obtain a CLT for some global geometric functionals, called the Lipschitz-Killing curvatures of excursion sets of Gaussian random fields, in an appropriate setting.