A CENTRAL LIMIT THEOREM FOR THE EULER CHARACTERISTIC OF A GAUSSIAN EXCURSION SET

• 成果类型: Article
• 署名作者: Estrade, Anne; Leon, Jose R.
• 署名单位: Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); University of Central Venezuela
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1062
• 发表日期: 2016
• 页码: 3849-3878
• 关键词: multiple stochastic integrals; random-fields; crossings; moment; number

摘要

We study the Euler characteristic of an excursion set of a stationary isotropic Gaussian random field X : Omega x R-d -> R. Let us fix a level u is an element of R and let us consider the excursion set above u, A(T, u) = {t is an element of T : X(t) >= u} where T is a bounded cube subset of R-d. The aim of this paper is to establish a central limit theorem for the Euler characteristic of A(T,u) as T grows to R-d, as conjectured by R. Adler more than ten years ago [Ann. Appl. Probab. 10 (2000) 1-74]. The required assumption on X is C-3 regularity of the trajectories, non degeneracy of the Gaussian vector X (t) and derivatives at any fixed point t is an element of R-d as well as integrability on R-d of the covariance function and its derivatives. The fact that X is C-3 is stronger than Geman's assumption traditionally used in dimension one. Nevertheless, our result extends what is known in dimension one to higher dimension. In that case, the Euler characteristic of A(T,u) equals the number of up-crossings of X at level u, plus eventually one if X is above u at the left bound of the interval T.