# A central limit theorem for the Euler characteristic of a Gaussian excursion set

Abstract : We study the Euler characteristic of an excursion set of a stationary isotropic Gaussian random field $X:\Omega\times\mathbb{R}^d\to\mathbb{R}$. Let us fix a level $u\in \R$ and let us consider the excursion set above $u$, $A(T,u)=\{t\in T:\,X(t)\ge u\}$ where $T$ is a bounded cube $\subset \R^d$. 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. 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 $t\in \R^d$ 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 $T$.
Type de document :
Article dans une revue
Annals of Probability, 2016, 44 (6), pp.3849-3878. <10.1214/15-AOP1062>
Domaine :

https://hal.archives-ouvertes.fr/hal-00943054
Soumis le : vendredi 10 avril 2015 - 17:18:00
Dernière modification le : jeudi 22 décembre 2016 - 10:28:12
Document(s) archivé(s) le : mardi 18 avril 2017 - 16:28:13

### Fichier

Fichiers produits par l'(les) auteur(s)

### Citation

Anne Estrade, José R. León. A central limit theorem for the Euler characteristic of a Gaussian excursion set. Annals of Probability, 2016, 44 (6), pp.3849-3878. <10.1214/15-AOP1062>. <hal-00943054v3>

Consultations de
la notice

## 359

Téléchargements du document