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

Anne Estrade (1) , José R. León (2)

1 MAP5 - Mathématiques Appliquées à Paris 5
2 UCV - Universidad Central de Venezuela

We study the Euler characteristic of an excursion set of a stationary Gaussian random field. Let $X:\Omega\times\mathbb{R}^d\to\mathbb{R}$ be a stationary isotropic Gaussian field having trajectories in $C^2(\mathbb{R}^d)$. Let us fix a level $u\in \R$ and consider the excursion set above $u$, $\{t\in \R^d:\,X(t)\ge u\}$. We take the restriction to a compact domain considering for any bounded rectangle $T\subset \R^d$, $A(T,u)=\{t\in T:\,X(t)\ge u\}~.$ 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 (R. Adler, Ann. Appl. Probab. 2000). The required assumption on $X$ is stronger than Geman's one in dimension one but weaker than having $C^3$ trajectories. Our result extends to higher dimension what is known in dimension one, since in that case, the Euler characteristic of $A(T,u)$ equals the number of up-crossings of $X$ at level $u$.

Domaines: Probabilités [math.PR]

Anne Estrade, José R. León. A central limit theorem for the Euler characteristic of a Gaussian excursion set. 2014. ⟨hal-00943054v1⟩