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November 2016 A central limit theorem for the Euler characteristic of a Gaussian excursion set
Anne Estrade, José R. León
Ann. Probab. 44(6): 3849-3878 (November 2016). DOI: 10.1214/15-AOP1062

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 $u\in\mathbb{R}$ 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 $\subset\mathbb{R}^{d}$. The aim of this paper is to establish a central limit theorem for the Euler characteristic of $A(T,u)$ as $T$ grows to $\mathbb{R}^{d}$, 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 $t\in\mathbb{R}^{d}$ as well as integrability on $\mathbb{R}^{d}$ 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 $T$.

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Anne Estrade. José R. León. "A central limit theorem for the Euler characteristic of a Gaussian excursion set." Ann. Probab. 44 (6) 3849 - 3878, November 2016. https://doi.org/10.1214/15-AOP1062

Information

Received: 1 June 2014; Revised: 1 April 2015; Published: November 2016
First available in Project Euclid: 14 November 2016

zbMATH: 1367.60016
MathSciNet: MR3572325
Digital Object Identifier: 10.1214/15-AOP1062

Subjects:
Primary: 60F05
Secondary: 53C65, 60G15, 60G60

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 6 • November 2016
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