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

We study the Euler characteristic of an excursion set of a stationary isotropic Gaussian random field X:..Rd.R. Let us fix a level u.R and let us consider the excursion set above u, A(T,u)={t.T:X(t).u} where T is a bounded cube .Rd. The aim of this paper is to establish a central limit theorem for the Euler characteristic of A(T,u) as T grows to Rd , as conjectured by R. Adler more than ten years ago [Ann. Appl. Probab. 10 (2000) 1.74]. The required assumption on X is C3 regularity of the trajectories, non degeneracy of the Gaussian vector X(t) and derivatives at any fixed point t.Rd as well as integrability on Rd of the covariance function and its derivatives. The fact that X is C3 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.