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

• Anne Estrade ^[1] ; José R. León ^[2]
  1. [1] Paris Descartes University

     Paris Descartes University

     París, Francia

     
  2. [2] Universidad Central de Venezuela

     Universidad Central de Venezuela

     Venezuela

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 6, 2016, págs. 3849-3878
• Idioma: inglés
• DOI: 10.1214/15-aop1062
• Enlaces
  • Texto completo
• Resumen

  • We study the Euler characteristic of an excursion set of a stationary isotropic Gaussian random field X:Ω×Rd→RX:Ω×Rd→R. Let us fix a level u∈Ru∈R and let us consider the excursion set above uu, A(T,u)={t∈T:X(t)≥u}A(T,u)={t∈T:X(t)≥u} where TT is a bounded cube ⊂Rd⊂Rd. The aim of this paper is to establish a central limit theorem for the Euler characteristic of A(T,u)A(T,u) as TT grows to RdRd, as conjectured by R. Adler more than ten years ago [Ann. Appl. Probab. 10 (2000) 1–74].

    The required assumption on XX is C3C3 regularity of the trajectories, non degeneracy of the Gaussian vector X(t)X(t) and derivatives at any fixed point t∈Rdt∈Rd as well as integrability on RdRd of the covariance function and its derivatives. The fact that XX is C3C3 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)A(T,u) equals the number of up-crossings of XX at level uu, plus eventually one if XX is above uu at the left bound of the interval TT.