On the distribution of the maximum of a Gaussian field with d parameters

• 成果类型: Article
• 署名作者: Azaïs, JM; Wschebor, M
• 署名单位: Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Centre National de la Recherche Scientifique (CNRS); Universidad de la Republica, Uruguay
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051604000000602
• 发表日期: 2005
• 页码: 254-278
• 关键词: density; number

摘要

Let I be a compact d-dimensional manifold, let X : I --> R be a Gaussian process with regular paths and let F-I (u), u c R, be the probability distribution function of sup(tis an element ofI) X (t). We prove that under certain regularity and nondegeneracy conditions, F-I is a C-1-function and satisfies a certain implicit equation that permits to give bounds for its values and to compute its asymptotic behavior as u --> +infinity. This is a partial extension of previous results by the authors in the case d = 1. Our methods use strongly the so-called Rice formulae for the moments of the number of roots of an equation of the form Z(t) = x, where Z : I --> R-d is a random field and x is a fixed point in R-d. We also give proofs for this kind of formulae, which have their own interest beyond the present application.