Open Access
February 2005 On the distribution of the maximum of a Gaussian field with d parameters
Jean-Marc Azaïs, Mario Wschebor
Ann. Appl. Probab. 15(1A): 254-278 (February 2005). DOI: 10.1214/105051604000000602


Let I be a compact d-dimensional manifold, let X:I→ℛ be a Gaussian process with regular paths and let FI(u), u∈ℛ, be the probability distribution function of sup tIX(t).

We prove that under certain regularity and nondegeneracy conditions, FI is a C1-function and satisfies a certain implicit equation that permits to give bounds for its values and to compute its asymptotic behavior as u→+∞. 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→ℛd is a random field and x is a fixed point in ℛd. We also give proofs for this kind of formulae, which have their own interest beyond the present application.


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Jean-Marc Azaïs. Mario Wschebor. "On the distribution of the maximum of a Gaussian field with d parameters." Ann. Appl. Probab. 15 (1A) 254 - 278, February 2005.


Published: February 2005
First available in Project Euclid: 28 January 2005

zbMATH: 1079.60031
MathSciNet: MR2115043
Digital Object Identifier: 10.1214/105051604000000602

Primary: 60G15 , 60G70

Keywords: Gaussian fields , regularity of the distribution of the maximum , Rice formula

Rights: Copyright © 2005 Institute of Mathematical Statistics

Vol.15 • No. 1A • February 2005
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