The Annals of Applied Probability

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

Jean-Marc Azaïs and Mario Wschebor

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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.

Article information

Ann. Appl. Probab., Volume 15, Number 1A (2005), 254-278.

First available in Project Euclid: 28 January 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 60G70: Extreme value theory; extremal processes

Gaussian fields Rice formula regularity of the distribution of the maximum


Azaïs, Jean-Marc; Wschebor, Mario. On the distribution of the maximum of a Gaussian field with d parameters. Ann. Appl. Probab. 15 (2005), no. 1A, 254--278. doi:10.1214/105051604000000602.

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