## The Annals of Applied Probability

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

#### Abstract

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

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

Dates
First available in Project Euclid: 28 January 2005

https://projecteuclid.org/euclid.aoap/1106922328

Digital Object Identifier
doi:10.1214/105051604000000602

Mathematical Reviews number (MathSciNet)
MR2115043

Zentralblatt MATH identifier
1079.60031

#### Citation

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. https://projecteuclid.org/euclid.aoap/1106922328

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