The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 15, Number 1A (2005), 254-278.
On the distribution of the maximum of a Gaussian field with d parameters
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 t∈IX(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.
Ann. Appl. Probab., Volume 15, Number 1A (2005), 254-278.
First available in Project Euclid: 28 January 2005
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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