On the distribution of the maximum of a Gaussian field with d parameters
Open Access
- 1 February 2005
- journal article
- Published by Institute of Mathematical Statistics in The Annals of Applied Probability
- Vol. 15 (1A) , 254-278
- https://doi.org/10.1214/105051604000000602
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 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.Keywords
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