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

Azais, Jean-Marc; Wschebor, Mario

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

Authors

Description

Let I be a compact d-dimensional manifold, let X:I\to R be a Gaussian process with regular paths and let F_I(u), u\in R, be the probability distribution function of sup_{t\in I}X(t). We prove that under certain regularity and nondegeneracy conditions, F_I is a C^1-function and satisfies a certain implicit equation that permits to give bounds for its values and to compute its asymptotic behavior as u\to +\infty. 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\to R^d is a random field and x is a fixed point in R^d. We also give proofs for this kind of formulae, which have their own interest beyond the present application.
Published at http://dx.doi.org/10.1214/105051604000000602 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)