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

By Jean-Marc Azaïs, Mario Wschebor. 2005

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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.

Subfields: Finance, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b30f75b2178c7450c9ae6fb7886f7b5289a9749f

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Citations in the corpus

On asymptotic constants in the theory of extremes for Gaussian processes (2012, background): “For sufficiently smooth Gaussian processes, various levelcrossing tools can be exploited [6, 25].”
Level crossings and other level functionals of stationary Gaussian processes (2006, background): “(see [157] and [2]), and by Azäıs and Wschebor (see [14]).”
A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail (2006, background): “We assume that S satisfies the hypothesis A1 A1 : • S is a compact subset of Rd • S is the disjoint union of Sd, Sd−1..., S0, where Sj is an orientable C3 manifold of dimension j without boundary.”
A central limit theorem for the Euler characteristic of a Gaussian excursion set (2016, unknown): “Actually our proof is inspired by [10] and [11].”