A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail

By Jean-Marc Azais Mario Wschebor. 2006

60 citations recorded by the source corpus.

We study the probability distribution F(u) of the maximum of smooth Gaussian fields defined on compact subsets of having some geometric regularity. Our main result is a general expression for the density of F. Even though this is an implicit formula, one can deduce from it explicit bounds for the density, and hence for the distribution, as well as improved expansions for 1-F(u) for large values of u. The main tool is the Rice formula for the moments of the number of roots of a random system of equations over the reals. This method enables also to study second-order properties of the expected Euler characteristic approximation using only elementary arguments and to extend these kinds of results to some interesting classes of Gaussian fields. We obtain more precise results for the "direct method" to compute the distribution of the maximum, using the spectral theory of GOE random matrices.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0a16477b07c62cb5638ed5a2e0033ced400c71d2

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Replica Symmetry Breaking Condition Exposed by Random Matrix Calculation of Landscape Complexity (2007, background): “the problem of determining the distribution of the maximum of a Gaussian random field, see [50].”
Density Formula and Concentration Inequalities with Malliavin Calculus (2009, methodology): “Expressing the density of Z , even implicitly, is a subject of study in its own right; in the case of differentiable random fields, geometric methods have been used by Azaïs and Wschebor, based on the so-called Rice-Kac formula, to express the density of Z in a way which allows them to derive sharp ”