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

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dc.creatorAzais, Jean-Marc
dc.creatorWschebor, Mario
dc.date2005-03-23
dc.date.accessioned2026-07-07T05:18:14Z
dc.date.available2026-07-07T05:18:14Z
dc.descriptionLet 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.
dc.descriptionPublished 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)
dc.identifierhttps://arxiv.org/abs/math/0503475
dc.identifierhttp://arxiv.org/abs/math/0503475
dc.identifierAnnals of Applied Probability 2005, Vol. 15, No. 1A, 254-278
dc.identifierdoi:10.1214/105051604000000602
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/74593
dc.subjectProbability
dc.subject60G15, 60G70. (Primary)
dc.titleOn the distribution of the maximum of a gaussian field with d parameters
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