Asymptotic expansions for the distribution of the maximum of Gaussian random fields
type: Item
instance of: scholarly article
title: Asymptotic expansions for the distribution of the maximum of Gaussian random fields
zbMATH Open document ID: 1035.60035
DOI: 10.1023/A:1022123321967
author: Jean-Marc Azaïs
author: Céline Delmas
published in: Extremes
publication date: 2004-03-16T00:00:00Z
review text: For a Gaussian random field \(X(t)\), \(t\in S\subset R^N\), with smooth enough trajectories and compact \(S\) the asymptotic expansions are derived of \(G(x)=\Pr\{\max_{t\in S} X(t)\geq u\}\) as \(u\to\infty\). For example, the following theorem is proved: \[ G(u)\sim\lambda(S)\sum_{j=0}^{[N/2]} k_{2j}\psi_{2j}(u), \] where \(\lambda(S)\) is the Lebesgue measure of \(S\), \[ k_{2j}=(-1)^j(\det\hbox{Var} X'(t))^{1/2}\binom{N}{2j}{(2j)!\over 2^j j!}, \] \[ \psi_{2j}(u)={1\over 2^{1+j}\pi^{(N+1)/2}} \int_{(u^2)/2}^{+\infty} x^{(N-1-2j)/2}e^{-x}dx. \] The remainders in these expansions are exponentially small. They are obtained by a Rice method. An application to the sea waves is considered.
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G70
zbMATH DE Number: 2059066
zbMATH Keywords: Rice method
zbMATH Keywords: extremes of Gaussian field
zbMATH Keywords: asymptotics of maximum
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2059066