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Numerical bounds for the distributions of the maxima of some one- and two-parameter Gaussian processes

Part of: Stochastic processes Nonparametric inference

Published online by Cambridge University Press:  01 July 2016

Cécile Mercadier
Cécile Mercadier*
Affiliation:
Université Paul Sabatier
*
∗ Current address: Equipe MODAL'X, Université Paris X - Nanterre, 200 avenue de la République, 92001 Nanterre cedex, France. Email address: cecile.mercadier@u-paris10.fr
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Abstract

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We consider the class of real-valued stochastic processes indexed on a compact subset of R or R2 with almost surely absolutely continuous sample paths. We obtain an implicit formula for the distributions of their maxima. The main result is the derivation of numerical bounds that turn out to be very accurate, in the Gaussian case, for levels that are not large. We also present the first explicit upper bound for the distribution tail of the maximum in the two-dimensional Gaussian framework. Numerical comparisons are performed with known tools such as the Rice upper bound and expansions based on the Euler characteristic. We deal numerically with the determination of the persistence exponent.

Keywords

Stochastic processrandom fielddistribution of the maximumabsolutely continuous sample pathfirst passage timeWAFO toolbox

MSC classification

Primary: 60G60: Random fields 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes 62G05: Estimation
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 1 , March 2006 , pp. 149 - 170
Copyright
Copyright © Applied Probability Trust 2006 

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