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Bounds and asymptotic expansions for the distribution of the Maximum of a smooth stationary Gaussian process

Published online by Cambridge University Press:  15 August 2002

Jean-Marc Azaïs ,
Christine Cierco-Ayrolles  and
Alain Croquette
Jean-Marc Azaïs
Affiliation:
Laboratoire de Statistique et Probabilités, UMR C55830 du CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France.
Christine Cierco-Ayrolles
Affiliation:
Laboratoire de Statistique et Probabilités, UMR C55830 du CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France. Institut National de la Recherche Agronomique, Unité de Biométrie et Intelligence Artificielle, BP. 27, Chemin de Borde-Rouge, 31326 Castanet-Tolosan Cedex, France.
Alain Croquette
Affiliation:
Laboratoire de Statistique et Probabilités, UMR C55830 du CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France.
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Abstract

This paper uses the Rice method [18] to give bounds to the distribution of the maximum of a smooth stationary Gaussian process. We give simpler expressions of the first two terms of the Rice series [3,13] for the distribution of the maximum. Our main contribution is a simpler form of the second factorial moment of the number of upcrossings which is in some sense a generalization of Steinberg et al.'s formula ([7] p. 212). Then, we present a numerical application and asymptotic expansions that give a new interpretation of a result by Piterbarg [15].

Keywords

Asymptotic expansionsextreme valuesstationary Gaussian processRice seriesupcrossings.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 3 , 1999 , pp. 107 - 129
Copyright
© EDP Sciences, SMAI, 1999

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