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On the tails of the distribution of the maximum of a smooth stationaryGaussian process

Published online by Cambridge University Press:  15 November 2002

Jean-Marc Azaïs ,
Jean-Marc Bardet  and
Mario Wschebor
Jean-Marc Azaïs
Affiliation:
Laboratoire de Statistique et de Probabilités, UMR 5583 du CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France.azais@cict.fr.
Jean-Marc Bardet
Affiliation:
Laboratoire de Statistique et de Probabilit és, UMR 5583 du CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulou se Cedex 4, France.bardet@cict.fr.
Mario Wschebor
Affiliation:
Centro de Matemática, Facultad de Ciencias, Universidad de la República, Calle Iguá , Montevideo, Uruguay; wscheb@fcien.edu.uy.
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Abstract

We study the tails of the distribution of the maximum of a stationary Gaussian process on a bounded interval of the real line. Under regularity conditions including the existence of the spectral moment of order 8, we give an additional term for this asymptotics. This widens the application of an expansion given originally by Piterbarg [CITE] for a sufficiently small interval.

Keywords

Tail of distribution of the maximumstationary Gaussian processes.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 6: New directions in Time Series Analysis (Guest Editor: Philippe Soulier)New directions in Time Series Analysis (Guest Editor: Philippe Soulier) , 2002 , pp. 177 - 184
Copyright
© EDP Sciences, SMAI, 2002

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References

M. Abramowitz and I.A. Stegun, Handbook of Mathematical functions with Formulas, graphs and mathematical Tables. Dover, New-York (1972).
R.J. Adler, An introduction to Continuity, Extrema and Related Topics for General Gaussian Processes. IMS, Hayward, CA (1990).
J.-M. Azaïs and J.-M. Bardet, Unpublished manuscript (2000).
J.-M. Azaïs and C. Cierco-Ayrolles, An asymptotic test for quantitative gene detection. Ann. Inst. H. Poincaré Probab. Statist. (to appear).
Azaïs, J.-M., Cierco-Ayrolles, C. and Croquette, A., Bounds and asymptotic expansions for the distribution of the maximum of a smooth stationary Gaussian process. ESAIM: P&S 3 (1999) 107-129. CrossRef
J.-M. Azaïs and M. Wschebor, The Distribution of the Maximum of a Gaussian Process: Rice Method Revisited, in In and out of equilibrium: Probability with a physical flavour. Birkhauser, Coll. Progress in Probability (2002) 321-348.
H. Cramér and M.R. Leadbetter, Stationary and Related Stochastic Processes. J. Wiley & Sons, New-York (1967).
Davies, R.B., Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 64 (1977) 247-254. CrossRef
J. Dieudonné, Calcul Infinitésimal. Hermann, Paris (1980).
Miroshin, R.N., Rice series in the theory of random functions. Vestn. Leningrad Univ. Math. 1 (1974) 143-155.
Piterbarg, V.I., Comparison of distribution functions of maxima of Gaussian processes. Theoret. Probab. Appl. 26 (1981) 687-705. CrossRef