On the tails of the distribution of the maximum of a smooth stationary gaussian process
ESAIM: Probability and Statistics, Tome 6 (2002), pp. 177-184.

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 [11] for a sufficiently small interval.

DOI : 10.1051/ps:2002010
Classification : 60Gxx, 60E05, 60G15, 65U05
Mots clés : tail of distribution of the maximum, stationary gaussian processes 
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     title = {On the tails of the distribution of the maximum of a smooth stationary gaussian process},
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     pages = {177--184},
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Azaïs, Jean-Marc; Bardet, Jean-Marc; Wschebor, Mario. On the tails of the distribution of the maximum of a smooth stationary gaussian process. ESAIM: Probability and Statistics, Tome 6 (2002), pp. 177-184. doi : 10.1051/ps:2002010. http://archive.numdam.org/articles/10.1051/ps:2002010/

[1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical functions with Formulas, graphs and mathematical Tables. Dover, New-York (1972). | MR | Zbl

[2] R.J. Adler, An introduction to Continuity, Extrema and Related Topics for General Gaussian Processes. IMS, Hayward, CA (1990). | MR | Zbl

[3] J.-M. Azaïs and J.-M. Bardet, Unpublished manuscript (2000).

[4] J.-M. Azaïs and C. Cierco-Ayrolles, An asymptotic test for quantitative gene detection. Ann. Inst. H. Poincaré Probab. Statist. (to appear). | Numdam | MR | Zbl

[5] J.-M. Azaïs, C. Cierco-Ayrolles and A. Croquette, Bounds and asymptotic expansions for the distribution of the maximum of a smooth stationary Gaussian process. ESAIM: P&S 3 (1999) 107-129. | Numdam | MR | Zbl

[6] 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. | MR | Zbl

[7] H. Cramér and M.R. Leadbetter, Stationary and Related Stochastic Processes. J. Wiley & Sons, New-York (1967). | MR | Zbl

[8] R.B. Davies, Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 64 (1977) 247-254. | MR | Zbl

[9] J. Dieudonné, Calcul Infinitésimal. Hermann, Paris (1980). | MR | Zbl

[10] R.N. Miroshin, Rice series in the theory of random functions. Vestn. Leningrad Univ. Math. 1 (1974) 143-155.

[11] V.I. Piterbarg, Comparison of distribution functions of maxima of Gaussian processes. Theoret. Probab. Appl. 26 (1981) 687-705. r | MR | Zbl

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