# On the tails of the distribution of the maximum of a smooth stationary gaussian process

Jean-Marc Azaïs; Jean-Marc Bardet; Mario Wschebor

ESAIM: Probability and Statistics (2002)

- Volume: 6, page 177-184
- ISSN: 1292-8100

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topAzaïs, Jean-Marc, Bardet, Jean-Marc, and Wschebor, Mario. "On the tails of the distribution of the maximum of a smooth stationary gaussian process." ESAIM: Probability and Statistics 6 (2002): 177-184. <http://eudml.org/doc/244829>.

@article{Azaïs2002,

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 [11] for a sufficiently small interval.},

author = {Azaïs, Jean-Marc, Bardet, Jean-Marc, Wschebor, Mario},

journal = {ESAIM: Probability and Statistics},

keywords = {tail of distribution of the maximum; stationary gaussian processes; stationary Gaussian processes; distribution of maximum; spectral moment},

language = {eng},

pages = {177-184},

publisher = {EDP-Sciences},

title = {On the tails of the distribution of the maximum of a smooth stationary gaussian process},

url = {http://eudml.org/doc/244829},

volume = {6},

year = {2002},

}

TY - JOUR

AU - Azaïs, Jean-Marc

AU - Bardet, Jean-Marc

AU - Wschebor, Mario

TI - On the tails of the distribution of the maximum of a smooth stationary gaussian process

JO - ESAIM: Probability and Statistics

PY - 2002

PB - EDP-Sciences

VL - 6

SP - 177

EP - 184

AB - 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.

LA - eng

KW - tail of distribution of the maximum; stationary gaussian processes; stationary Gaussian processes; distribution of maximum; spectral moment

UR - http://eudml.org/doc/244829

ER -

## References

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- [3] J.-M. Azaïs and J.-M. Bardet, Unpublished manuscript (2000).
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- [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. Zbl0933.60032MR1716124
- [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. Zbl1018.60036MR1901961
- [7] H. Cramér and M.R. Leadbetter, Stationary and Related Stochastic Processes. J. Wiley & Sons, New-York (1967). Zbl0162.21102MR217860
- [8] R.B. Davies, Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 64 (1977) 247-254. Zbl0362.62026MR501523
- [9] J. Dieudonné, Calcul Infinitésimal. Hermann, Paris (1980). Zbl0497.26004MR226971
- [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 Zbl0488.60051MR636766

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