# 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 (2010)

- 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 (2010): 177-184. <http://eudml.org/doc/104286>.

@article{Azaïs2010,

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.
},

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},

month = {3},

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/104286},

volume = {6},

year = {2010},

}

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

DA - 2010/3//

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 [CITE] 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/104286

ER -

## References

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- R.J. Adler, An introduction to Continuity, Extrema and Related Topics for General Gaussian Processes. IMS, Hayward, CA (1990). Zbl0747.60039
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- 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&S3 (1999) 107-129.
- 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. Zbl1018.60036
- H. Cramér and M.R. Leadbetter, Stationary and Related Stochastic Processes. J. Wiley & Sons, New-York (1967). Zbl0162.21102
- R.B. Davies, Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika64 (1977) 247-254. Zbl0362.62026
- J. Dieudonné, Calcul Infinitésimal. Hermann, Paris (1980).
- R.N. Miroshin, Rice series in the theory of random functions. Vestn. Leningrad Univ. Math.1 (1974) 143-155.
- V.I. Piterbarg, Comparison of distribution functions of maxima of Gaussian processes. Theoret. Probab. Appl.26 (1981) 687-705. Zbl0488.60051

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