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

Abstract

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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.

How to cite

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Azaïs, 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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  1. M. Abramowitz and I.A. Stegun, Handbook of Mathematical functions with Formulas, graphs and mathematical Tables. Dover, New-York (1972).  Zbl0543.33001
  2. R.J. Adler, An introduction to Continuity, Extrema and Related Topics for General Gaussian Processes. IMS, Hayward, CA (1990).  Zbl0747.60039
  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).  
  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&S3 (1999) 107-129.  
  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.  Zbl1018.60036
  7. H. Cramér and M.R. Leadbetter, Stationary and Related Stochastic Processes. J. Wiley & Sons, New-York (1967).  Zbl0162.21102
  8. R.B. Davies, Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika64 (1977) 247-254.  Zbl0362.62026
  9. J. Dieudonné, Calcul Infinitésimal. Hermann, Paris (1980).  
  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.  Zbl0488.60051

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