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

Abstract

top
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.

How to cite

top

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 (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

top
  1. [1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical functions with Formulas, graphs and mathematical Tables. Dover, New-York (1972). Zbl0543.33001MR208797
  2. [2] R.J. Adler, An introduction to Continuity, Extrema and Related Topics for General Gaussian Processes. IMS, Hayward, CA (1990). Zbl0747.60039MR1088478
  3. [3] J.-M. Azaïs and J.-M. Bardet, Unpublished manuscript (2000). 
  4. [4] J.-M. Azaïs and C. Cierco-Ayrolles, An asymptotic test for quantitative gene detection. Ann. Inst. H. Poincaré Probab. Statist. (to appear). Zbl1011.62113MR1955355
  5. [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. [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. [7] H. Cramér and M.R. Leadbetter, Stationary and Related Stochastic Processes. J. Wiley & Sons, New-York (1967). Zbl0162.21102MR217860
  8. [8] R.B. Davies, Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 64 (1977) 247-254. Zbl0362.62026MR501523
  9. [9] J. Dieudonné, Calcul Infinitésimal. Hermann, Paris (1980). Zbl0497.26004MR226971
  10. [10] R.N. Miroshin, Rice series in the theory of random functions. Vestn. Leningrad Univ. Math. 1 (1974) 143-155. 
  11. [11] V.I. Piterbarg, Comparison of distribution functions of maxima of Gaussian processes. Theoret. Probab. Appl. 26 (1981) 687-705. r Zbl0488.60051MR636766

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.