A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail

Stochastic Processes and their Applications

Volumn 118, Issue 7, 2008, Pages 1190-1218

  Azaïs, Jean Marc ^a   Wschebor, Mario ^b  

^a INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

^b UNIVERSIDAD DE LA REPÚBLICA   (Uruguay)

Author keywords

Density of the maximum; Distribution of the maximum; Euler Poincar characteristic; Gaussian fields; Random matrices; Rice formula

Indexed keywords

APPROXIMATION THEORY; BOUNDARY CONDITIONS; MATRIX ALGEBRA; NUMERICAL METHODS; PROBABILITY DISTRIBUTIONS; SET THEORY;

DENSITY OF THE MAXIMUM; DISTRIBUTION OF THE MAXIMUM; GAUSSIAN FIELDS; RANDOM MATRICES; RICE FORMULA;

RANDOM PROCESSES;

EID: 44449122948     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.spa.2007.07.016     Document Type: Article

References (35)

1. Adler R.J. The Geometry of Random Fields (1981), Wiley, New York
2. Adler R.J. An Introduction to Continuity, Extrema and Related Topics for General Gaussian Processes (1990), IMS, Hayward, CA
3. Adler R.J., and Taylor J.E. Random Fields and Geometry (2007), Springer
4. Azaïs J.-M., Bardet J.-M., and Wschebor M. On the Tails of the distribution of the maximum of a smooth stationary Gaussian Process. ESAIM: Probab. Stat. 6 (2002) 177-184
5. Azaïs J.-M., and Delmas C. Asymptotic expansions for the distribution of the maximum of a Gaussian random fields. Extremes 5 2 (2002) 181-212
6. J.-M. Azaïs, M. Wschebor, 2002. The distribution of the maximum of a gaussian process: Rice method revisited, in: In and out of equilibrium: Probability with a physical flavour, in: Progress in Probability, Birkhaüser, pp. 321-348
7. Azaïs J.-M., and Wschebor M. On the regularity of the distribution of the Maximum of one parameter Gaussian processes. Probab. Theory Related Fields 119 (2001) 70-98
8. Azaïs J.-M., and Wschebor M. On the distribution of the maximum of a Gaussian field with d parameters. Ann. Appl. Probab. 15 1A (2005) 254-278
9. J.-M. Azaïs, M. Wschebor, A self contained proof of the Rice formula for random fields, 2006. Preprint available at: http://www.lsp.ups-tlse.fr/Azais/publi/completeproof.pdf
10. Belyaev Y. On the number of intersections of a level by a Gaussian stochastic process. Theory Probab. Appl. 11 (1966) 106-113
11. Berman S.M. An asymptotic formula for the distribution of the maximum of a Gaussian process with stationary increments. J. Appl. Probab. 22 (1985) 454-460
12. S.M. Berman, 1992. Sojourns and extremes of stochastic processes, in: The Wadworth and Brooks, Probability Series
13. Borell C. The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30 (1975) 207-216
14. Borell C. The Ehrhard inequality. C.R. Acad. Sci. Paris, Sér. I 337 (2003) 663-666
15. Cramér H., and Leadbetter M.R. Stationary and Related Stochastic Processes (1967), J. Wiley & Sons, New York
16. Cucker F., and Wschebor M. On the expected condition number of linear programming problems. Numer. Math. 94 (2003) 419-478
17. Fernique X. Régularité des trajectoires des fonctions aléatoires gaussiennes. Ecole d'Eté de Probabilités de Saint Flour (1974). Lecture Notes in Mathematics vol. 480 (1975), Springer-Verlag, New York
18. Fyodorov Y. Complexity of Random Energy Landscapes, Glass Transition and Absolute Value of Spectral Determinant of Random Matrices. Phys. Rev. Lett. 92 (2004) 240601
19. Fyodorov Y. Complexity of Random Energy Landscapes, Glass Transition and Absolute Value of Spectral Determinant of Random Matrices. Phys. Rev. Lett. 93 (2004) 149901 (1page) (erratum)
20. Kendall M.G., Stuart A., and Ord J.K. The Advanced Theory of Statistics vol. 3 (1987)
21. Landau H.J., and Shepp L.A. On the supremum of a Gaussian process. Sankya Ser. A 32 (1970) 369-378
22. Ledoux M. The Concentration of Measure Phenomenon (2001), American Math. Soc., Providence, RI
23. Ledoux M., and Talagrand M. Probability in Banach Spaces (1991), Springer-Verlag, New York
24. Marcus M.B. Level Crossings of a Stochastic Process with Absolutely Continuous Sample Paths. Ann. Probab. 5 (1977) 52-71
25. Marcus M.B., and Shepp L.A. Sample behaviour of Gaussian processes. Proc. Sixth Berkeley Symp. Math. Statist. Prob. vol. 2 (1972) 423-442
26. Mehta M.L. Random Matrices. 3rd edition (2004), Academic Press
27. Mercadier C. Numerical bounds for the distribution of the maximum of one- and two-dimensional processes. Adv. Appl. Probab. 38 1 (2006) 149-170
28. Piterbarg V.I. Comparison of distribution functions of maxima of Gaussian processes. Theory Probab. Appl. 26 (1981) 687-705
29. Piterbarg V.I. Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996), American Mathematical Society, Providence, Rhode Island
30. Rychlik I. New bounds for the first passage, wave-length and amplitude densities. Stochastic Process. Appl. 34 (1990) 313-339
31. Sudakov V.N., and Tsirelson B.S. Extremal properties of half spaces for spherically invariant measures. Zap. Nauchn. Sem. LOMI 45 (1974) 75-82 (in Russian)
32. Sun J. Tail Probabilities of the Maxima of Gaussian Random Fields. Ann. Probab. 21 (1993) 34-71
33. Talagrand M. Majorising measures: the general chaining. Ann. Probab. 24 (1996) 1049-1103
34. Taylor J.E., Takemura A., and Adler R.J. Validity of the expected Euler Characteristic heuristic. Ann. Probab. 33 4 (2005) 1362-1396
35. Wschebor M. Surfaces aléatoires. Mesure géometrique des ensembles de niveau. Lecture Notes in Mathematics vol. 1147 (1985), Springer-Verlag