Comptes Rendus
no. 1
Statistics/Probability Theory
Tail behavior of anisotropic norms for Gaussian random fields
[Comportement des queues pour les normes anisotropes des champs aléatoires gaussiens]

Présenté par : Paul Deheuvels

Mikhail Lifshits1 ; Alexander Nazarov1 ; Yakov Nikitin1
1 Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg 198504, Russia
Comptes Rendus. Mathématique, Volume 336 (2003) no. 1, pp. 85-88.

Nous étudions les grandes déviations logarithmiques pour les normes anisotropes des champs gaussiens aléatoires de deux variables. Le problème est résolu en calculant des normes anisotropes pour les opérateurs intégraux engendrés par les covariances. Nous trouvons des valeurs exactes de telles normes pour quelques classes importantes de champs gaussiens.

We investigate the logarithmic large deviation asymptotics for anisotropic norms of Gaussian random functions of two variables. The problem is solved by the evaluation of the anisotropic norms of corresponding integral covariance operators. We find the exact values of such norms for some important classes of Gaussian fields.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)00013-4

Mikhail Lifshits 1 ; Alexander Nazarov 1 ; Yakov Nikitin 1

1 Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg 198504, Russia 
@article{CRMATH_2003__336_1_85_0,
     author = {Mikhail Lifshits and Alexander Nazarov and Yakov Nikitin},
     title = {Tail behavior of anisotropic norms for {Gaussian} random fields},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {85--88},
     publisher = {Elsevier},
     volume = {336},
     number = {1},
     year = {2003},
     doi = {10.1016/S1631-073X(02)00013-4},
     language = {en},
}
TY  - JOUR
AU  - Mikhail Lifshits
AU  - Alexander Nazarov
AU  - Yakov Nikitin
TI  - Tail behavior of anisotropic norms for Gaussian random fields
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 85
EP  - 88
VL  - 336
IS  - 1
PB  - Elsevier
DO  - 10.1016/S1631-073X(02)00013-4
LA  - en
ID  - CRMATH_2003__336_1_85_0
ER  - 
%0 Journal Article
%A Mikhail Lifshits
%A Alexander Nazarov
%A Yakov Nikitin
%T Tail behavior of anisotropic norms for Gaussian random fields
%J Comptes Rendus. Mathématique
%D 2003
%P 85-88
%V 336
%N 1
%I Elsevier
%R 10.1016/S1631-073X(02)00013-4
%G en
%F CRMATH_2003__336_1_85_0
Mikhail Lifshits; Alexander Nazarov; Yakov Nikitin. Tail behavior of anisotropic norms for Gaussian random fields. Comptes Rendus. Mathématique, Volume 336 (2003) no. 1, pp. 85-88. doi : 10.1016/S1631-073X(02)00013-4. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)00013-4/

[1] G. Bennett Schur multipliers, Duke Math. J., Volume 44 (1977), pp. 603-639

[2] A.P. Buslaev; V.A. Kondrat'ev; A.I. Nazarov On a family of extremum problems and the properties of an integral, Math. Notes, Volume 64 (1998), pp. 719-725

[3] N. Henze; Ya.Yu. Nikitin A new approach to goodness-of-fit testing based on the integrated empirical process, J. Nonparametr. Statist., Volume 12 (2000), pp. 391-416

[4] L.V. Kantorovich; G.P. Akilov Functional Analysis, Pergamon Press, 1982

[5] W.V. Li Comparison results for the lower tail of Gaussian seminorms, J. Theoret. Probab., Volume 5 (1992), pp. 1-31

[6] M.A. Lifshits Gaussian Random Functions, Kluwer Academic Publishers, 1995

[7] A.I. Nazarov On exact constant in the generalized Poincaré inequality, Probl. Math. Anal., Volume 24 (2002), pp. 155-180 (in Russian) J. Math. Sci., 112, 2002, pp. 4029-4047

[8] A.I. Nazarov; Ya.Yu. Nikitin Some extremal problems for Gaussian and empirical random fields, Proc. St. Petersburg Math. Soc., Volume 8 (2000), pp. 214-230 (in Russian) Amer. Math. Soc. Trans. Ser. 2, 205, 2002, pp. 189-202

[9] Ya.Yu. Nikitin Asymptotic Efficiency of Nonparametric Tests, Cambridge University Press, 1995

[10] H.P. Rosenthal; S.J. Sharek On tensor products of operators from Lp to Lq, Lecture Notes in Math., 1470, Springer, 1991, pp. 108-132

[11] E.D. Rothmann Tests of coordinate independence for a bivariate sample on a torus, Ann. Math. Statist., Volume 42 (1971), pp. 1962-1969

  • D. E. Apushkinskaya; A. A. Arkhipova; A. I. Nazarov; V. G. Osmolovskii; N. N. Uraltseva A Survey of Results of St. Petersburg State University Research School on Nonlinear Partial Differential Equations. I, Vestnik St. Petersburg University, Mathematics, Volume 57 (2024) no. 1, p. 1 | DOI:10.1134/s1063454124010035
  • Alexander Nazarov; Yulia Petrova L2-small ball asymptotics for Gaussian random functions: A survey, Probability Surveys, Volume 20 (2023) no. none | DOI:10.1214/23-ps20
  • V. R. Fatalov On the Laplace Method for Gaussian Measures in a Banach Space, Theory of Probability Its Applications, Volume 58 (2014) no. 2, p. 216 | DOI:10.1137/s0040585x97986539
  • Norbert Henze; Yakov Nikitin; Bruno Ebner Integral distribution-free statistics of -type and their asymptotic comparison, Computational Statistics Data Analysis, Volume 53 (2009) no. 9, p. 3426 | DOI:10.1016/j.csda.2009.02.018
  • Александр Семенович Холево; Alexander Semenovich Holevo Мультипликативность p-норм вполне положительных отображений и проблема аддитивности в квантовой теории информации, Успехи математических наук, Volume 61 (2006) no. 2, p. 113 | DOI:10.4213/rm1709
  • Вадим Роландович Фаталов; Vadim Rolandovich Fatalov Константы в асимптотиках вероятностей малых уклонений для гауссовских процессов и полей, Успехи математических наук, Volume 58 (2003) no. 4, p. 89 | DOI:10.4213/rm643

Cité par 6 documents. Sources : Crossref

Commentaires - Politique