On the suprema of Bernoulli processes

Witold Bednorz; Rafał Latała

doi:10.1016/j.crma.2013.02.013

Probability Theory

On the suprema of Bernoulli processes

Présenté par : Michel Talagrand

Witold Bednorz ¹ ; Rafał Latała ¹

¹ Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Comptes Rendus. Mathématique, Volume 351 (2013) no. 3-4, pp. 131-134.

Résumé
Abstract

Dans cette note, nous annonçons la solution positive de la « conjecture de Bernoulli » concernant la caractérisation des processus de Bernoulli bornés. Nous en présentons des applications et discutons de questions ouvertes qui lui sont liées.

In this note we announce the affirmative solution of the so-called Bernoulli Conjecture concerning the characterization of the sample boundedness of Bernoulli processes. We also present some applications and discuss related open problems.

Reçu le : 2013-02-25
Accepté le : 2013-02-27
Publié le : 2013-02-01

DOI : 10.1016/j.crma.2013.02.013

Affiliations des auteurs :

Witold Bednorz ¹ ; Rafał Latała ¹

¹ Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

@article{CRMATH_2013__351_3-4_131_0,
     author = {Witold Bednorz and Rafa{\l} Lata{\l}a},
     title = {On the suprema of {Bernoulli} processes},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {131--134},
     publisher = {Elsevier},
     volume = {351},
     number = {3-4},
     year = {2013},
     doi = {10.1016/j.crma.2013.02.013},
     language = {en},
}

TY  - JOUR
AU  - Witold Bednorz
AU  - Rafał Latała
TI  - On the suprema of Bernoulli processes
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 131
EP  - 134
VL  - 351
IS  - 3-4
PB  - Elsevier
DO  - 10.1016/j.crma.2013.02.013
LA  - en
ID  - CRMATH_2013__351_3-4_131_0
ER  -

%0 Journal Article
%A Witold Bednorz
%A Rafał Latała
%T On the suprema of Bernoulli processes
%J Comptes Rendus. Mathématique
%D 2013
%P 131-134
%V 351
%N 3-4
%I Elsevier
%R 10.1016/j.crma.2013.02.013
%G en
%F CRMATH_2013__351_3-4_131_0

Witold Bednorz; Rafał Latała. On the suprema of Bernoulli processes. Comptes Rendus. Mathématique, Volume 351 (2013) no. 3-4, pp. 131-134. doi : 10.1016/j.crma.2013.02.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.02.013/

Bibliographie
Cité par

[1] W. Bednorz A theorem on majorizing measures, Ann. Probab., Volume 34 (2006), pp. 1771-1781

[2] W. Bednorz, R. Latała, On the boundedness of Bernoulli processes, preprint.

[3] R.M. Dudley The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Funct. Anal., Volume 1 (1967), pp. 290-330

[4] X. Fernique Regularité des trajectoires des fonctions aléatoires gaussiennes, École dʼété de probabilités de Saint-Flour, IV-1974, Lectures Notes in Math., vol. 480, Springer, Berlin, 1975, pp. 1-96

[5] R. Latała Sudakov minoration principle and supremum of some processes, Geom. Funct. Anal., Volume 7 (1997), pp. 936-953

[6] R. Latała A note on the maximal inequalities for VC classes, Atlanta, GA, 1997 (Contemp. Math.), Volume vol. 234, Amer. Math. Soc., Providence, RI (1999), pp. 125-134

[7] R. Latała On the boundedness of Bernoulli processes over thin sets, Electron. Commun. Probab., Volume 13 (2008), pp. 175-186

[8] R. Latała On weak tail domination of random vectors, Bull. Pol. Acad. Sci. Math., Volume 57 (2009), pp. 75-80

[9] M. Ledoux; M. Talagrand Probability in Banach Spaces. Isoperimetry and Processes, Springer-Verlag, Berlin, 1991

[10] M. Talagrand Regularity of Gaussian processes, Acta Math., Volume 159 (1987), pp. 99-149

[11] M. Talagrand The supremum of some canonical processes, Amer. J. Math., Volume 116 (1994), pp. 284-325

[12] M. Talagrand Majorizing measures without measures, Ann. Probab., Volume 29 (2001), pp. 411-417

[13] M. Talagrand The Generic Chaining. Upper and Lower Bounds of Stochastic Processes, Springer-Verlag, Berlin, 2005

[14] M. Talagrand, Chaining and the geometry of stochastic processes, in: Proceedings of the 6th European Congress of Mathematics, in press.

[15] M. Talagrand, Upper and Lower Bounds for Stochastic Processes, Modern Methods and Classical Problems, Ergebnisse der Mathematik, Springer-Verlag, in press.

Cité par Sources :

^☆ Research supported by the NCN grant DEC-2012/05/B/ST1/00412.

Commentaires - Politique