Probability Theory
On the suprema of Bernoulli processes
Comptes Rendus. Mathématique, Volume 351 (2013) no. 3-4, pp. 131-134.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.02.013
Bednorz, Witold 1; Latała, Rafał 1

1 Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
@article{CRMATH_2013__351_3-4_131_0,
     author = {Bednorz, Witold and Lata{\l}a, Rafa{\l}},
     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},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2013.02.013/}
}
TY  - JOUR
AU  - Bednorz, Witold
AU  - Latała, Rafał
TI  - On the suprema of Bernoulli processes
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 131
EP  - 134
VL  - 351
IS  - 3-4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2013.02.013/
DO  - 10.1016/j.crma.2013.02.013
LA  - en
ID  - CRMATH_2013__351_3-4_131_0
ER  - 
%0 Journal Article
%A Bednorz, Witold
%A Latała, Rafał
%T On the suprema of Bernoulli processes
%J Comptes Rendus. Mathématique
%D 2013
%P 131-134
%V 351
%N 3-4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2013.02.013/
%R 10.1016/j.crma.2013.02.013
%G en
%F CRMATH_2013__351_3-4_131_0
Bednorz, Witold; Latała, Rafał. 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. http://www.numdam.org/articles/10.1016/j.crma.2013.02.013/

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

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

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

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

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

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

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

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

[13] Talagrand, M. 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.

Cited by Sources:

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