On the boundedness of Bernoulli processes

We present a positive solution to the so-called Bernoulli Conjecture concerning the characterization of sample boundedness of Bernoulli processes. We also discuss some applications and related open problems.

[1]  R. Dudley.  The Sizes of Compact Subsets of Hilbert Space and Continuity of Gaussian Processes , 1967 .

[2]  X. Fernique,et al.  Ecole D'Ete de Probabilites de Saint-Flour Iv 1974 , 1975 .

[3]  X. Fernique.  Regularite des trajectoires des fonctions aleatoires gaussiennes , 1975 .

[4]  G. Pisier,et al.  Random Fourier Series with Applications to Harmonic Analysis. , 1981 .

[5]  RANDOM FOURIER SERIES WITH APPLICATIONS TO HARMONIC ANALYSIS (Annals of Mathematics Studies, 101) , 1982 .

[6]  Review: Michael B. Marcus and Gilles Pisier, Random Fourier series with applications to harmonic analysis , 1983 .

[7]  M. Talagrand.  Regularity of gaussian processes , 1987 .

[8]  M. Talagrand.  An isoperimetric theorem on the cube and the Kintchine-Kahane inequalities , 1988 .

[9]  M. Talagrand,et al.  Probability in Banach Spaces: Isoperimetry and Processes , 1991 .

[10]  M. Talagrand,et al.  Probability in Banach spaces , 1991 .

[11]  M. Talagrand.  A simple proof of the majorizing measure theorem , 1992 .

[12]  Michel Talagrand,et al.  Regularity of Infinitely Divisible Processes , 1993 .

[13]  Michel Talagrand.  Constructions of majorizing measures Bernoulli processes and cotype , 1994 .

[14]  M. Talagrand.  THE SUPREMUM OF SOME CANONICAL PROCESSES , 1994 .

[15]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[16]  M. Talagrand.  Majorizing measures: the generic chaining , 1996 .

[17]  R. Latala,et al.  Sudakov Minoration Principle and Supremum of Some Processes , 1997 .

[18]  A note on the maximal inequalities for VC-classes , 1999 .

[19]  M. Ledoux.  The concentration of measure phenomenon , 2001 .

[20]  A. W. van der Vaart,et al.  Uniform Central Limit Theorems , 2001 .

[21]  M. Talagrand.  Majorizing measures without measures , 2001 .

[22]  M. Talagrand.  The Generic Chaining , 2005 .

[23]  M. Talagrand.  The Generic chaining : upper and lower bounds of stochastic processes , 2005 .

[24]  A theorem on majorizing measures , 2005, math/0510373.

[25]  On the boundedness of Bernoulli processes over thin sets , 2008 .

[26]  R. Lata.  On the boundedness of Bernoulli processes over thin sets , 2008 .

[27]  Michel Talagrand,et al.  Are many small sets explicitly small? , 2010, STOC '10.

[28]  R. Latala,et al.  On the suprema of Bernoulli processes , 2013 .

[29]  M. Talagrand.  Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems , 2014 .

[30]  Chaining and the Geometry of Stochastic Processes , 2014 .

[31]  M. Talagrand.  Upper and Lower Bounds for Stochastic Processes , 2021, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics.