The Annals of Probability

Concentration around the mean for maxima of empirical processes

T. Klein and E. Rio

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In this paper we give optimal constants in Talagrand’s concentration inequalities for maxima of empirical processes associated to independent and eventually nonidentically distributed random variables. Our approach is based on the entropy method introduced by Ledoux.

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Ann. Probab. Volume 33, Number 3 (2005), 1060-1077.

First available in Project Euclid: 6 May 2005

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Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings 60F10: Large deviations

Empirical processes Rademacher processes Talagrand’s inequality tensorization of entropy moderate deviations Bennett’s inequality concentration around the mean


Klein, T.; Rio, E. Concentration around the mean for maxima of empirical processes. Ann. Probab. 33 (2005), no. 3, 1060--1077. doi:10.1214/009117905000000044.

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  • Bobkov, S. (1996). Some extremal properties of the Bernoulli distribution. Theory Probab. Appl. 41 748--755.
  • Bousquet, O. (2003). Concentration inequalities for sub-additive functions using the entropy method. Stochastic Inequalities and Applications 56 213--247.
  • Klein, T. (2002). Une inégalité de concentration à gauche pour les processus empiriques. C. R. Acad. Sci. Paris Sér. I Math. 334 495--500.
  • Ledoux, M. (1996). On Talagrand's deviation inequalities for product measures. ESAIM Probab. Statist. 1 63--87.
  • Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Isoperimetry and Processes. Springer, Berlin.
  • Massart, P. (2000). About the constants in Talagrand's concentration inequalities for empirical processes. Ann. Probab. 28 863--884.
  • Panchenko, D. (2001). A note on Talagrand's concentration inequality. Electron. Comm. Probab. 6 55--65.
  • Panchenko, D. (2003). Symmetrization approach to concentration inequalities for empirical processes. Ann. Probab. 31 2068--2081.
  • Rio, E. (2000). Théorie asymptotique des processus aléatoires faiblement dépendants. In Mathématiques et Applications (J. M. Ghidaglia et X. Guyon, eds.) 31. Springer, Berlin.
  • Rio, E. (2001). Inégalités de concentration pour les processus empiriques de classes de parties. Probab. Theory Related Fields 119 163--175.
  • Rio, E. (2002). Une inégalité de Bennett pour les maxima de processus empiriques. Ann. Inst. H. Poincaré Probab. Statist. 38 1053--1057.
  • Talagrand, M. (1996). New concentration inequalities in product spaces. Invent. Math. 126 503--563.