A Bennett concentration inequality and its application to suprema of empirical processes
[Une inégalité de concentration de type Bennett et son application aux maxima de processus empiriques]
Bousquet, Olivier1
1 CMAP, École polytechnique, 91128 Palaiseau, France
Comptes Rendus. Mathématique, Tome 334 (2002) no. 6, pp. 495-500.

Nous proposons deux inégalités de concentration pour des fonctions de n variables indépen-dantes. L'une d'elles permet d'obtenir une inégalité de déviation de type Bennett pour les processus empiriques indexés par des classes de fonctions bornées à droite. Cela améliore la version donnée par Rio [6] de l'inégalité de Talagrand [7] pour des observations équi-distribuées.

We introduce new concentration inequalities for functions on product spaces. They allow to obtain a Bennett type deviation bound for suprema of empirical processes indexed by upper bounded functions. The result is an improvement on Rio's version [6] of Talagrand's inequality [7] for equidistributed variables.

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DOI : 10.1016/S1631-073X(02)02292-6
Bousquet, Olivier 1

1 CMAP, École polytechnique, 91128 Palaiseau, France 
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Bousquet, Olivier. A Bennett concentration inequality and its application to suprema of empirical processes. Comptes Rendus. Mathématique, Tome 334 (2002) no. 6, pp. 495-500. doi : 10.1016/S1631-073X(02)02292-6. http://www.numdam.org/articles/10.1016/S1631-073X(02)02292-6/

[1] Boucheron, S.; Lugosi, G.; Massart, P. A sharp concentration inequality with applications, Random Structures Algorithms, Volume 16 (2000) no. 3, pp. 277-292

[2] S. Boucheron, G. Lugosi, P. Massart, Concentration of measure based on logarithmic Sobolev inequalities, 2001 (submitted)

[3] Ledoux, M. On Talagrand's deviation inequalities for product measures, Probab. Statist., Volume 1 (1996), pp. 63-87

[4] Massart, P. About the constants in Talagrand's inequality for empirical processes, Ann. Probab., Volume 29 (2000) no. 2, pp. 863-884

[5] Rio, E. Inégalités de concentration pour les processus empiriques de classes de parties, Probab. Theory Related Fields, Volume 119 (2000), pp. 163-175

[6] E. Rio, Une inegalité de Bennett pour les maxima de processus empiriques, Colloque en l'honneur de J. Bretagnolle, D. Dacunha-Castelle et I. Ibragimov, 2001 (to appear)

[7] Talagrand, M. New concentration inequalities in product spaces, Invent. Math., Volume 126 (1996), pp. 503-563

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