Comptes Rendus
no. 6
Probability theory/Statistics
An exponential inequality for suprema of empirical processes with heavy tails on the left
[Une inégalité exponentielle pour les suprema de processus empiriques avec queues lourdes sur la gauche]

Présenté par : Michel Talagrand

Antoine Marchina1, 2
1 Université Paris-Est, LAMA (UMR 8050), UPEM, UPEC, CNRS, 77454 Marne-la-Vallée, France
2 Laboratoire de mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 78035 Versailles, France
Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 537-544.

Dans cette Note, nous donnons des inégalités exponentielles pour les suprema de processus empiriques avec queues lourdes sur la gauche. Notre approche est basée sur une décomposition en martingale, associée à des inégalités de comparaison sur un cône de fonctions convexes, initialement introduit par Pinelis. Les constantes données sont explicites.

In this Note, we provide exponential inequalities for suprema of empirical processes with heavy tails on the left. Our approach is based on a martingale decomposition, associated with comparison inequalities over a cone of convex functions originally introduced by Pinelis. Furthermore, the constants in our inequalities are explicit.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2019.06.008

Antoine Marchina 1, 2

1 Université Paris-Est, LAMA (UMR 8050), UPEM, UPEC, CNRS, 77454 Marne-la-Vallée, France
2 Laboratoire de mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 78035 Versailles, France 
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Antoine Marchina. An exponential inequality for suprema of empirical processes with heavy tails on the left. Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 537-544. doi : 10.1016/j.crma.2019.06.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.06.008/

[1] R. Adamczak A tail inequality for suprema of unbounded empirical processes with applications to Markov chains, Electron. J. Probab., Volume 13 (2008) no. 34, pp. 1000-1034

[2] S. Arlot; M. Lerasle Choice of V for V-fold cross-validation in least-squares density estimation, J. Mach. Learn. Res., Volume 17 (2016) no. 208, pp. 1-50

[3] V. Bentkus Bounds for the stop loss premium for unbounded risks under the variance constraints, 2010 https://www.math.uni-bielefeld.de/sfb701/preprints/view/423 (Preprint on)

[4] S. Boucheron; O. Bousquet; G. Lugosi; P. Massart Moment inequalities for functions of independent random variables, Ann. Probab., Volume 33 (2005) no. 2, pp. 514-560 (p. 03)

[5] S. Boucheron; G. Lugosi; P. Massart Concentration Inequalities: A Nonasymptotic Theory of Independence, Oxford University Press, Oxford, UK, 2013

[6] S. Foss; D. Korshunov; S. Zachary An Introduction to Heavy-Tailed and Subexponential Distributions, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2013

[7] A. Goldenshluger; O. Lepski Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality, Ann. Stat., Volume 39 (2011) no. 3, pp. 1608-1632

[8] W. Hoeffding Probability inequalities for sums of bounded random variables, J. Amer. Stat. Assoc., Volume 58 (1963), pp. 13-30

[9] V. Koltchinskii Oracle Inequalities in Empirical Risk Minimization and Sparse Recovery Problems, École d'été de probabilités de Saint-Flour XXXVIII-2008, vol. 2033, Springer Science & Business Media, 2011

[10] C. Lacour; P. Massart; V. Rivoirard Estimator selection: a new method with applications to kernel density estimation, Sankhya, Ser. A, Volume 79 (2017) no. 2, pp. 298-335

[11] J. Lederer; S. van de Geer New concentration inequalities for suprema of empirical processes, Bernoulli, Volume 20 (2014) no. 4, pp. 2020-2038

[12] M. Ledoux On Talagrand's deviation inequalities for product measures, ESAIM Probab. Stat., Volume 1 (1995–1997), pp. 63-87

[13] A. Marchina Concentration inequalities for suprema of unbounded empirical processes, 2017 (Preprint on) | HAL

[14] A. Marchina Concentration inequalities for separately convex functions, Bernoulli, Volume 24 (2018), pp. 2906-2933

[15] P. Massart France, Lectures from the 33rd Summer School on Probability Theory Held in Saint-Flour, Lecture Notes in Mathematics, vol. 1896, Springer, Berlin, 6–23 July 2007 (with a foreword by Jean Picard)

[16] I. Pinelis Optimal tail comparison based on comparison of moments, High Dimensional Probability, Springer, 1998, pp. 297-314

[17] I. Pinelis Fractional sums and integrals of r-concave tails and applications to comparison probability inequalities, Adv. Stoch. Inequal., Volume 234 (1999), pp. 149-168

[18] I. Pinelis On normal domination of (super)martingales, Electron. J. Probab., Volume 11 (2006) no. 39, pp. 1049-1070

[19] I. Pinelis On the Bennett-Hoeffding inequality, Ann. Inst. Henri Poincaré Probab. Stat., Volume 50 (2014) no. 1, pp. 15-27

[20] E. Rio Inégalités exponentielles pour les processus empiriques, C. R. Acad. Sci. Paris, Ser. I, Volume 330 (2000), pp. 597-600

[21] E. Rio Inégalités de concentration pour les processus empiriques de classes de parties, Probab. Theory Relat. Fields, Volume 119 (2001) no. 2, pp. 163-175

[22] E. Rio Une inégalité de Bennett pour les maxima de processus empiriques. En l'honneur de J. Bretagnolle, D. Dacunha-Castelle, I. Ibragimov, Ann. Inst. Henri Poincaré Probab. Stat., Volume 38 (2002) no. 6, pp. 1053-1057

[23] M. Talagrand New concentration inequalities in product spaces, Invent. Math., Volume 126 (1996) no. 3, pp. 505-563

[24] A.W. van der Vaart; J.A. Wellner Weak Convergence and Empirical Processes: With Applications to Statistics, Springer Series in Statistics, Springer, New York, 1996

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