The Annals of Probability

About the constants in Talagrand's concentration inequalities for empirical processes

Pascal Massart

Full-text: Open access


We propose some explicit values for the constants involved in the exponential concentration inequalities for empirical processes which are due to Talagrand. It has been shown by Ledoux that deviation inequalities for empirical processes could be obtained by iteration of logarithmic Sobolev type inequalities. Our approach follows closely that of Ledoux. The improvements that we get with respect to Ledoux’s work are based on refinements of his entropy inequalities and computations.

Article information

Ann. Probab. Volume 28, Number 2 (2000), 863-884.

First available in Project Euclid: 18 April 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60F10: Large deviations 94A17

Concentration of measure concentration inequalities deviation inequalities empirical processes


Massart, Pascal. About the constants in Talagrand's concentration inequalities for empirical processes. Ann. Probab. 28 (2000), no. 2, 863--884. doi:10.1214/aop/1019160263.

Export citation


  • [1] Baraud, Y. (1997). Model selection for regression on a fixed design. Probab. Theory Related Fields. To appear.
  • [2] Bennett, G. (1962). Probability inequalities for the sum of independent random variables. J. Amer. Statist. Assoc. 57 33-45.
  • [3] Birg´e, L. and Massart, P. (1997). From model selection to adaptive estimation. In Festschrift for Lucien LeCam: Research Papers in Probability and Statistics (D. Pollard, E. Torgersen and G. Yang, eds.) 55-87. Springer, NewYork.
  • [4] Birg´e, L. and Massart, P. (1998). Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli 4 329-375.
  • [5] Bobkov, S. (1995). On Gross' and Talagrand's inequalities on the discrete cube. Vestnik Syktyvkar Univ. Ser. 1 1 12-19. (in Russian.)
  • [6] Borell, C. (1975). The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30 207-216.
  • [7] Boucheron, S., Lugosi, G. and Massart, P. (1999). A sharp concentration inequality with applications. Random Structures Algorithms. To appear.
  • [8] Cirel'son, B. S., Ibragimov, I. A. and Sudakov, V. N. (1976). Norm of Gaussian sample function. Proceedings of the Third Japan-U.S.S.R. Symposium on Probability Theory. Lecture Notes in Math. 550 20-41. Springer, Berlin.
  • [9] Cirel'son, B. S. and Sudakov, V. N. (1978). Extremal properties of half spaces for spherically invariant measures. J. Soviet. Math. 9 9-18. [Translated from Zap. Nauch. Sem.
  • L.O.M.I. 41 14-24 (1974).]
  • [10] Davies, E. B. and Simon, B. (1984). Ultracontractivity and the heat kernel for Schr¨odinger operators and Dirichlet Laplacians. J. Funct. Anal. 59 335-395.
  • [11] Dembo, A. (1997). Information inequalities and concentration of measure. Ann. Probab. 25 927-939.
  • [12] Gross, L. (1975). Logarithmic Sobolev inequalities. Amer. J. Math. 97 1061-1083.
  • [13] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13-30.
  • [14] Holley, R. and Stroock, D. (1987). Logarithmic Sobolev inequalities and stochastic Ising models. J. Statist. Phys. 46 1159-1194.
  • [15] Ledoux, M. (1996). Isoperimetry and Gaussian analysis. Lectures on Probability Theory and Statistics. Ecole d'Et´e de Probabiliti´es de Saint Flour XXIV 165-294. Springer, Berlin.
  • [16] Ledoux, M. (1996). Talagrand deviation inequalities for product measures. ESAIM: Probab. Statist. 1 63-87. Available at
  • [17] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces (Isoperimetry and Processes). Springer, Berlin.
  • [18] Marton, K. (1996). Bounding d-distance by information divergence: a method to prove measure concentration. Ann. Probab. 24 927-939.
  • [19] Samson, P. M. (1997). In´egalit´es de concentration de la mesure pour des chaines de Markov et des processus -m´elangeants. Th´ese de l'Universit´e Paul Sabatier de Toulouse.
  • [20] Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Publications Math´ematiques de l'I.H.E.S. 81 73-205.
  • [21] Talagrand, M. (1996). Newconcentration inequalities in product spaces. Invent. Math. 126 505-563.