Sankhya A

, Volume 79, Issue 2, pp 355–383 | Cite as

The Bennett-Orlicz Norm

  • Jon A. WellnerEmail author


van de Geer and Lederer (Probab. Theory Related Fields 157(1-2), 225–250, 2013) introduced a new Orlicz norm, the Bernstein-Orlicz norm, which is connected to Bernstein type inequalities. Here we introduce another Orlicz norm, the Bennett-Orlicz norm, which is connected to Bennett type inequalities. The new Bennett-Orlicz norm yields inequalities for expectations of maxima which are potentially somewhat tighter than those resulting from the Bernstein-Orlicz norm when they are both applicable. We discuss cross connections between these norms, exponential inequalities of the Bernstein, Bennett, and Prokhorov types, and make comparisons with results of Talagrand (Ann. Probab., 17(4), 1546–1570, 1989, 1991), and Boucheron et al. (2013).


Bennett’s inequality Exponential bound Maximal inequality Orlicz norm Poisson Prokhorov’s inequality 

AMS (2000) subject classification

Primary: 60E15 60F10 Secondary: 60G50 33E20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



I owe thanks to Evan Greene and Johannes Lederer for several helpful conversations and suggestions. Thanks are also due to Richard Nickl for a query concerning Prokhorov’s inequality.


  1. Arcones, M. A. and Giné, E. 1995 On the law of the iterated logarithm for canonical U-statistics and processes. Stochastic Process. Appl., 58(2), 217–245.Google Scholar
  2. Bennett, G. (1962) Probability inequalities for the sum of independent random variables. Journal of the American Statistical Association, 57, 33–45.Google Scholar
  3. Birgé, L. and Massart, P. (1998) Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli, 4(3), 329–375.Google Scholar
  4. Boucheron S., Lugosi G. and Massart P. (2013). Concentration Inequalities. Oxford University Press, Oxford.CrossRefzbMATHGoogle Scholar
  5. Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. and Knuth, D. E. (1996) On the Lambert W function. Adv. Comput. Math., 5(4), 329–359.Google Scholar
  6. de la Peña, V. H. and Giné, E. (1999) Decoupling; From dependence to independence. Probability and its Applications (New York). Springer-Verlag, New York.Google Scholar
  7. Dudley, R. M. (1999) Uniform Central Limit Theorems, volume 63 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge.Google Scholar
  8. Ghosh, S. and Goldstein, L. (2011a) Applications of size biased couplings for concentration of measures. Electron. Commun. Probab., 16, 70–83.Google Scholar
  9. Ghosh, S. and Goldstein, L. (2011b) Concentration of measures via size-biased couplings. Probab. Theory Related Fields, 149(1-2), 271–278.Google Scholar
  10. Goldstein, L. and Iṡlak, Ü. (2014) Concentration inequalities via zero bias couplings. Statist. Probab. Lett., 86, 17–23.Google Scholar
  11. Hewitt, E. and Stromberg, K. (1975) Real and Abstract Analysis. Springer-Verlag, New York-Heidelberg. A modern treatment of the theory of functions of a real variable, Third printing, Graduate Texts in Mathematics, No. 25.Google Scholar
  12. Johnson, W. B., Schechtman, G. and Zinn, J. (1985) Best constants in moment inequalities for linear combinations of independent and exchangeable random variables. Ann. Probab., 13(1), 234–253.Google Scholar
  13. Krasnosel\(^{\prime }\)skiı̆ M. A. and Rutickiı̆, J. B. (1961) Convex Functions and Orlicz Spaces. Translated from the first Russian edition by Leo F. Boron. P. Noordhoff Ltd., Groningen.Google Scholar
  14. Kruglov, V. M. (2006) Strengthening of Prokhorov’s arcsine inequality. Theor. Probab. Appl., 50, 677–684. Transl. from Strengthening the Prokhorov arcsine inequality, Teor. Veroyatn. Primen., 50, (2005).Google Scholar
  15. Ledoux, M. (2001) The Concentration of Measure Phenomenon, volume 89 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI.Google Scholar
  16. Massart, P. (2000) About the constants in Talagrand’s concentration inequalities for empirical processes. Ann. Probab., 28(2), 863–884.Google Scholar
  17. Pisier, G. (1983) Some applications of the metric entropy condition to harmonic analysis. In Banach spaces, harmonic analysis, and probability theory (Storrs, Conn., 1980/1981), volume 995 of Lecture Notes in Math., pages 123–154. Springer, Berlin.Google Scholar
  18. Pollard, D. (1990) Empirical Processes: Theory and Applications. NSF-CBMS Regional Conference Series in Probability and Statistics, 2. Institute of Mathematical Statistics, Hayward CA; American Statistical Association, Alexandria, VA.Google Scholar
  19. Prokhorov, Y. V. (1959) An extremal problem in probability theory. Theor. Probability Appl., 4, 201–203.Google Scholar
  20. Rao, M. M. and Ren, Z. D. (1991). Theory of Orlicz spaces, volume 146 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York.Google Scholar
  21. Roy, R. and Olver, F. W. J. (2010) Elementary functions. IN NIST handbook of mathematical functions, pages 103-134. U.S. Dept. Commerce, Washington, DC.Google Scholar
  22. Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Inc., New York.zbMATHGoogle Scholar
  23. Stout, W. F. (1974) Almost Sure Convergence. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London. Probability and Mathematical Statistics, Vol. 24.Google Scholar
  24. Talagrand, M. (1989) Isoperimetry and integrability of the sum of independent Banach-space valued random variables. Ann. Probab., 17(4), 1546–1570.Google Scholar
  25. Talagrand, M. (1991) A new isoperimetric inequality and the concentration of measure phenomenon. In Geometric aspects of functional analysis (1989–90), volume 1469 of Lecture Notes in Math., pages 94–124. Springer, Berlin.Google Scholar
  26. Talagrand, M. (1994) Sharper bounds for Gaussian and empirical processes. Ann. Probab., 22(1), 28–76.Google Scholar
  27. Talagrand, M. (1995) Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math., 81, 73–205.Google Scholar
  28. van de Geer, S. and Lederer, J. (2013) The Bernstein-Orlicz norm and deviation inequalities. Probab. Theory Related Fields, 157(1-2), 225–250.Google Scholar
  29. van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer-Verlag, New York.CrossRefzbMATHGoogle Scholar

Copyright information

© Indian Statistical Institute 2017

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of WashingtonSeattleUSA

Personalised recommendations