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Rates of convergence in the central limit theorem for empirical processes

  • Pascal Massart
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1193)

Abstract

In this paper we study we uniform behavior of the empirical brownian bridge over families of functions F bounded by a function F (the observations are independent with common distribution P). Under some suitable entropy conditions which were already used by Kolčinskii and Pollard, we prove exponential inequalities in the uniformly bounded case where F is a constant (the classical Kiefer's inequality (1961) is improved), as well as weak and strong invariance principles with rates of convergence in the case where F belongs to L2+δ(P) with δε]0,1] (our results improve on Dudley, Philipp's results (1983) whenever F is a Vapnik-Červonenkis class in the uniformly bounded case and are new in the unbounded case).

Key words and phrases

Invariance principles empirical processes gaussian processes exponential bounds 

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References

  1. [1]
    ALEXANDER, K. (1982). Ph. D. Dissertation, Mass. Inst. Tech. (1982).Google Scholar
  2. [2]
    ALEXANDER, K. Probability inequalities for empirical processes and a law of iterated logarithm. Annals of Probability (1984), vol. 12, No4, 1041–1067.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    ASSOUAD, P. Densité et dimension, Ann. Inst. Fourier, Grenoble 33, 3 (1983) 233–282.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    BAKHVALOV, N.S. On approximate calculation of multiple integrals (in Russian). Vestnik Mosk. Ser. Mat. Mekh. Astron. Fiz. Khim. (1959) 4, 3. 18.MathSciNetGoogle Scholar
  5. [5]
    BENNETT, G. Probability inequalities for sums of independent random variables. J. Amer. Statist. Assoc. (1962), 57, 33–45.CrossRefzbMATHGoogle Scholar
  6. [6]
    BERKES, I., PHILIPP, W. Approximation theorems for independent and weakly dependent random vectors. Ann. Probability (1979), 7, 29–54.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    BILLINGSLEY, P. Convergence of probability measures. Wiley, New York.Google Scholar
  8. [8]
    BORISOV, I.S. Abstracts of the Colloquium on non parametric statistical inference, Budapest (1980), 77–87.Google Scholar
  9. [9]
    BREIMAN, L. On the tail behavior of sums of independent random variables. Z. Warschein. Verw. Geb. (1967), 9, 20–25.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    BREIMAN, L. Probability. Reading Mass. Addison-Wesley (1968).Google Scholar
  11. [11]
    CABAÑA, E. On the transition density of a multidimensional parameter Wiener process with one barrier. J. Appl. Prob. (1984), 21, 197–200.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    CABAÑA, E., WSCHEBOR, M. The two-parameter Brownian bridge. Pub. Univ. Simon Bolivar.Google Scholar
  13. [13]
    COHN, D.L. (1980). Measure theory. Birkhaüser, Boston (1980).CrossRefzbMATHGoogle Scholar
  14. [14]
    CSÖRGO, M., RÉVÉSZ, P. A new method to prove Strassen type laws of invariance principle II. Z. Warschein. Verw. Geb. (1975), 31, 261–269.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    DEHLING, H. Limit theorems for sums of weakly dependent Banach space valued random variables. Z. Warschein. Verw. Geb. (1983), 391–432.Google Scholar
  16. [16]
    DEVROYE, L. Bounds for the uniform deviations of empirical measures. J. of Multivar. Anal. (1982), 12, 72–79.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    INCHI HU, A uniform bound for the tail probability of Kolmogorov-Smirnov statistics. The Annals of Statistics (1985), Vol. 13, No2, 821–826.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    DUDLEY, R.M. The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Functional Analysis (1967), 1, 290–330.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    DUDLEY, R.M. Metric entropy of some classes of sets with differential boundaries. J. Approximation Theory (1974), 10, 227–236.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    DUDLEY, R.M. Central limit theorems for empirical measures. Ann. Probability (1978), 6, p. 899–929; correction 7 (1979), 909–911.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    DUDLEY, R.M. Saint Flour 1982. Lecture Notes in Mathematics no1097.Google Scholar
  22. [22]
    DUDLEY, R.M., DURST. Empirical Processes, Vapnik-Červonenkis classes and Poisson processes. Proba. and Math. Stat. (Wroclaw)1, 109–115 (1981).MathSciNetzbMATHGoogle Scholar
  23. [23]
    DUDLEY, R.M., PHILIPP, W. Invariance principles for sums of Banach spaces valued random elements and empirical processes. Z. Warschein. Veiw. Geb. (1983), 82, 509–552.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    DVORETZKY, A., KIEFER, J.C., WOLFOWITZ, J. Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Ann. Math. Stat. (1956), 33, 642–669.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    FERNIQUE, X. Régularité de processus gaussiens. Invent. Math. (1971), 12, 304–320.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    GAENSSLER, P., STUTE, W. Empirical processes: a survey of results for independent and identically distributed random variables. Ann. Proba.7, 193–243.Google Scholar
  27. [27]
    GINE, M.E., ZINN, J. On the central limit theorem for empirical processes. Annals of Probability (1984), vol. 12, no 4, 929–989.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    GOODMAN, V. Distribution estimations for functionals of the two parameter Wiener process. Annals of Probability (1976), vol. 4, no 6, 977–982.CrossRefzbMATHGoogle Scholar
  29. [29]
    HOEFFDING, W. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. (1963), 58, p. 13–30.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    IBRAGIMOV, I.A., KHASMINSKII, R.Z. On the non-parametric density estimates. Zap. Naucha. Semin. LOMI 108, p. 73–81 (1981). In Russian.MathSciNetGoogle Scholar
  31. [31]
    JAIN, N., MARCUS, M.B. Central limit theorem for C(S)-valued random variables. J. Functional Analysis (1975), 19, p. 216–231.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    KAHANE, J.P. Some random series of functions. Lexington, Mass. D.C. Heuth (1968).zbMATHGoogle Scholar
  33. [33]
    KARLIN, S. TAYLOR, H.M. A first course in Stochastic Processes (1971). Academic Press, New York.zbMATHGoogle Scholar
  34. [34]
    KIEFER, J.C. On large deviations of the empirical d.f. of vector chance variables and a law of iterated logarithm. Pacific J. Math. (1961), 11, 649–660.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    KIEFER, J.C., WOLFOWITZ, J. On the deviations of the empiric distribution function of vector chance variables. Trans. Amer. Math. Soc. (1958), 87, p. 173–186.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    KOLMOGOROV, A.N., TIKHOMIROV, V.M. ε-entropy and ε-capacity of sets in functional spaces. Amer. Math. Soc. Transl. (Ser. 2) (1961), 17, 277–364.Google Scholar
  37. [37]
    KOMLOS, J., MAJOR, P., TUSNADY, G. An approximation of partial sums of independent RV's and the sample DF. I. Z. Warschein. Verw. Geb. (1975), 32, p. 111–131.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    LE CAM, L. (1983). A remark on empirical measures. In A Festscheiht for Erich L. Lehmann in Honor of his Sixty-Fifth Birthday (1983), 305–327 Wadsworth, Belmont, California.Google Scholar
  39. [39]
    MAJOR, P. The approximation of partial sums of independent RV's. Z. Warschein. Verw. Geb. (1976), 35, 213–220.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    MASSART, P. Vitesse de convergence dans le théorème de la limite centrale pour le processus empirique. Thèse de 3e cycle no 3545 de l'Université de Paris-Sud (1983).Google Scholar
  41. [41]
    MASSART, P. Vitesses de convergence dans le théorème central limite pour des processus empiriques. Note aux C.R.A.S., t. 296 (20 juin 1983) Série I, 937–940.Google Scholar
  42. [42]
    MASSART, P. Rates of convergence in the central limit theorem for empirical processes (April 1985), submitted to Ann. Inst. Henri Poincaré.Google Scholar
  43. [43]
    PHILIPP, W. Almost sure invariance principles for sums of B-valued random variables. Lecture Notes in Mathematics709, 171–193.Google Scholar
  44. [44]
    POLLARD, D. A central limit theorem for empirical processes. J. Australian Math. Soc. Ser. A (1982), 33, 235–248.MathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    POLLARD, D. Rates of strong uniform convergence (1982). Preprint.Google Scholar
  46. [46]
    SERFLING, R.J. Probability inequalities for the sum in sampling without replacement. Ann. Stat. (1974), vol. 2, no 1, 39–48.MathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    SION, M. On uniformization of sets in topological spaces. Trans. Amer. Math. Soc. (1960), 96, 237–245.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    SKOROHOD, A.V. Theory Prob. Appl. (1976), 21, 628–632.CrossRefGoogle Scholar
  49. [49]
    STRASSEN, V. The existence of probability measures with given marginals. Ann. Math. Stat. (1965), 36, 423–439.MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    TUSNADY, G. A remark on the approximation of the sample DF in the multidimensional case. Periodica Math. Hung. (1977), 8, 53–55.MathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    VAPNIK, V.N. ČERVONENKIS, A.Y. On the uniform convergence of relative frequencies of events to their probabilities. Theor. Prob. Appl. (1971), 16, 264–28.CrossRefzbMATHGoogle Scholar
  52. [52]
    YURINSKII, V.V. A smoothing inequality for estimates of the Lévy-Prohorov distance. Theory Prob. Appl. (1975), 20, 1–10.CrossRefGoogle Scholar
  53. [53]
    YURINSKII, V.V. On the error of the gaussian approximation for convolutions. Theor. Prob. Appl. (1977), 22, 236–247.MathSciNetCrossRefGoogle Scholar
  54. [54]
    YUKICH, J.E. Uniform exponential bounds for the normalized empirical process (1985). Preprint.Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Pascal Massart
    • 1
  1. 1.Université Paris-Sud U.A. CNRS 743 "Statistique Appliquée"OrsayFrance

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