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Geometrical and Statistical Aspects of Probability in Banach Spaces pp 73–109Cite as

Rates of convergence in the central limit theorem for empirical processes

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Part of the book series: Lecture Notes in Mathematics ((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 L 2+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).

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References

  1. ALEXANDER, K. (1982). Ph. D. Dissertation, Mass. Inst. Tech. (1982).

    Google Scholar 

  2. ALEXANDER, K. Probability inequalities for empirical processes and a law of iterated logarithm. Annals of Probability (1984), vol. 12, No4, 1041–1067.

    Article  MathSciNet  MATH  Google Scholar 

  3. ASSOUAD, P. Densité et dimension, Ann. Inst. Fourier, Grenoble 33, 3 (1983) 233–282.

    Article  MathSciNet  MATH  Google Scholar 

  4. BAKHVALOV, N.S. On approximate calculation of multiple integrals (in Russian). Vestnik Mosk. Ser. Mat. Mekh. Astron. Fiz. Khim. (1959) 4, 3. 18.

    MathSciNet  Google Scholar 

  5. BENNETT, G. Probability inequalities for sums of independent random variables. J. Amer. Statist. Assoc. (1962), 57, 33–45.

    Article  MATH  Google Scholar 

  6. BERKES, I., PHILIPP, W. Approximation theorems for independent and weakly dependent random vectors. Ann. Probability (1979), 7, 29–54.

    Article  MathSciNet  MATH  Google Scholar 

  7. BILLINGSLEY, P. Convergence of probability measures. Wiley, New York.

    Google Scholar 

  8. BORISOV, I.S. Abstracts of the Colloquium on non parametric statistical inference, Budapest (1980), 77–87.

    Google Scholar 

  9. BREIMAN, L. On the tail behavior of sums of independent random variables. Z. Warschein. Verw. Geb. (1967), 9, 20–25.

    Article  MathSciNet  MATH  Google Scholar 

  10. BREIMAN, L. Probability. Reading Mass. Addison-Wesley (1968).

    Google Scholar 

  11. CABAÑA, E. On the transition density of a multidimensional parameter Wiener process with one barrier. J. Appl. Prob. (1984), 21, 197–200.

    Article  MathSciNet  MATH  Google Scholar 

  12. CABAÑA, E., WSCHEBOR, M. The two-parameter Brownian bridge. Pub. Univ. Simon Bolivar.

    Google Scholar 

  13. COHN, D.L. (1980). Measure theory. Birkhaüser, Boston (1980).

    Book  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. DEVROYE, L. Bounds for the uniform deviations of empirical measures. J. of Multivar. Anal. (1982), 12, 72–79.

    Article  MathSciNet  MATH  Google Scholar 

  17. INCHI HU, A uniform bound for the tail probability of Kolmogorov-Smirnov statistics. The Annals of Statistics (1985), Vol. 13, No2, 821–826.

    Article  MathSciNet  MATH  Google Scholar 

  18. DUDLEY, R.M. The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Functional Analysis (1967), 1, 290–330.

    Article  MathSciNet  MATH  Google Scholar 

  19. DUDLEY, R.M. Metric entropy of some classes of sets with differential boundaries. J. Approximation Theory (1974), 10, 227–236.

    Article  MathSciNet  MATH  Google Scholar 

  20. DUDLEY, R.M. Central limit theorems for empirical measures. Ann. Probability (1978), 6, p. 899–929; correction 7 (1979), 909–911.

    Article  MathSciNet  MATH  Google Scholar 

  21. DUDLEY, R.M. Saint Flour 1982. Lecture Notes in Mathematics no1097.

    Google Scholar 

  22. DUDLEY, R.M., DURST. Empirical Processes, Vapnik-Červonenkis classes and Poisson processes. Proba. and Math. Stat. (Wroclaw)1, 109–115 (1981).

    MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  25. FERNIQUE, X. Régularité de processus gaussiens. Invent. Math. (1971), 12, 304–320.

    Article  MathSciNet  MATH  Google Scholar 

  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. GINE, M.E., ZINN, J. On the central limit theorem for empirical processes. Annals of Probability (1984), vol. 12, no 4, 929–989.

    Article  MathSciNet  MATH  Google Scholar 

  28. GOODMAN, V. Distribution estimations for functionals of the two parameter Wiener process. Annals of Probability (1976), vol. 4, no 6, 977–982.

    Article  MATH  Google Scholar 

  29. HOEFFDING, W. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. (1963), 58, p. 13–30.

    Article  MathSciNet  MATH  Google Scholar 

  30. IBRAGIMOV, I.A., KHASMINSKII, R.Z. On the non-parametric density estimates. Zap. Naucha. Semin. LOMI 108, p. 73–81 (1981). In Russian.

    MathSciNet  Google Scholar 

  31. JAIN, N., MARCUS, M.B. Central limit theorem for C(S)-valued random variables. J. Functional Analysis (1975), 19, p. 216–231.

    Article  MathSciNet  MATH  Google Scholar 

  32. KAHANE, J.P. Some random series of functions. Lexington, Mass. D.C. Heuth (1968).

    MATH  Google Scholar 

  33. KARLIN, S. TAYLOR, H.M. A first course in Stochastic Processes (1971). Academic Press, New York.

    MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  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. MAJOR, P. The approximation of partial sums of independent RV's. Z. Warschein. Verw. Geb. (1976), 35, 213–220.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. MASSART, P. Rates of convergence in the central limit theorem for empirical processes (April 1985), submitted to Ann. Inst. Henri Poincaré.

    Google Scholar 

  43. PHILIPP, W. Almost sure invariance principles for sums of B-valued random variables. Lecture Notes in Mathematics709, 171–193.

    Google Scholar 

  44. POLLARD, D. A central limit theorem for empirical processes. J. Australian Math. Soc. Ser. A (1982), 33, 235–248.

    Article  MathSciNet  MATH  Google Scholar 

  45. POLLARD, D. Rates of strong uniform convergence (1982). Preprint.

    Google Scholar 

  46. SERFLING, R.J. Probability inequalities for the sum in sampling without replacement. Ann. Stat. (1974), vol. 2, no 1, 39–48.

    Article  MathSciNet  MATH  Google Scholar 

  47. SION, M. On uniformization of sets in topological spaces. Trans. Amer. Math. Soc. (1960), 96, 237–245.

    Article  MathSciNet  MATH  Google Scholar 

  48. SKOROHOD, A.V. Theory Prob. Appl. (1976), 21, 628–632.

    Article  Google Scholar 

  49. STRASSEN, V. The existence of probability measures with given marginals. Ann. Math. Stat. (1965), 36, 423–439.

    Article  MathSciNet  MATH  Google Scholar 

  50. TUSNADY, G. A remark on the approximation of the sample DF in the multidimensional case. Periodica Math. Hung. (1977), 8, 53–55.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  52. YURINSKII, V.V. A smoothing inequality for estimates of the Lévy-Prohorov distance. Theory Prob. Appl. (1975), 20, 1–10.

    Article  Google Scholar 

  53. YURINSKII, V.V. On the error of the gaussian approximation for convolutions. Theor. Prob. Appl. (1977), 22, 236–247.

    Article  MathSciNet  Google Scholar 

  54. YUKICH, J.E. Uniform exponential bounds for the normalized empirical process (1985). Preprint.

    Google Scholar 

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Authors and Affiliations

  1. Université Paris-Sud U.A. CNRS 743 "Statistique Appliquée", Mathématiques, Bât. 425, 91405, Orsay, France

    Pascal Massart

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  1. Pascal Massart
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Xavier Fernique Bernard Heinkel Paul-André Meyer Michael B. Marcus

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© 1986 Springer-Verlag

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Massart, P. (1986). Rates of convergence in the central limit theorem for empirical processes. In: Fernique, X., Heinkel, B., Meyer, PA., Marcus, M.B. (eds) Geometrical and Statistical Aspects of Probability in Banach Spaces. Lecture Notes in Mathematics, vol 1193. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0077101

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  • DOI: https://doi.org/10.1007/BFb0077101

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  • Online ISBN: 978-3-540-39826-4

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