Rates of convergence in the central limit theorem for empirical processes

Massart, Pascal

Annales de l'I.H.P. Probabilités et statistiques, Tome 22 (1986) no. 4, pp. 381-423

[1] K. Alexander, Ph. D. Dissertation, Mass. Inst. Tech., 1982.

[2] K. Alexander, Probability inequalities for empirical processes and a law of iterated logarithm. Annals of Probability, t. 12, 4, 1984, p. 1041-1067. | Zbl | MR

[3] P. Assouad, Densité et dimension, Ann. Inst. Fourier, Grenoble, t. 33, 3, 1983, p. 233- 282. | Zbl | MR | Numdam

[4] N.S. Bakhvalov, On approximate calculation of multiple integrals (in Russian). Vestnik Mosk. Ser. Mat. Mekh. Astron. Fiz. Khim., t. 4, 1959, p. 3-18. | Zbl

[5] G. Bennett, Probability inequalities for sums of independent random variables. J. Amer. Statist. Assoc., t. 57, 1962, p. 33-45. | Zbl

[6] I. Berkes, W. Philipp, Approximation theorems for independent and weakly dependent random vectors. Ann. Probability, t. 7, 1979, p. 29-54. | Zbl | MR

[7] P. Billingsley, Convergence of probability measures. Wiley, New York. | Zbl | MR

[8] I.S. Borisov, Abstracts of the Colloquium on non parametric statistical inference, Bundapest, 1980, p. 77-87.

[9] L. Breiman, On the tail behavior of sums of independent random variables. Z. Warschein. Verw. Geb., t. 9, 1967, p. 20-25. | Zbl | MR

[10] L. Breiman, Probability. Reading Mass., Addison-Wesley, 1968. | Zbl | MR

[11] E. Cabaña, On the transition density of a multidimensional parameter Wiener process with one barrier. J. Appl. Prob., t. 21, 1984, p. 197-200. | Zbl | MR

[12] E. Cabana, M. Wschebor, The two-parameter Brownian bridge. Annals of Probability, t. 10, 2, 1982, p. 289-302. | Zbl | MR

[13] D.L. Cohn, Measure theory. Birkhaiiser, Boston, 1980. | Zbl | MR

[14] M. Csörgo, P.A. Révész, A new method to prove Strassen type laws of invariance principle II. Z. Warschein. Verw. Geb., t. 31, 1975, p. 261-269. | Zbl

[15] H. Dehling, Limit theorems for sums of weakly dependent Banach space valued random variables. Z. Warschein. Verw. Geb., 1983, p. 391-432. | Zbl | MR

[16] L. Devroye, Bounds for the uniform deviations of empirical measures. J. of Multivar. Anal., t. 13, 1982, p. 72-79. | Zbl | MR

[17] Hu Inchi, A uniform bound for the tail probability of Kolmogorov-Smirnov statistics. The Annals of Statistics, t. 13, 2, 1985, p. 821-826. | Zbl | MR

[18] R.M. Dudley, The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Functional Analysis, t. 1, 1967, p. 290-330. | Zbl | MR

[19] R.M. Dudley, Metric entropy of some classes of sets with differential boundaries. J. Approximation Theory, t. 10, 1974, p. 227-236. | Zbl | MR

[20] R.M. Dudley, Central limit theorems for empirical measures. Ann. Probability, t. 6, 1978, p. 899-929 ; correction, t. 7, 1979, p. 909-911. | Zbl | MR

[21] R.M. Dudley, Saint-Flour, 1982. Lecture Notes in Mathematics n° 1097. | Zbl

[22] R.M. Dudley, Durst, Empirical Processes, Vapnik-Cervonenkis classes and Poisson processes. Proba. and Math. Stat. (Wroclaw), t. 1, 1981, p. 109-115. | Zbl | MR

[23] R.M. Dudley, W. Philipp, Invariance principles for sums of Banach spaces valued random elements and empirical processes. Z. Warschein. Verw. Geb., t. 82, 1983, p. 509-552. | Zbl | MR

[24] A. Dvoretzky, J.C. Kiefer, J. Wolfowitz, Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Ann. Math. Stat., t. 33, 1956, p. 642-669. | Zbl | MR

[25] X. Fernique, Régularité de processus gaussiens. Invent. Math., t. 12, 1971, p. 304-320. | Zbl | MR | EuDML

[26] P. Gaenssler, W. Stute, Empirical processes: a survey of results for independent and identically distributed random variables. Ann. Proba., t. 7, p. 193-243. | Zbl | MR

[27] M.E. Gine, J. Zinn, On the central limit theorem for empirical processes. Annals of Probability, t. 12, 4, 1984, p. 929-989. | Zbl

[28] V. Goodman, Distribution estimations for functionals of the two parameter Wiener process. Annals of Probability, t. 4, 6, 1976, p. 977-982. | Zbl | MR

[29] W. Hoeffding, Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc., t. 58, 1963, p. 13-30. | Zbl | MR

[30] I.A. Ibragimov, R.Z. Khasminskii, On the non-parametric density estimates. Zap. Naucha. Semin. LOMI, t. 108, 1981, p. 73-81. In Russian.

[31] N. Jain, M.B. Marcus, Central limit theorem for C(S)-valued random variables. J. Functional Analysis, t. 19, 1975, p. 216-231. | Zbl | MR

[32] J.P. Kahane, Some random series of functions. Lexington, Mass. D. C. Heuth, 1968. | Zbl | MR

[33] S. Karlin, H.M. Taylor, A first course in Stochastic Processes, 1971, Academic Press, New-York. | Zbl

[34] J.C. Kiefer, On large deviations of the empirical d. f. of vector chance variables and a law of iterated logarithm. Pacific J. Math., t. 11, 1961, p. 649-660. | Zbl | MR

[35] J.C. Kiefer, J. Wolfowitz, On the deviations of the empiric distribution function of vector variables. Trans. Amer. Math. Soc., t. 87, 1958, p. 173-186. | Zbl | MR

[36] A.N. Kolmogorov, V.M. Tikhomirov, ∈-entropy and ∈-capacity of sets in functional spaces. Amer. Math. Soc. Transl. ser. 2, t. 17, 1961, p. 277-364. | Zbl | MR

[37] J. Komlos, P. Major, G. Tusnady, An approximation of partial sums of independent RV' s and the sample DF. 1. Z. Warschein. Verw. Geb., t. 32, 1975, p. 111-131. | Zbl | MR

[38] L. Le Cam, A remark on empirical measures. In A Fetscheift for Erich L. Lehmam in Honor of his Sixty-Fifth Birthday, 1983, p. 305-327 Wadsworth, Belmont, California. | Zbl | MR

[39] P. Major, The approximation of partial sums of independent RV' s. Z. Warschein. Verw. Geb., t. 35, 1976, p. 213-220. | Zbl | MR

[40] P. Massart, Vitesse de convergence dans le théorème de la limite centrale pour le processus empirique. Thèse de 3e cycle n° 3545 de l'Université de Paris-Sud, 1983.

[41] P. Massart, Vitesses de convergence dans le théorème central limite pour des processus empiriques. Note aux C. R. A. S., t. 296, 20 juin 1983, Serie I, p. 937-940. | Zbl | MR

[42] P.A. Meyer, Martingales and stochastic integrals I. Lecture Notes in Mathematics, t. 284. | Zbl

[43] W. Philipp, Almost sure invariance principles for sums of B-valued random variables. Lecture Notes in Mathematics, t. 709, p. 171-193. | Zbl | MR

[44] D. Pollard, A central limit theorem for empirical processes. J. Australian Math. Soc. Ser. A, t. 33, 1982, p. 235-248. | Zbl | MR

[45] D. Pollard, Rates of strong uniform convergence, 1982. Preprint.

[46] R.J. Serfling, Probability inequalities for the sum in sampling without replacement. Ann. Stat., t. 2, 1, 1974, p. 39-48. | Zbl | MR

[47] M. Sion, On uniformization of sets in topological spaces. Trans. Amer. Math. Soc., t. 96, 1960, p. 237-245. | Zbl | MR

[48] A.V. Skorohod, Theory Prob. Appl., t. 21, 1976, p. 628-632. | Zbl

[49] V. Strassen, The existence of probability measures with given marginals. Ann. Math. Stat., t. 36, 1965, p. 423-439. | Zbl | MR

[50] G. Tusnady, A remark on the approximation of the sample DF in the multidimensional case. Periodica Math. Hung., t. 8, 1977, p. 53-55. | Zbl | MR

[51] V.N. Vapnik, A.Y. Cervonenkis, On the uniform convergence of relative frequencies of events to their probabilities. Theor. Prob. Appl., t. 16, 1971, p. 264-328. | Zbl

[52] V.V. Yurinskii, A smoothing inequality for estimates of the Lévy-Prohorov distance. Theory Prob. Appl., t. 20, 1975, p. 1-10. | Zbl | MR

[53] V.V. Yurinskii, On the error of the gaussian approximation for convolutions. Theor. Prob. Appl., t. 22, 1977, p. 236-247. | Zbl | MR

[54] J.E. Yukich, Uniform exponential bounds for the normalized empirical process, 1985. Preprint. | MR | Zbl