Metric entropy and the central limit theorem in Banach spaces

  • J. E. Yukich
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1193)


Banach Space Central Limit Theorem Separable Hilbert Space Empirical Process Iterate Logarithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alexander, K. S. (1984). Rates of growth and sample moduli for weighted empirical processes indexed by sets, preprint.Google Scholar
  2. 2.
    Borisov, I. S. (1983). Problem of accuracy of approximation in the central limit theorem for empirical measures, Siberskij Matematicheskij Zhurnal, v. 24, no. 6, pp. 14–25 = Siberian Mathematical Journal, July issue, 1984, pp. 833–843.Google Scholar
  3. 3.
    DeHardt, J. (1971). Generalization of the Glivenko-Cantelli theorem, Ann. Math. Statist., 42, pp. 2050–2055.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dudley, R. M. (1978). Central limit theorems for empirical measures, Ann. Prob. 6, pp. 899–929; correction 7, (1979) pp. 909–911.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dudley, R. M. (1981). Donsker classes of functions, Statistics and Related Topics (Proc. Symp. Ottawa 1980), pp. 341–352, New York, Amsterdam: North Holland.Google Scholar
  6. 6.
    Dudley, R. M. (1984). An extended Wichura theorem, definitions of Dansker class, and weighted empirical distributions, preprint.Google Scholar
  7. 7.
    Dudley, R. M. (1984). A course on empirical processes, Lecture Notes in Mathematics, no. 1097.Google Scholar
  8. 8.
    Dudley, R. M. and Walter Philipp (1983). Invariance principles for sums of Banach space valued random elements and empirical processes, Z. Wahrschein. verw. Geb., 62, pp. 509–552.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Durst, Mark and R. M. Dudley (1981). Empirical processes, Vapnik-Chervonenkis classes and Poisson processes, Prob. Math. Statist. (Wrocław), 1, no. 2, pp. 109–115.MathSciNetzbMATHGoogle Scholar
  10. 10.
    Gaenssler, P. (1983). Empirical processes. Institute of Mathematical Statistics Lecture Notes — Monograph Series 3.Google Scholar
  11. 11.
    Giné, E. and J. Zinn (1984). Some limit theorems for empirical processes, Ann. Prob., 12, pp. 929–989.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jain, N. (1977). Central limit theorem and related guestions in Banach space, Proceedings of Symposia in Pure Mathematics, vol. 31, pp. 55–65.CrossRefGoogle Scholar
  13. 13.
    Jain, N. and M. B. Marcus (1975). Central limit theorems for C(S) valued random variables, J. Functional Analysis, 19, pp. 216–231.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kolčinskii, V. I. (1981). On the central limit theorem for empirical measures, Theor. Probability Math. Statist., 24, pp. 71–82 = Teor. Verojatnost. i. Mat. Statist., 24, pp. 63–75.MathSciNetGoogle Scholar
  15. 15.
    Mourier, Edith (1951). Lois des grands nombres dans un espace de Banach, C. R. Acad. Sci. Paris, 232, pp. 923–925.MathSciNetzbMATHGoogle Scholar
  16. 16.
    Pisier, G. (1975). Le theórème de la limite centrale et la loi du logarithme itéré dans les espaces de Banach, suite et fin, Sém. Maurey-Schwartz 1975–76, Exposé IV, Ecole Polytechnique, Palaiseau.Google Scholar
  17. 17.
    Pollard, David B. (1982). A central limit theorem for empirical processes, J. Australian Math. Soc., Ser. A. 33, pp. 235–248.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pyke, Ronald (1984). Asymptotic results for empirical and partialsum processes: A review, The Canadian Journal of Statistics, 12, no. 4, pp. 241–264.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Yukich, J. E. (1982). Convergence of empirical probability measures, Ph.D. thesis (M.I.T.)Google Scholar
  20. 20.
    Yukich, J. E. (1984). The law of the iterated logarithm and the empirical characteristic function, preprint.Google Scholar
  21. 21.
    Yukich, J. E. (1985). Théorème limite central et entropie métrique dans les espaces de Banach, C. R. Acad. Sci. Paris, to appear.Google Scholar
  22. 22.
    Yukich, J. E. (1985). Weak convergence of the empirical characteristic function, Proc. of Amer. Math. Soc., to appear.Google Scholar
  23. 23.
    Yukich, J. E. (1985). Rates of convergence for classes of functions, preprint.Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • J. E. Yukich
    • 1
  1. 1.Institut de Recherche Mathématique AvancéeUniversité Louis PasteurStrasbourgFrance

Personalised recommendations