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The Law of the Iterated Logarithm for Empirical Processes

  • J. E. Yukich
Part of the Progress in Probability book series (PRPR, volume 20)

Abstract

Let (A,A,P) be a probability space and F ⊂ L2(A,A,P). Let X i , i ≥ 1, be a sequence of i.i.d. random variables with distribution P and let P n = n−1(δX 1 + ? + δX n ) be the n’th empirical measure for P. Using the methods employed to describe functional Donsker classes, we characterize when the normalized empirical process
$${{v}_{n}}(f):={{n}^{1/2}}\int{f(d{{P}_{n}}-dP)},f\in F$$
satisfies the compact and bounded law of the iterated logarithm (LIL) uniformly over F. Sufficient conditions implying the bounded LIL are obtained. In particular, we obtain two new metric entropy integral conditions implying the bounded LIL. Moreover, the integral condition is essentially the best possible.

Keywords

Central Limit Theorem Empirical Measure Empirical Process Iterate Logarithm Functional Limit Theorem 
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.

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Reference

  1. [1]
    K.S. Alexander, Probability inequalities for empirical processes and a law of the iterated logarithm, Ann. Prob. 12 (1984), 1041–1067.MATHCrossRefGoogle Scholar
  2. [2]
    N.T. Anderson, E. Giné, M. Ossiander and J. Zinn, The central limit theorem and the law of the iterated logarithm for empirical processes under local conditions, Prob. Theory Relat. Fields 77 (1988), 271–305.CrossRefGoogle Scholar
  3. [3]
    I.S. Borisov, Problem of accuracy of approximation in the central limit theorem for empirical measures, Siberskij Matematicheskig Zhurnal 24, No. 6 (1983), 14–25.Google Scholar
  4. I.S. Borisov, Problem of accuracy of approximation in the central limit theorem for empirical measures, Siberian Mathematical Journal, July issue, 1984, pp. 833–843.Google Scholar
  5. [4]
    R.M. Dudley, Central limit theorems for empirical measures, Ann. Prob. 6 (1978), 899–929.MathSciNetMATHCrossRefGoogle Scholar
  6. R.M. Dudley, Central limit theorems for empirical measures, Correction 7 (1979), pp. 909–911.MathSciNetMATHGoogle Scholar
  7. [5]
    R.M. Dudley, A course on empirical processes, pp. 1–142 in “École d’Été de Probabilités Saint-Flour XII-1982”, Lecture Notes in Math. 1097 (1984). Springer Verlag 1984.CrossRefGoogle Scholar
  8. [6]
    R.M. Dudley, An extended Wichura theorem, definition of Donsker class, and weighted empirical distributions, pp. 141–178 in “Probability in Banach Spaces V”, Lecture Notes in Math. 1153 (1984). Springer Verlag 1985.CrossRefGoogle Scholar
  9. [7]
    R.M. Dudley and W. Philipp, Invariance principles for sums of Banach space valued random elements and empirical processes, Z. Wahr. v. Geb. 62 (1983), 509–552.MathSciNetMATHCrossRefGoogle Scholar
  10. [8]
    M. Durst and R.M. Dudley, Empirical processes, Vapnik-Chervonenkis classes and Poisson processes, Prob. math. Statist. (Wroclaw) 1, No. 2 (1981), 109–115.MathSciNetGoogle Scholar
  11. [9]
    X. Fernique, Regularité des trajectoires des fonctions aléatoires gaussiennes, pp. 1–96 in “École d’Été de Probabilités Saint-Flour IV-1974”, Lecture Notes in Math. 480 (1974). Springer Verlag 1975.Google Scholar
  12. [10]
    H. Finkelstein, The law of the iterated logarithm for empirical distributions, Ann. Math. Statist. 42 (1971), 607–615.MathSciNetMATHCrossRefGoogle Scholar
  13. [11]
    E. Giné and J. Zinn, Some limit theorems for empirical processes, Ann. Prob. 12 (1984), 929–989.MATHCrossRefGoogle Scholar
  14. [12]
    E. Giné and J. Zinn, Lectures on the central limit theorem for empirical processes, pp. 50–113 in “Probability and Banach Spaces”, Proceedings Zaragoza 1985, Lecture Notes in Math. Springer Verlag 1986.Google Scholar
  15. [13]
    V.I. Kolcinskii, On the law of the iterated logarithm in the Strassen form for empirical measures, Theor. Prob. and Math. Stat. 25 (1982), 43–49.Google Scholar
  16. [14]
    J. Kuelbs, Kolmogorov’s law of the iterated logarithm for Banach space valued random variables, Ill. J. Math. 21 (1977), 784–800.MathSciNetMATHGoogle Scholar
  17. [15]
    J. Kuelbs and R.M. Dudley, Log log laws for empirical measures, Ann. Prob. 8 (1980), 405–418.MathSciNetMATHCrossRefGoogle Scholar
  18. [16]
    M. Ledoux, Loi du logarithme itéré dans C(S) et fonction caracteristique empirique, Z. Wahr. v. Geb. 60 (1982), 425–435.MathSciNetMATHCrossRefGoogle Scholar
  19. [17]
    M. Ledoux and M. Talagrand, Characterization of the law of the iterated logarithm in Banach spaces, Ann. Prob. 16 (1988), 1242–1264.MathSciNetMATHCrossRefGoogle Scholar
  20. [18]
    M. Ossiander, A central limit theorem under metric entropy with L2 bracketing, Ann. Prob. 15 (1987), 897–919.MathSciNetMATHCrossRefGoogle Scholar
  21. [19]
    D. Pollard, A central limit theorem for empirical processes, J. Australian Math. Soc., Ser. A 33 (1982), 235–248.MathSciNetMATHCrossRefGoogle Scholar
  22. [20]
    D. Pollard, Limit theorems for empirical processes, Z. Wahr. v. Geb. 57 (1981), 181–195.MathSciNetMATHCrossRefGoogle Scholar
  23. [21]
    M. Talagrand, Donsker classes and random geometry, Ann. Prob. 15 (1987), 897–919.CrossRefGoogle Scholar
  24. [22]
    M. Talagrand, Regularité des processus gaussiens, C.R. Acad.Sc. Paris, t. 301, Serie I, No. 7 (1985), 379–381.MathSciNetGoogle Scholar
  25. [23]
    J.E. Yukich, Weak convergence of the empirical characteristic function, Proc. Amer. Math. Soc. 95 (1985), 470–473.MathSciNetMATHCrossRefGoogle Scholar
  26. [24]
    J.E. Yukich, Uniform exponential bounds for the normalized empirical process, Studia Mathematics 84 (1986), 71–78.MathSciNetMATHGoogle Scholar
  27. [25]
    J.E. Yukich, Théorème limite centrale et l’entropie metrique dans les espaces de Banach, C.R. Acad. Sci., Paris, t. 301, Serie I, no. 6 (1985), 333–335.MathSciNetMATHGoogle Scholar
  28. [26]
    J.E. Yukich, Metric entropy and the central limit theorem in Banach spaces, pp. 113–128 in “Geometrical and Statistical Aspects of Probability in Banach Spaces”, Lecture Notes in Math. 1193. Springer Verlag 1986.CrossRefGoogle Scholar
  29. [27]
    J.E. Yukich, Convergence rates for function classes with applications to the empirical characteristic function, IIlinios Journal Math. 32 (1988), 81–97.MathSciNetMATHGoogle Scholar

Copyright information

© Birkhäuser Boston 1990

Authors and Affiliations

  • J. E. Yukich
    • 1
  1. 1.Department of MathematicsLehigh UniversityBethlehemUSA

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