An Extended Wichura Theorem, Definitions of Donsker Class, and Weighted Empirical Distributions

  • R. M. Dudley
Part of the Selected Works in Probability and Statistics book series (SWPS)


Let \(({\text{X}}, a >,{\text{P}})\) be a probability space, x(1),x(2),··· coordinates for the countable product of copies of X, and \( \mathcal{F} > \subset \mathcal{L} > ^2 ({\text{X}}, a >,{\text{P}})\). Let ν n be the normalized empirical measure n−1/2x(1)+···+δx(n)−nP). Let \(\ell ^\infty ( \mathcal{F} > )\) be the set of all bounded real functions G on \( \mathcal{F} > \) with norm \(||{\text{G}}||_{ \mathcal{F} > } : = \sup _{{\text{f}}{\varepsilon } \mathcal{F} > } |{\text{G(f)|}}\). Let μ(f) := ∫ fdμ for any measure μ. We are interested in central limit theorems where ν n converges in law for \(|| \cdot ||_{ \mathcal{F} > } \). The limit is a Gaussian process GP with mean 0 and covariance
$${\text{EG}}_{\text{P}} ({\text{f}}){\text{G}}_{\text{P}} {\text{(g)}} = {\text{P(fg)}} - {\text{P(f)P(g)}}: = ({\text{f,g}})_{{\text{P}},0},$$

\({\text{f,g}} \in \mathcal{F} > \). Say \( \mathcal{F} > \) is GPBUC if GP can be chosen to have each sample function bounded and uniformly continuous for (·,·)P, 0. Say the central limit theorem holds for \( \mathcal{F} > \) if \( \mathcal{F} > \) is GPBUC and we have convergence of upper integrals \(\lim _{{\text{n}} \to \infty } \int_{}^{\ast} {\text{H}} ( v > _{\text{n}} ){\text{d}}{\text{P}}^{\text{n}} = {\text{EH}}({\text{G}}_{\text{P}} )\) for every bounded continuous real function H on \(\ell ^\infty ( \mathcal{F} > )\). (Thus, as noted by J. Hoffmann-Jørgensen, convergence “in law” does not require definition of laws.) In Sec. 4 an extended Wichura’s theorem is proved: given such convergence in law, there exist almost surely convergent realizations. Sec. 5 shows that the new definition of central limit theorem holding is equivalent to the previous definition of “functional Donsker class.” Secs. 6–7 treat the case where \( \mathcal{F} > = \left\{ {{\text{M}}1_{{\text{n}} \geq {\text{M}}} :0 < {\text{M}} < \infty } \right\}\) for a random variable h. This case reduces to the study of weighted empirical distribution functions. Conditions for the central limit theorem are collected and made precise.


Statistical Theory Probability Space Central Limit Theorem Gaussian Process Real Function 
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  1. Chibisov, D. M. (1964). Some theorems on the limiting behavior of the empirical distribution function. Trudy Mat. Inst. Steklov (Moscow) 71 104–112; Selected Transls. Math. Statist. Prob. 6 147–156.MATHGoogle Scholar
  2. Cohn, Donald L. (1980). Measure Theory. Birkhäuser, Boston.MATHGoogle Scholar
  3. Csörgő, M. (1984). Review of Shorack and Wellner (1982). Math. Revs. 84f:60041.Google Scholar
  4. Donsker, M. D. (1952). Justification and extension of Doob’s heuristic approach to the Kolmogorov-Smirnov theorems, Ann. Math. Statist. 23 pp. 277–281.MATHCrossRefMathSciNetGoogle Scholar
  5. Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals. North-Holland, Amsterdam.MATHGoogle Scholar
  6. Dudley, R. M. (1966). Weak convergence of probabilities on nonseparable metric spaces and empirical measures on Euclidean spaces. Illinois J. Math. 10 109–126.MATHMathSciNetGoogle Scholar
  7. _____ (1967a). Measures on non-separable metric spaces. Ibid. 11 449–453.Google Scholar
  8. _____ (1967b). The sizes of compact subsets of Hilbert spaces and continuity of Gaussian processes. J. Functional Analysis 1 290–330.Google Scholar
  9. _____ (1968). Distances of probability measures and random variables. Ann. Math. Statist. 39 1563–1572.Google Scholar
  10. _____ (1973). Sample functions of the Gaussian process. Ann. Probab. 1 66–103.Google Scholar
  11. _____ (1978). Central limit theorems for empirical measures. Ibid. 6 899–929; Correction, ibid. 7 (1979) 909–911.Google Scholar
  12. _____ (1984). A course on empirical processes. Ecole d’été de probabilités de St.-Flour, 1982. Lecture Notes in Math. 1097 2–142.Google Scholar
  13. _____, and Walter Philipp (1983). Invariance principles for sums of Banach space valued random elements and empirical processes. Z. Wahrsch. verw. Geb. 62 509–552.Google Scholar
  14. Erdös, P. (1942). On the law of the iterated logarithm. Ann. Math. 43 419–436.CrossRefGoogle Scholar
  15. Feldman, Jacob (1971). Sets of boundedness and continuity for the canonical normal process. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 2, 357–368. Univ. Calif. Press.Google Scholar
  16. Fernandez, Pedro J. (1974). Almost surely convergent versions of sequences which converge weakly. Bol. Soc. Brasil. Math. 5 51–61.MATHCrossRefGoogle Scholar
  17. Halmos, P. (1950). Measure Theory. Princeton, Van Nostrand. 2d. printing, Springer, N. Y. 1974.Google Scholar
  18. Hoffmann-Jørgensen, J., and Niels Trolle Andersen (1984). Personal communication.Google Scholar
  19. _____ (1984). Envelopes and perfect random variables (preprint, section of forthcoming book).Google Scholar
  20. Itô, K., and H. P. McKean Jr. (1974). Diffusion processes and their sample paths. Springer, N. Y. (2d. printing, corrected).Google Scholar
  21. Marczewski, E., and R. Sikorski (1948). Measures in nonseparable metric spaces. Colloq. Math. 1 133–139.MATHMathSciNetGoogle Scholar
  22. O’Reilly, N. E. (1974). On the weak convergence of empirical processes in sup-norm metrics. Ann. Probab. 2 642–651.MATHCrossRefMathSciNetGoogle Scholar
  23. Petrovskii, I. G. (1935). Zur ersten Randwertaufgabe der Wärmeleitungagleichung. Compositio Math. 1 383–419.MATHMathSciNetGoogle Scholar
  24. Pollard, D. B. (1982). A central limit theorem for empirical processes. J. Austral. Math. Soc. Ser. A 33 235–248.MATHCrossRefMathSciNetGoogle Scholar
  25. Pyke, R. (1969). Applications of almost surely convergent constructions of weakly convergent processes. Lecture Notes in Math. 89 187–200.CrossRefMathSciNetGoogle Scholar
  26. Shorack, G. R. (1979). Weak convergence of empirical and quantile processes in sup-norm metrics via KMT-constructions. Stochastic Processes Applics. 9 95–98.MATHCrossRefMathSciNetGoogle Scholar
  27. _____, and J. Wellner (1982). Limit theorems and inequalities for the uniform empirical process indexed by intervals. Ann. Probab. 10 639–652.Google Scholar
  28. Skorohod, A. V. (1956). Limit theorems for stochastic processes. Theor. Prob. Appls. 1 261–290 (English), 289–319 (Russian).Google Scholar
  29. Stute, W. (1982). The oscillation behavior of empirical processes, Ann. Probab. 10 86–107.MATHCrossRefMathSciNetGoogle Scholar
  30. Talagrand, M. (1984). The Glivenko-Cantelli problem (preprint).Google Scholar
  31. Wichura, M. J. (1970). On the construction of almost uniformly convergent random variables with given weakly convergent image laws. Ann. Math. Statist. 41 284–291.MATHCrossRefMathSciNetGoogle Scholar

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Authors and Affiliations

  • R. M. Dudley
    • 1
  1. 1.MITCambridgeUSA

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