Special Invited Paper

Central Limit Theorems for Empirical Measures
  • R. M. Dudley
Part of the Selected Works in Probability and Statistics book series (SWPS)


The statistics used in Kolmogorov-Smirnov test are suprema of normalized empirical measures \({n^\frac{1}{2}\left(P_n-P\right)}\) or \(\left(mn\right)^\frac{1}{2}{\left({m+n}\right)}^{-\frac{1}{2}}\left({P_m}-{Q_n}\right)\) over a class \(\mathcal{C}\) of sets, namely the interval \(]-{\infty},a],a \in {\mathbb{R}}.\) Donsker (1952) Showed here that \({n^\frac{1}{2}\left(P_n-P\right)}\) converges in law, in the spacel \(l^{\infty}\left({\mathcal{C}}\right)\) of all bounded functions products of interval parallel to the axes in \(\mathbb{R}^k\) (Dudley (1966), (1967a)). Since \(\ell^\infty\mathcal{C}\) in the supremum norm is nonseparable, some measurablity problems (overlooked by Dansker) had to be trated. Recently Révész (1976) proved an iterated logarithm law for a much more general class of sets
$$\bigcap_{1\leq i\leq k}\left\{x:{f}_i\left\{\left(x_j:\neq i\right)\right\}< {x}_i < {g}_i\left(\left\{ {x_j:j\neq i}\right\}\right)\right\}$$
where \({f}_i\) and g i have a fixed bound on their partial derivatives of orders \(\leq k,\) and \(\mathbf{P}\) is the uniform measure on the unit cube. This paper will consider extensions of Donsker’s theorem to suitable classes of sets in general probability spaces.

Key words and phrases

Central limit theorems empirical measures Donsker classes Effros Borel structure metric entropy with inclusion two-sample case Vapnik-Červonenkis classes 


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  1. Aumann, R. J. (1961). Borel structures for function spaces. Illinois J. Math. 5 614–630.MATHMathSciNetGoogle Scholar
  2. Bennett, G. (1962). Probability inequalities for the sum of independent random variables. J. Amer. Statist. Assoc. 57 33–45.MATHCrossRefGoogle Scholar
  3. Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.MATHGoogle Scholar
  4. Bolthausen, E. (1976). On weak convergence of an empirical process indexed by the closed convex subsets of.2. (Preprint).Google Scholar
  5. Chernoff, H. (1952). A measure of asymptotic efficiency for tests based on the sum of observations. Ann. Math. Statist. 23 493–507.MATHCrossRefMathSciNetGoogle Scholar
  6. Christensen, J. P. R. (1971). On some properties of Effros Borel structure on spaces of closed subsets. Math. Ann. 195 17–23.MATHCrossRefMathSciNetGoogle Scholar
  7. Christensen, J. P. R. (1974). Topology and Borel Structure. North-Holland, Amsterdam; American Elsevier, New York.MATHGoogle Scholar
  8. Clements, G. F. (1963). Entropies of several sets of real valued functions. Pacific J. Math. 13 1085–1095.MATHMathSciNetGoogle Scholar
  9. Cover, T. M. (1965). Geometric and statistical properties of systems of linear inequalities with applications to pattern recognition. IEEE Trans. Elec. Comp. EC-14 326–334.CrossRefGoogle Scholar
  10. Dehardt, J. (1971). Generalizations of the Glivenko-Cantelli theorem. Ann. Math. Statist. 42 2050– 2055.MATHCrossRefMathSciNetGoogle Scholar
  11. Donsker, M. D. (1952). Justification and extension of Doob’s heuristic approach to the Kolmogorov- Smirnov theorems. Ann. Math. Statist. 23 277–281.MATHCrossRefMathSciNetGoogle Scholar
  12. 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
  13. Dudley, R. M. (1967a). Measures on non-separable metric spaces. Illinois J. Math. 11 449–453.MATHMathSciNetGoogle Scholar
  14. Dudley, R. M. (1967b). The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Functional Analysis 1 290–330.MATHCrossRefMathSciNetGoogle Scholar
  15. Dudley, R. M. (1973). Sample functions of the Gaussian process. Ann. Probability 1 66–103.MATHCrossRefMathSciNetGoogle Scholar
  16. Dudley, R. M. (1974). Metric entropy of some classes of sets with differentiable boundaries. J. Approximation Theory 10 227–236.MATHCrossRefMathSciNetGoogle Scholar
  17. Effros, E. G. (1965). Convergence of closed subsets in a topological space. Proc. Amer. Math. Soc. 16 929–931.MATHMathSciNetGoogle Scholar
  18. Federer, H. (1969). Geometric Measure Theory. Springer, Berlin.MATHGoogle Scholar
  19. Freedman, D. (1966). On two equivalence relations between measures. Ann. Math. Statist. 37 686–689.MATHCrossRefMathSciNetGoogle Scholar
  20. Harding, E. F. (1967). The number of partitions of a set of N points in k dimensions induced by hyperplanes. Proc. Edinburgh Math. Soc. (Ser. II) 15 285–289.MATHCrossRefMathSciNetGoogle Scholar
  21. Håusdorff, F. (1937). Set Theory, 3rd ed. Transl. by J. Aumann et al. (New York, Chelsea, 1962).Google Scholar
  22. Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.MATHCrossRefMathSciNetGoogle Scholar
  23. Jogdeo, K. and Samuels, S. M. (1968). Monotone convergence of binomial probabilities and a generalization of Ramanujan’s equation. Ann. Math. Statist. 39 1191–1195.MATHCrossRefGoogle Scholar
  24. Kelley, J. L. (1955). General Topology. Van Nostrand, Princeton.MATHGoogle Scholar
  25. Kolmogorov, A. N. (1956). On Skorohod convergence. Theor. Probability Appl. 1 215–222. (Teor. Verojatnost. i Primenen. 1 239–247, in Russian.)CrossRefGoogle Scholar
  26. Kolmogorov, A. N. and Tihomirov, V. M. (1959). ε-entropy and ε-capacity of sets in functional spaces. Uspehi Mat. Nauk 14, #2 (86), 3–86 (in Russian); (1961) Amer. Math. Soc. Transl. (Ser. 2), 17 277–364.MathSciNetGoogle Scholar
  27. Kuratowski, K. (1966). Topology, 1. Academic Press, New York.Google Scholar
  28. McShane, E. J. (1934). Extension of range of functions. Bull. Amer. Math. Soc. 40 837–842.CrossRefMathSciNetGoogle Scholar
  29. Neuhaus, G. (1971). On weak convergence of stochastic processes with multidimensional time parameter. Ann. Math. Statist. 42 1285–1295.MATHCrossRefMathSciNetGoogle Scholar
  30. Okamoto, Masashi (1958). Some inequalities relating to the partial sum of binomial probabilities. Ann. Inst. Statist. Math. 10 29–35.MATHCrossRefMathSciNetGoogle Scholar
  31. Parthasarathy, K. R. (1967). Probability Measures on Metric Spaces. Academic Press, New York.MATHGoogle Scholar
  32. Philipp, W. (1973). Empirical distribution functions and uniform distribution mod 1. In Diophantine Approximation and its Applications (C. F. Osgood, ed.) 211–234. Academic Press, New York.Google Scholar
  33. Pyke, R. and Shorack, G. (1968). Weak convergence of a two-sample empirical process and a new approach to Chernoff-Savage theorems. Ann. Math. Statist. 39 755–771.MATHCrossRefMathSciNetGoogle Scholar
  34. Rao, B. V. (1971). Borel structures for function spaces. Colloq. Math. 23 33–38.MATHMathSciNetGoogle Scholar
  35. Révész, P. (1976). Three theorems of multivariate empirical process. Lecture Notes in Math. 566 106–126.CrossRefGoogle Scholar
  36. Schläfli, Ludwig (1901, posth.). Theorie der vielfachen Kontinuität, in Gesammelte Math. Abhandlungen I (Basel, Birkhäuser, 1950).Google Scholar
  37. Sion, M. (1960). On uniformization of sets in topological spaces. Trans. Amer. Math. Soc. 96 237–245.MATHMathSciNetGoogle Scholar
  38. Skorohod, A. V. (1955). On passage to the limit from sequences of sums of independent random variables to a homogeneous random process with independent increments. Dokl. Akad. Nauk. SSSR 104 364–367 (in Russian).MathSciNetGoogle Scholar
  39. Steele, J. M. (1978). Empirical discrepancies and subadditive processes. Ann. Probability 6 118–127.MATHCrossRefMathSciNetGoogle Scholar
  40. Steiner, J. (1826). Einige Gesetze über die Theilung der Ebene und des Raumes. J. Reine Angew. Math. 1 349–364.MATHGoogle Scholar
  41. Stone, A. H. (1962). Non-separable Borel sets. Rozprawy Mat. 28 (41 pp.).Google Scholar
  42. Straf, M. L. (1972). Weak convergence of stochastic processes with several parameters. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 2 187–221. Univ. of California Press.MathSciNetGoogle Scholar
  43. Sun, Tze-Gong (1976). Ph.D. dissertation, Dept. of Mathematics, Univ. of Washington, Seattle.Google Scholar
  44. Szpilrajn, E. (1938). Ensembles indépendants et mesures non séparables. C. R. Acad. Sci. Paris 207 768–770.Google Scholar
  45. Talagrand, M. (1978). Les boules peuvent elles engendrer la tribu borélienne d’un espace métrisable non séparable? (Preprint).Google Scholar
  46. Uhlmann, W. (1966). Vergleich der hypergeometrischen mit der Binomial-Verteilung. Metrika 10 145–158.MATHCrossRefMathSciNetGoogle Scholar
  47. Vapnik, V. N. and Cervonenkis, A. Ya. (1971). On the uniform convergence of relative frequencies of events to their probabilities. Theor. Probability Appl. 16 264–280. (Teor. Verojatnost. i Primenen. 16 264–279, in Russian.)MATHCrossRefMathSciNetGoogle Scholar
  48. Vapnik, V. N. and Cervonenkis, A. Ya. (1974). Theory of Pattern Recognition (in Russian). Nauka, Moscow.Google Scholar
  49. Watson, D. (1969). On partitions of n points. Proc. Edinburgh Math. Soc. 16 263–264.MATHCrossRefMathSciNetGoogle Scholar
  50. Wichura, Michael 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

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • R. M. Dudley
    • 1
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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