Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-03T02:33:01.623Z Has data issue: false hasContentIssue false
  • English
  • Français

A central limit theorem for empirical processes

Part of: Limit theorems

Published online by Cambridge University Press:  09 April 2009

David Pollard
David Pollard
Affiliation:
Department of Statistics Yale UniversityNew Haven, Connecticut 06520, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The empirical measure Pn for independent sampling on a distribution P is formed by placing mass n−1 at each of the first n sample points. In this paper, n½(Pn − P) is regarded as a stochastic process indexed by a family of square integrable functions. A functional central limit theorem is proved for this process. The statement of this theorem involves a new form of combinatorial entropy, definable for classes of square integrable functions.

Keywords

empirical measureDonsker class of functionscombinatorial entropyweak convergence of empirical processescentral limit theoremsymmetrizationseparable stochastic processes.

MSC classification

Secondary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 33 , Issue 2 , October 1982 , pp. 235 - 248
Copyright
Copyright © Australian Mathematical Society 1982

References

Billingsley, P. (1968), Convergence of probability measures (Wiley).Google Scholar
Bolthausen, E. (1978), ‘Weak convergence of an empirical process indexed by the closed convex subsets of I2’, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 43, 173–181.Google Scholar
De Finetti, B. (1972), Probability, induction and statistics (Wiley).Google Scholar
Dudley, R. M. (1966), ‘Weak convergence of probabilities on nonseparable metric spaces and empirical measures on euclidean spaces’, Illinois J. Math. 10, 109–126.Google Scholar
Dudley, R. M. (1967), ‘Measures on non-separable metric spaces’, Illinois J. Math. 11, 449–453.CrossRefGoogle Scholar
Dudley, R. M. (1973), ‘Sample functions of the gaussian process’, Ann. Probability 1, 66–103.Google Scholar
Dudley, R. M. (1978), ‘Central limit theorems for empirical measures’, Ann. Probability 6, 899–929.CrossRefGoogle Scholar
Correction, Ann. Probability 7 (1979), 909–911.Google Scholar
Dudley, R. M. (1981a), ‘Donsker classes of functions’, Statistics and related topics, Proc. Symp. Ottawa 1980 (North-Holland), 341–352.Google Scholar
Dudley, R. M. (1981b), ‘Vapnik-Červonenkis Donsker classes of functions’, Proc. Colloque CNRS St. Flour 1980 (CNRS Paris), 251–269.Google Scholar
Durst, M. and Dudley, R. M. (1981), ‘Empirical processes, Vapnik-Červonenkis classes and Poisson processes’, Probability and Mathematical Statistics (Wroclaw) 1.Google Scholar
Gnedenko, B. V. (1968), The theori of probability (Chelsea, fourth edition).Google Scholar
Hoeffding, W. (1963), ‘Probability inequalities for sums of bounded random variables’, J. Amer. Statist. Assoc. 58, 13–30.CrossRefGoogle Scholar
Pollard, D. (1981), ‘Limit theorems for empirical processes’, Z. Wahrscheinlichkeitstheorie und Verw. Gehiete 57, 181–195.Google Scholar
Pollard, D. (1982), ‘A central limit theorem for k-means clustering’, Ann. Probability (to appear).CrossRefGoogle Scholar
Vapnik, V. N. and Červonenkis, A. Ya. (1971), ‘On the uniform convergence of relative frequencies to their probabilities’, Theor. Probability Appl. 16, 264–280.Google Scholar