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November, 1984 Some Limit Theorems for Empirical Processes
Evarist Gine, Joel Zinn
Ann. Probab. 12(4): 929-989 (November, 1984). DOI: 10.1214/aop/1176993138


In this paper we provide a general framework for the study of the central limit theorem (CLT) for empirical processes indexed by uniformly bounded families of functions $\mathscr{F}$. From this we obtain essentially all known results for the CLT in this case; we improve Dudley's (1982) theorem on entropy with bracketing and Kolcinskii's (1981) CLT under random entropy conditions. One of our main results is that a combinatorial condition together with the existence of the limiting Gaussian process are necessary and sufficient for the CLT for a class of sets (modulo a measurability condition). The case of unbounded $\mathscr{F}$ is also considered; a general CLT as well as necessary and sufficient conditions for the law of large numbers are obtained in this case. The results for empiricals also yield some new CLT's in $C\lbrack 0, 1\rbrack$ and $D\lbrack 0, 1\rbrack$.


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Evarist Gine. Joel Zinn. "Some Limit Theorems for Empirical Processes." Ann. Probab. 12 (4) 929 - 989, November, 1984.


Published: November, 1984
First available in Project Euclid: 19 April 2007

zbMATH: 0553.60037
MathSciNet: MR757767
Digital Object Identifier: 10.1214/aop/1176993138

Primary: 60F17
Secondary: 60B12 , 60F05 , 62E20

Keywords: central limit theorems , Empirical processes , functional Donsker classes , Gaussian processes , laws of large numbers , Metric entropy

Rights: Copyright © 1984 Institute of Mathematical Statistics

Vol.12 • No. 4 • November, 1984
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