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 $D\lbrack 0, 1\rbrack$.
Publié le : 1984-11-14
Classification:
Central limit theorems,
empirical processes,
functional Donsker classes,
Gaussian processes,
metric entropy,
laws of large numbers,
60F17,
60B12,
60F05,
62E20
@article{1176993138,
author = {Gine, Evarist and Zinn, Joel},
title = {Some Limit Theorems for Empirical Processes},
journal = {Ann. Probab.},
volume = {12},
number = {4},
year = {1984},
pages = { 929-989},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993138}
}
Gine, Evarist; Zinn, Joel. Some Limit Theorems for Empirical Processes. Ann. Probab., Tome 12 (1984) no. 4, pp. 929-989. http://gdmltest.u-ga.fr/item/1176993138/