It is proved that, for classes of functions $\mathscr{F}$ satisfying some measurability, the empirical processes indexed by $\mathscr{F}$ and based on $P \in \mathscr{P}(S)$ satisfy the central limit theorem uniformly in $P \in \mathscr{P}(S)$ if and only if the $P$-Brownian bridges $G_p$ indexed by $\mathscr{F}$ are sample bounded and $\rho_p$ uniformly continuous uniformly in $P \in \mathscr{P}(S)$. Uniform exponential bounds for empirical processes indexed by universal bounded Donsker and uniform Donsker classes of functions are also obtained.
Publié le : 1991-04-14
Classification:
Empirical processes,
uniformity in $P$ in the central limit theorem,
uniform Donsker classes of functions,
uniformly pregaussian classes of functions,
exponential inequalities,
60F17,
60B12,
62E20
@article{1176990450,
author = {Gine, Evarist and Zinn, Joel},
title = {Gaussian Characterization of Uniform Donsker Classes of Functions},
journal = {Ann. Probab.},
volume = {19},
number = {4},
year = {1991},
pages = { 758-782},
language = {en},
url = {http://dml.mathdoc.fr/item/1176990450}
}
Gine, Evarist; Zinn, Joel. Gaussian Characterization of Uniform Donsker Classes of Functions. Ann. Probab., Tome 19 (1991) no. 4, pp. 758-782. http://gdmltest.u-ga.fr/item/1176990450/