Abstract
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.
Citation
Evarist Gine. Joel Zinn. "Gaussian Characterization of Uniform Donsker Classes of Functions." Ann. Probab. 19 (2) 758 - 782, April, 1991. https://doi.org/10.1214/aop/1176990450
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