## The Annals of Probability

### Gaussian Characterization of Uniform Donsker Classes of Functions

#### 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.

#### Article information

Source
Ann. Probab. Volume 19, Number 2 (1991), 758-782.

Dates
First available in Project Euclid: 19 April 2007

http://projecteuclid.org/euclid.aop/1176990450

Digital Object Identifier
doi:10.1214/aop/1176990450

Mathematical Reviews number (MathSciNet)
MR1106285

Zentralblatt MATH identifier
0734.60007

JSTOR