## The Annals of Probability

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

### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/aop/1176990450

**Mathematical Reviews number (MathSciNet)**

MR1106285

**Zentralblatt MATH identifier**

0734.60007

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F17: Functional limit theorems; invariance principles

Secondary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 62E20: Asymptotic distribution theory

**Keywords**

Empirical processes uniformity in $P$ in the central limit theorem uniform Donsker classes of functions uniformly pregaussian classes of functions exponential inequalities

#### Citation

Gine, Evarist; Zinn, Joel. Gaussian Characterization of Uniform Donsker Classes of Functions. Ann. Probab. 19 (1991), no. 2, 758--782. doi:10.1214/aop/1176990450. https://projecteuclid.org/euclid.aop/1176990450