The Annals of Probability

Gaussian Characterization of Uniform Donsker Classes of Functions

Evarist Gine and Joel Zinn

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


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