## The Annals of Probability

- Ann. Probab.
- Volume 12, Number 4 (1984), 929-989.

### Some Limit Theorems for Empirical Processes

#### Abstract

In this paper we provide a general framework for the study of the central limit theorem (CLT) for empirical processes indexed by uniformly bounded families of functions $\mathscr{F}$. From this we obtain essentially all known results for the CLT in this case; we improve Dudley's (1982) theorem on entropy with bracketing and Kolcinskii's (1981) CLT under random entropy conditions. One of our main results is that a combinatorial condition together with the existence of the limiting Gaussian process are necessary and sufficient for the CLT for a class of sets (modulo a measurability condition). The case of unbounded $\mathscr{F}$ is also considered; a general CLT as well as necessary and sufficient conditions for the law of large numbers are obtained in this case. The results for empiricals also yield some new CLT's in $C\lbrack 0, 1\rbrack$ and $D\lbrack 0, 1\rbrack$.

#### Article information

**Source**

Ann. Probab. Volume 12, Number 4 (1984), 929-989.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176993138

**Digital Object Identifier**

doi:10.1214/aop/1176993138

**Mathematical Reviews number (MathSciNet)**

MR757767

**Zentralblatt MATH identifier**

0553.60037

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F17: Functional limit theorems; invariance principles

Secondary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60F05: Central limit and other weak theorems 62E20: Asymptotic distribution theory

**Keywords**

Central limit theorems empirical processes functional Donsker classes Gaussian processes metric entropy laws of large numbers

#### Citation

Gine, Evarist; Zinn, Joel. Some Limit Theorems for Empirical Processes. Ann. Probab. 12 (1984), no. 4, 929--989. doi:10.1214/aop/1176993138. http://projecteuclid.org/euclid.aop/1176993138.