## The Annals of Probability

- Ann. Probab.
- Volume 15, Number 3 (1987), 897-919.

### A Central Limit Theorem Under Metric Entropy with $L_2$ Bracketing

#### Abstract

Let $(\mathbf{S}, \rho)$ be a metric space, $(\mathbf{V}, \mathscr{V}, \mu)$ be a probability space, and $f: \mathbf{S} \times \mathbf{V} \rightarrow \mathbb{R}$ be a real-valued function on $\mathbf{S} \times \mathbf{V}$ which has mean zero and is Lipschitz in $L_2(\mu)$ with respect to $\rho$. Let $V$ be a random variable defined on $(\mathbf{V}, \mathscr{V}, \mu)$, and let $\{V_i: i \geq 1\}$ be a sequence of independent copies of $V$. The limiting behavior of the process $S_n(s) = n^{-1/2}\sum^n_{i=1} f(s, V_i)$ is studied under an integrability condition on the metric entropy with bracketing in $L_2(\mu)$. This metric entropy condition is analogous to one which implies the continuity of the limiting Gaussian process. A tightness result is derived which, in conjunction with the results of Andersen and Dobric (1987), shows that a central limit theorem holds for the $S_n$-process. This result generalizes those of Dudley (1978), Dudley (1981) and Jain and Marcus (1975).

#### Article information

**Source**

Ann. Probab. Volume 15, Number 3 (1987), 897-919.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176992072

**Digital Object Identifier**

doi:10.1214/aop/1176992072

**Mathematical Reviews number (MathSciNet)**

MR893905

**Zentralblatt MATH identifier**

0665.60036

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F17: Functional limit theorems; invariance principles

Secondary: 60F05: Central limit and other weak theorems

**Keywords**

Weak convergence functional central limit theorems empirical processes invariance principles law of the iterated logarithm

#### Citation

Ossiander, Mina. A Central Limit Theorem Under Metric Entropy with $L_2$ Bracketing. Ann. Probab. 15 (1987), no. 3, 897--919. doi:10.1214/aop/1176992072. http://projecteuclid.org/euclid.aop/1176992072.