## The Annals of Probability

- Ann. Probab.
- Volume 2, Number 4 (1974), 629-641.

### On the Central Limit Theorem for Sample Continuous Processes

#### Abstract

Let $\{X_k\}^\infty_{k = 1}$ be a sequence of independent centered random variables with values in $C(S)$ (i.e., sample continuous processes in $S$), $(S,d)$ being a compact metric space. This sequence is said to satisfy the central limit theorem if there exists a sample continuous Gaussian process on $S, Z$, such that $\mathscr{L}(\sum^n_{k = 1}X_k/n^{\frac{1}{2}})\rightarrow_{w^\ast} \mathscr{L}(Z)$ in $C'(C(S))$. In this paper some sufficient conditions are given for the central limit theorem to hold for $\{X_k\}^\infty_{k = 1}$; these conditions are on the modulus of continuity of the processes $X_k$ and they are expressed in terms of the metric entropy of distances associated to $\{X_k\}$. Then, in order to give some insight on these theorems, several results on the central limit theorem for particular processes (random Fourier and Taylor series, as well as more general processes on [0, 1]) are deduced.

#### Article information

**Source**

Ann. Probab., Volume 2, Number 4 (1974), 629-641.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996609

**Digital Object Identifier**

doi:10.1214/aop/1176996609

**Mathematical Reviews number (MathSciNet)**

MR370695

**Zentralblatt MATH identifier**

0288.60017

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60G99: None of the above, but in this section

**Keywords**

Central limit theorem for sample continuous processes $\epsilon$-entropy (metric) random Fourier series random Taylor series

#### Citation

M., Evarist Gine. On the Central Limit Theorem for Sample Continuous Processes. Ann. Probab. 2 (1974), no. 4, 629--641. doi:10.1214/aop/1176996609. https://projecteuclid.org/euclid.aop/1176996609