## The Annals of Probability

- Ann. Probab.
- Volume 7, Number 4 (1979), 662-671.

### Asymptotic Independence in the Multivariate Central Limit Theorem

William N. Hudson and Howard G. Tucker

#### Abstract

Necessary and sufficient conditions are given for asymptotic independence in the multivariate central limit theorem. If $\{X_n\}$ is a sequence of independent, identically distributed random variables whose common distribution is symmetric, and if the distribution of $X^2_1$ is in the domain of attraction of a stable distribution of characteristic exponent $\alpha$, then $\bar{X}$ and $s^2$ are asymptotically independent if and only if $1 \leqslant \alpha \leqslant 2$. If the components of a multivariate infinitely divisible distribution are pairwise independent, then they are independent.

#### Article information

**Source**

Ann. Probab., Volume 7, Number 4 (1979), 662-671.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994989

**Digital Object Identifier**

doi:10.1214/aop/1176994989

**Mathematical Reviews number (MathSciNet)**

MR537213

**Zentralblatt MATH identifier**

0411.60028

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

**Keywords**

Multivariate central limit theorem asymptotic independence stable distributions and their domains of attraction

#### Citation

Hudson, William N.; Tucker, Howard G. Asymptotic Independence in the Multivariate Central Limit Theorem. Ann. Probab. 7 (1979), no. 4, 662--671. doi:10.1214/aop/1176994989. https://projecteuclid.org/euclid.aop/1176994989