## The Annals of Probability

### The Central Limit Theorem for Exchangeable Random Variables Without Moments

#### Abstract

If $\{X_n, n \geq 1\}$ is an exchangeable sequence with $(1/b_n(\sum^n_1X_i - a_n)) \rightarrow N(0, 1)$ for some constants $a_n$ and $0 < b_n \rightarrow \infty$ then $b_n/n^\alpha$ is slowly varying with $\alpha = 1$ or $\frac{1}{2}$ and necessary conditions (depending on $\alpha$) which are also sufficient, are obtained. Three such examples are given, one with infinite mean, one with no positive moments, and the third with almost all conditional distributions belonging to no domain of attraction of any law.

#### Article information

Source
Ann. Probab., Volume 15, Number 1 (1987), 138-153.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992260

Digital Object Identifier
doi:10.1214/aop/1176992260

Mathematical Reviews number (MathSciNet)
MR877594

Zentralblatt MATH identifier
0619.60024

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems

#### Citation

Klass, Michael; Teicher, Henry. The Central Limit Theorem for Exchangeable Random Variables Without Moments. Ann. Probab. 15 (1987), no. 1, 138--153. doi:10.1214/aop/1176992260. https://projecteuclid.org/euclid.aop/1176992260