## The Annals of Probability

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

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

Michael Klass and Henry Teicher

#### 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

**Keywords**

Central limit theorem exchangeable symmetrized r.v.'s tightness

#### 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