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.