Asymptotic Independence in the Multivariate Central Limit Theorem

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.