Central Limit Theorems for Mixing Sequences of Random Variables Under Minimal Conditions

Abstract

Let $\{X_j, j \geq 1\}$ be a strictly stationary sequence of random variables with mean zero, finite variance, and satisfying a strong mixing condition. Denote by $S_n$ the $n$th partial sum and suppose that $\operatorname{Var} S_n$ is regularly varying of order 1. We prove that if $S_n (\operatorname{Var} S_n)^{-1/2}$ does not converge to zero in $L^1$, then $\{X_j, j \geq 1\}$ is in the domain of partial attraction of a Gaussian law. If, however, no subsequence of $\{S_n(\operatorname{Var} S_n)^{-1/2}, n \geq 1\}$ converges to zero in $L^1$ and if $E|S_n|$ is regularly varying of order $\frac{1}{2}$, then $\{X_j, j \geq 1\}$ is in the domain of attraction to a Gaussian law. In each case the norming constant can be chosen as $E|S_n|$.