  In this paper, we derive a central limit theorem for collections of weakly
correlated random variables indexed by discrete metric spaces, where the
correlation decays in the distance of the indices. The correlation structure we
study depends solely on the separability of mixed moments. Our investigation
yields a new proof for the CLT for $\alpha$-mixing random variables, but also
non-$\alpha$-mixing random variables fit within our framework, such as
MA($\infty$) processes. In particular, our results can be applied to
ARMA($p,q$) process with independent white noise.
