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 $α$-mixing random variables, but also non-$α$-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.