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In this paper, we derive a central limit theorem for collections of weaklycorrelated random variables indexed by discrete metric spaces, where thecorrelation decays in the distance of the indices. The correlation structure westudy depends solely on the separability of mixed moments. Our investigationyields a new proof for the CLT for $\alpha$-mixing random variables, but alsonon-$\alpha$-mixing random variables fit within our framework, such asMA($\infty$) processes. In particular, our results can be applied toARMA($p,q$) process with independent white noise.

The Central Limit Theorem for Weakly Dependent Random Variables by the Moment Method