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

TL;DR

This paper establishes a central limit theorem for weakly dependent random variables with decaying correlations, extending classical results to broader classes including non-$\alpha$-mixing processes like ARMA models.

Contribution

It provides a new proof for the CLT under weak dependence conditions and includes non-$\alpha$-mixing processes such as MA($\infty$) and ARMA($p,q$).

Findings

Proves CLT for weakly correlated variables with decaying dependence.

Extends CLT applicability to non-$\alpha$-mixing processes.

Includes applications to ARMA processes with white noise.

Abstract

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.