General Covariance-Based Conditions for Central Limit Theorems with Dependent Triangular Arrays

TL;DR

This paper introduces a covariance-based central limit theorem for dependent triangular arrays, broadening applicability to various dependent structures using simple, intuitive conditions.

Contribution

It provides a novel, covariance-focused CLT that encompasses diverse dependence types, simplifying assumptions compared to traditional mixing conditions.

Findings

Enables asymptotic normality in treatment effect estimation with spillovers.

Applies to covariance matrices, epidemic spread, and spatial processes.

Unifies various dependence structures under a common CLT framework.

Abstract

We present a general central limit theorem with simple, easy-to-check covariance-based sufficient conditions for triangular arrays of random vectors when all variables could be interdependent. The result is constructed from Stein's method, but the conditions are distinct from related work. We show that these covariance conditions nest standard assumptions studied in the literature such as $M$-dependence, mixing random fields, non-mixing autoregressive processes, and dependency graphs, which themselves need not imply each other. This permits researchers to work with high-level but intuitive conditions based on overall correlation instead of more complicated and restrictive conditions such as strong mixing in random fields that may not have any obvious micro-foundation. As examples of the implications, we show how the theorem implies asymptotic normality in estimating: treatment effects with spillovers in more settings than previously admitted, covariance matrices, processes with global dependencies such as epidemic spread and information diffusion, and spatial process with Mat\'{e}rn dependencies.