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.

翻译：我们提出一个一般性的中心极限定理，该定理针对所有变量可能存在相互依赖的随机向量三角阵列，给出了简单且易于检验的基于协方差的充分条件。该结果通过Stein方法构造，但条件与相关研究截然不同。我们证明这些协方差条件囊括了文献中研究的标准假设，如$M$相依性、混合随机场、非混合自回归过程以及依赖图，这些假设本身未必相互蕴含。这使得研究者能够使用基于整体相关性的高层次直观条件，而非随机场中可能缺乏明显微观基础的复杂且限制性条件（如强混合性）。作为应用实例，我们展示该定理如何证明以下情形中的渐近正态性：在比以往更多场景中具有溢出效应的处理效应估计、协方差矩阵估计、具有全局依赖性的过程（如流行病传播与信息扩散），以及具有Matérn依赖性的空间过程。