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：我们提出了一个一般的中心极限定理，该定理具有简单且易于检验的基于协方差的充分条件，适用于所有变量可能相互依赖的三角随机向量阵列。该结果通过Stein方法构建，但其条件与相关研究不同。我们证明，这些协方差条件涵盖了文献中研究的标准假设，例如$M$-依赖性、混合随机场、非混合自回归过程以及依赖图，而这些假设本身并不相互蕴含。这使得研究人员能够基于整体相关性使用高层次但直观的条件，而非像随机场中可能缺乏明显微观基础的强混合等更复杂且限制性的条件。作为该定理应用的示例，我们展示了它如何在以下场景中估计渐近正态性：在比先前所承认的更多设置下具有溢出效应的处理效应、协方差矩阵、具有全局依赖性的过程（如流行病传播和信息扩散），以及具有Mat\'{e}rn依赖性的空间过程。