In this work, we provide a $1/\sqrt{n}$-rate finite sample Berry-Esseen bound for $m$-dependent high-dimensional random vectors over the class of hyper-rectangles. This bound imposes minimal assumptions on the random vectors such as nondegenerate covariances and finite third moments. The proof uses inductive relationships between anti-concentration inequalities and Berry-Esseen bounds, which are inspired by the classical Lindeberg swapping method and the concentration inequality approach for dependent data. Performing a dual induction based on the relationships, we obtain tight Berry-Esseen bounds for dependent samples.

翻译：本文在超矩形类上给出了m相依高维随机向量的$1/\sqrt{n}$速率有限样本Berry-Esseen界。该界对随机向量施加了最小假设，如非退化协方差和有限三阶矩。证明利用了反集中不等式与Berry-Esseen界之间的归纳关系，该关系受经典Lindeberg交换法和相依数据集中不等式方法的启发。基于这些关系进行对偶归纳，我们得到了相依样本的紧致Berry-Esseen界。