In this work, we provide a $(n/m)^{-1/2}$-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 telescoping method of Chen and Shao (2004) and the recursion method of Kuchibhotla and Rinaldo (2020). Performing a dual induction based on the relationships, we obtain tight Berry-Esseen bounds for dependent samples.

翻译：在这项工作中,我们提供了一笔美元(n/m) ⁇ -1/2}美元(美元)的有限样本Berry-Esseen在超矩类上为依赖美元的高维随机矢量而约束的Berry-Esseen,该捆绑对随机矢量,如非变性共变和有限第三秒规定了最低的假设。证据使用了抗浓缩不平等和Berry-Esseen界限之间的感应关系,这些关系受陈晓(2004年)和Shao(2004年)的电话覆盖法以及Kuchibhotla和Rinaldo(2020年)的循环法的启发。根据关系进行双重感应,我们获得了依赖样本的紧紧的Berry-Esseen界限。