We derive novel and sharp high-dimensional Berry--Esseen bounds for the sum of $m$-dependent random vectors over the class of hyper-rectangles exhibiting only a poly-logarithmic dependence in the dimension. Our results hold under minimal assumptions, such as non-degenerate covariances and finite third moments, and yield a sample complexity of order $\sqrt{m/n}$, aside from logarithmic terms, matching the optimal rates established in the univariate case. When specialized to the sums of independent non-degenerate random vectors, we obtain sharp rates under the weakest possible conditions. On the technical side, we develop an inductive relationship between anti-concentration inequalities and Berry--Esseen bounds, inspired by the classical Lindeberg swapping method and the concentration inequality approach for dependent data, that may be of independent interest.

翻译：本文针对$m$相依随机向量之和在超矩形类上建立了新颖且尖锐的高维Berry–Esseen界，该界仅与维度呈多对数依赖关系。我们的结果在最小假设下成立，例如非退化协方差和有限三阶矩，并得到阶为$\sqrt{m/n}$（忽略对数项）的样本复杂度，与单变量情形下的最优率相匹配。当特化为独立非退化随机向量之和时，我们在最弱条件下获得了尖锐的收敛速率。在技术层面，我们受经典Lindeberg交换方法和相依数据集中不等式方法启发，推导了反集中不等式与Berry–Esseen界之间的归纳关系，这一关系可能具有独立研究价值。