Dual Induction CLT for High-dimensional m-dependent Data

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.