Dual induction CLT for high-dimensional m-dependent data

We derive novel and sharp high-dimensional Berry–Esseen bounds for the sum of m-dependent random vectors over the class of hyperrectangles exhibiting only a poly-logarithmic dependence in the dimension. Our results hold under minimal assumptions, such as nondegenerate covariances and finite third moments, and exhibit an optimal sample complexity of order m(q−1)/(q−2)/n. Aside from logarithmic terms, the resulting rates match the optimal rates established in the univariate case. When specialized to the sums of independent nondegenerate random vectors, our results produce sharp and, in some cases, optimal rates under the weakest possible conditions. We develop a novel inductive relationship between anticoncentration 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.