Dual Induction CLT for High-dimensional m-dependent Data (2306.14299v2)

Abstract: 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.