Dual Induction CLT for High-dimensional m-dependent Data
Abstract
We derive novel and sharp high-dimensional Berry--Esseen bounds for the sum of -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 exhibit an optimal sample complexity of order . Aside from logarithmic terms, the resulting rates match the optimal rates established in the univariate case. When specialized to the sums of independent non-degenerate random vectors, our results produce sharp and, in some cases, optimal rates under the weakest possible conditions. We develop a novel 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.
Cite
@article{arxiv.2306.14299,
title = {Dual Induction CLT for High-dimensional m-dependent Data},
author = {Heejong Bong and Arun Kumar Kuchibhotla and Alessandro Rinaldo},
journal= {arXiv preprint arXiv:2306.14299},
year = {2025}
}
Comments
We have completed a major revision of the manuscript: the assumptions were reformulated to align with the style of results in the existing literature, the proofs were reorganized so that each induction hypothesis is invoked in a more clear and streamlined way, finally, a new bootstrap procedure was introduced for m-dependent data