Central Limit Theorem in High Dimensions : The Optimal Bound on Dimension Growth Rate

In this article, we try to give an answer to the simple question: ``textit{What is the critical growth rate of the dimension $p$ as a function of the sample size $n$ for which the Central Limit Theorem holds uniformly over the collection of $p$-dimensional hyper-rectangles ?}. Specifically, we are interested in the normal approximation of suitably scal