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 scaled versions of the sum $\sum_{i=1}^{n}X_i$ in $\mathcal{R}^p$ uniformly over the class of hyper-rectangles $\mathcal{A}^{re}=\{\prod_{j=1}^{p}[a_j,b_j]\cap\mathcal{R}:-\infty\leq a_j\leq b_j \leq \infty, j=1,\ldots,p\}$, where $X_1,\dots,X_n$ are independent $p-$dimensional random vectors with each having independent and identically distributed (iid) components. We investigate the critical cut-off rate of $\log p$ below which the uniform central limit theorem (CLT) holds and above which it fails. According to some recent results of Chernozukov et al. (2017), it is well known that the CLT holds uniformly over $\mathcal{A}^{re}$ if $\log p=o\big(n^{1/7}\big)$. They also conjectured that for CLT to hold uniformly over $\mathcal{A}^{re}$, the optimal rate is $\log p = o\big(n^{1/3}\big)$. We show instead that under some conditions, the CLT holds uniformly over $\mathcal{A}^{re}$, when $\log p=o\big(n^{1/2}\big)$. More precisely, we show that if $\log p =\epsilon \sqrt{n}$ for some sufficiently small $\epsilon>0$, the normal approximation is valid with an error $\epsilon$, uniformly over $\mathcal{A}^{re}$. Further, we show by an example that the uniform CLT over $\mathcal{A}^{re}$ fails if $\limsup_{t\rightarrow \infty} n^{-(1/2+\delta)} \log p >0$ for some $\delta>0$. Hence the critical rate of the growth of $p$ for the validity of the CLT is given by $\log p=o\big(n^{1/2}\big)$.