Central Limit Theorem in high dimensions: The optimal bound on dimension growth rate

Abstract

In this article, we try to give an answer to the simple question: “What is the optimal growth rate of the dimensionppas a function of the sample sizennfor which the Central Limit Theorem (CLT) holds uniformly over the collection ofpp-dimensional hyper-rectangles ?”. Specifically, we are interested in the normal approximation of suitably scaled versions of the sum∑i=1nXi\sum _{i=1}^{n}X_iinRp\mathcal {R}^puniformly over the class of hyper-rectanglesAre={∏j=1p[aj,bj]∩R:−∞≤aj≤bj≤∞,j=1,…,p}\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\}, whereX1,…,XnX_1,\dots ,X_nare independentp−p-dimensional random vectors with each having independent and identically distributed (iid) components. We investigate the optimal cut-off rate oflog⁡p\log pbelow which the uniform CLT holds and above which it fails. According to some recent results of Chernozukov et al. [Ann. Probab. 45 (2017), pp. 2309–2352], it is well known that the CLT holds uniformly overAre\mathcal {A}^{re}iflog⁡p=o(n1/7)\log p=o\big (n^{1/7}\big ). They also conjectured that for CLT to hold uniformly overAre\mathcal {A}^{re}, the optimal rate islog⁡p=o(n1/3)\log p = o\big (n^{1/3}\big ). We show instead that under some suitable conditions on the even moments and under vanishing odd moments, the CLT holds uniformly overAre\mathcal {A}^{re}, whenlog⁡p=o(n1/2)\log p=o\big (n^{1/2}\big ). More precisely, we show that iflog⁡p=ϵn\log p =\epsilon \sqrt {n}for some sufficiently smallϵ>0\epsilon >0, the normal approximation is valid with an errorϵ\epsilon, uniformly overAre\mathcal {A}^{re}. Further, we show by an example that the uniform CLT overAre\mathcal {A}^{re}fails iflim supn→∞n−(1/2+δ)log⁡p>0\limsup _{ n\rightarrow \infty } n^{-(1/2+\delta )} \log p >0for someδ>0\delta >0. Therefore, with some moment conditions the optimal rate of the growth ofppfor the validity of the CLT is given bylog⁡p=o(n1/2)\log p=o\big (n^{1/2}\big ).