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: "What is the optimal growth rate of the dimension p as a function of the sample size n for which the Central Limit Theorem (CLT) 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 n i=1 Xi in Rp uniformly over the class of hyper-rectangles Are = {p j=1[aj, bj ]R : -∞ ≤ aj ≤ bj ≤ ∞, j = 1, . . . ,p}, where X1, . . . , Xn are independent p-dimensional random vectors with each having independent and identically distributed (iid) components. We investigate the optimal cut-off rate of log p below 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 over Are if log p = on1/7. They also conjectured that for CLT to hold uniformly over Are, the optimal rate is log p = on1/3. We show instead that under some suitable conditions on the even moments and under vanishing odd moments, the CLT holds uniformly over Are, when logp = on1/2. More precisely, we show that if log p = √ n for some sufficiently small > 0, the normal approximation is valid with an error , uniformly over Are. Further, we show by an example that the uniform CLT over Are fails if lim supn→∞ n-(1/2+δ) logp > 0 for some δ > 0. Therefore, with some moment conditions the optimal rate of the growth of p for the validity of the CLT is given by log p = on1/2.