Necessary and sufficient conditions for high dimensional Central Limit Theorem under moment conditions

TL;DR

This paper establishes the precise moment conditions necessary and sufficient for high dimensional CLTs to hold under independence, across different classes of tail behaviors, and identifies optimal dimension growth rates.

Contribution

It provides the first comprehensive characterization of necessary and sufficient moment conditions for high dimensional CLTs under independence, covering various tail decay regimes.

Findings

Optimal dimension growth rates identified for each moment class.

Necessary and sufficient moment conditions derived for CLT validity.

CLTs can hold under weaker conditions than previously known.

Abstract

High dimensional central limit theorems (the CLTs) have been extensively studied in recent years under a variety of sufficient moment conditions connecting the dimension growth rate with the tail decay rate. In this article, we investigate whether the existing moment conditions are also necessary under the independence of the components. We consider four exhaustive classes, viz. when underlying random variables (I) have all polynomial moments, (II) have some polynomial moment of order higher than two, (III) have only second moment but no polynomial moment higher than two exists, and (IV) have infinite second moment, but belong to the domain of attraction of normal distribution. We find the optimal growth rate of the dimension with respect to sample size in the high dimensional CLTs over hyper-rectangles. More precisely, we derive necessary and sufficient moment conditions for the validity of the the CLT over hyper-rectangles in each of the four regimes listed above, showing that the CLT may hold under much weaker conditions compared to those considered in the existing literature.