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

TL;DR

This work analyzes high-dimensional CLTs where the dimension $p$ grows with the sample size $n$ under independent components. It partitions the underlying distribution into four regimes (I–IV) based on tail behavior and moment existence, and derives necessary and sufficient moment conditions for the CLT to hold over hyper-rectangles, yielding regime-specific growth rates for $\log p$ in terms of $n$. The main contributions are sharp, regime-dependent bounds (e.g., $\log p \le \Lambda_n^2/2$ in Class I with $\Lambda_n$ solving $h(n^{1/2}\Lambda_n)=\Lambda_n^2$) and analogous results for polynomial-tailed, near-domain-of-attraction, and infinite-variance cases, demonstrating that CLTs can hold under substantially weaker assumptions than previously known. The proofs leverage a partition-based approach combined with one-dimensional non-uniform Berry-Esseen inequalities and zones of normal attraction, delivering both sufficiency and necessity results and highlighting the roles of tail decay and truncated moments. These findings advance high-dimensional Gaussian approximation theory, informing statistical inference when $p$ grows rapidly relative to $n$ under minimal moment conditions.

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.

Theorems & Definitions (11)

• Theorem 1
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• Lemma 1
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• Lemma 6