Optimal dimension growth rates in the high dimensional Central Limit Theorems
Abstract
We investigate the optimal dimension growth rates for the high-dimensional Central Limit Theorem (CLT) over half spaces, Euclidean balls and Borel convex sets, under the existence of fourth moments. It has been observed in the literature that d=o(nc) with 0< c≤ 1/2 is sufficient for the CLT to hold over Euclidean balls. As a consequence, the CLT over half spaces should ideally hold under similar dimension growth rates. In case of Borel convex sets, however d= o(nc) with 0< c≤ 2/5 is generally sufficient. The most recent results for CLT over Euclidean balls and Borel convex sets with respective rates d= o(√n) and d= o(n2/5) is due to Fang and Koike (2024) and under the existence of fourth moments. In this paper, we investigate whether these sufficient growth rates of d with respect to n are also necessary. Thus, our results completely characterize the sharpness of dimension growth rates under existence of fourth moments across half-spaces, Euclidean balls, and Borel convex sets.
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BibTeX
@misc{choudhury2026optimal,
title = {{Optimal dimension growth rates in the high dimensional Central Limit Theorems}},
author = {Mayukh Choudhury and Debraj Das},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24183},
eprint = {2610.24183},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24183},
}
RIS
TY - UNPB
TI - Optimal dimension growth rates in the high dimensional Central Limit Theorems
AU - Mayukh Choudhury
AU - Debraj Das
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24183
UR - https://arxiv.org/abs/2610.24183
AB - We investigate the optimal dimension growth rates for the high-dimensional Central Limit Theorem (CLT) over half spaces, Euclidean balls and Borel convex sets, under the existence of fourth moments. It has been observed in the literature that d=o(n^c) with 0< c≤ 1/2 is sufficient for the CLT to hold over Euclidean balls. As a consequence, the CLT over half spaces should ideally hold under similar dimension growth rates. In case of Borel convex sets, however d= o(n^c) with 0< c≤ 2/5 is generally sufficient. The most recent results for CLT over Euclidean balls and Borel convex sets with respective rates d= o(√n) and d= o(n^2/5) is due to Fang and Koike (2024) and under the existence of fourth moments. In this paper, we investigate whether these sufficient growth rates of d with respect to n are also necessary. Thus, our results completely characterize the sharpness of dimension growth rates under existence of fourth moments across half-spaces, Euclidean balls, and Borel convex sets.
ER -