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Optimal central limit theorem for bounded random variables in high dimensions
Abstract
Let W=n−1/2∑i=1n Xi, where the Xi are independent centered random vectors in ℝp with |Xij|≤ B almost surely. Suppose that Cov(W) has unit diagonal and smallest eigenvalue at least b2>0. We prove that the distance between W and a Gaussian vector with the same covariance, uniformly over axis-aligned rectangles, is at most Cmin{1,b−2Bn−1/2log3/2(ep)}. For fixed b, the dependence on summand size and dimension matches known lower bounds in growing-dimensional regimes. The proof combines a concentration estimate near rectangle boundaries with a carefully chosen Gaussian comparison.
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BibTeX
@misc{aronow2026optimal,
title = {{Optimal central limit theorem for bounded random variables in high dimensions}},
author = {P. M. Aronow and Patrick Lopatto},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07067},
eprint = {2609.07067},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07067},
}
RIS
TY - UNPB
TI - Optimal central limit theorem for bounded random variables in high dimensions
AU - P. M. Aronow
AU - Patrick Lopatto
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07067
UR - https://arxiv.org/abs/2609.07067
AB - Let W=n^−1/2∑_i=1^n X_i, where the X_i are independent centered random vectors in ℝ^p with |X_ij|≤ B almost surely. Suppose that Cov(W) has unit diagonal and smallest eigenvalue at least b^2>0. We prove that the distance between W and a Gaussian vector with the same covariance, uniformly over axis-aligned rectangles, is at most Cmin{1,b^−2Bn^−1/2log^3/2(ep)}. For fixed b, the dependence on summand size and dimension matches known lower bounds in growing-dimensional regimes. The proof combines a concentration estimate near rectangle boundaries with a carefully chosen Gaussian comparison.
ER -