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Necessary and sufficient conditions for the self-normalized central limit theorem for independent triangular arrays
Abstract
For a rowwise independent triangular array {Xn,i:1≤ i≤ n}, put Sn=∑i=1nXn,i and Vn2=∑i=1nXn,i2. We prove that Sn/VnxrightarrowdN(0,1) implies maxi≤ n|Xn,i|/Vnxrightarrowℙ0. The same assumption gives Vn2/bn2xrightarrowℙ1 for any choice of medians bn2 of Vn2. Combined with the results of Shao (2018), this yields necessary and sufficient conditions for the self-normalized central limit theorem and resolves his conjecture for independent sequences. The proof identifies the joint limits of the sum and its sum of squares, then uses the standard normal ratio to remove every non-Gaussian component.
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BibTeX
@misc{li2026necessary,
title = {{Necessary and sufficient conditions for the self-normalized central limit theorem for independent triangular arrays}},
author = {Xiang Li and Song-Hao Liu and Qi-Man Shao and Hao Shi and Jing-Yu Xu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08137},
eprint = {2611.08137},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08137},
}
RIS
TY - UNPB
TI - Necessary and sufficient conditions for the self-normalized central limit theorem for independent triangular arrays
AU - Xiang Li
AU - Song-Hao Liu
AU - Qi-Man Shao
AU - Hao Shi
AU - Jing-Yu Xu
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08137
UR - https://arxiv.org/abs/2611.08137
AB - For a rowwise independent triangular array {X_n,i:1≤ i≤ n}, put S_n=∑_i=1^nX_n,i and V_n^2=∑_i=1^nX_n,i^2. We prove that S_n/V_nxrightarrowdN(0,1) implies max_i≤ n|X_n,i|/V_nxrightarrowℙ0. The same assumption gives V_n^2/b_n^2xrightarrowℙ1 for any choice of medians b_n^2 of V_n^2. Combined with the results of Shao (2018), this yields necessary and sufficient conditions for the self-normalized central limit theorem and resolves his conjecture for independent sequences. The proof identifies the joint limits of the sum and its sum of squares, then uses the standard normal ratio to remove every non-Gaussian component.
ER -