The Central Limit Theorem and Berry--Esseen bound for logarithmic law of random determinants
Abstract
Let A=(An)n≥2 be a triangular array of random matrices, where An=(aij)1≤ i,j≤ n is an n× n random matrix with independent real entries satisfying 𝔼 aij=0 and 𝔼aij2=1, and put ℒn=log|det An| and Wnd(An):=(ℒn − 1/2log(n−1)!)/(√(1/2log n)), Wne(An):= (ℒn−𝔼 ℒn)/(√(1/2log n)). We prove that Wnd(An) ⇒ 𝒩(0,1), whenever the family {|aij|4/(√(log(e+|aij|))) }n≥ 2;1≤ i,j≤ n is uniformly integrable. If, in addition, the entries have uniformly bounded densities, then Wne(An) ⇒ 𝒩(0,1) whenever the family $\left\{\frac{|a_{ij}|^{4}}{\log(e+|a_{ij}|)}\right\}_{n\geq 2;1\leq i,j\leq n} $ is uniformly integrable. These two conditions are optimal at the level of universal moment assumptions. We further establish the corresponding Berry--Esseen bounds, and show that for 0<δ≤1/2, if supnmax1≤ i,j≤ n𝔼 |aij|4/({log(e+|aij|)}1/2−δ)<∞, then dK(Wnd(An),𝒩(0,1))≤ C(log n)−δ. For 0<γ≤1, if supnmax1≤ i,j≤ n𝔼 |aij|4/({log(e+|aij|)}1−γ)<∞ and the entries have uniformly bounded densities, then dK(Wne(An),𝒩(0,1))≤ C(log n)−γ. When δ= 1/2 and γ= 1, the bounds (log n)−1/2 and (log n)−1 are optimal, respectively. Our results improve the earlier Central Limit Theorem by \cite{BaoPanZhou2015} and the Berry--Esseen bound by \cite{NguyenVu2014}.
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BibTeX
@misc{liu2026central,
title = {{The Central Limit Theorem and Berry--Esseen bound for logarithmic law of random determinants}},
author = {Song-Hao Liu and Qi-Man Shao and Jing-Yu Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11688},
eprint = {2609.11688},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11688},
}
RIS
TY - UNPB
TI - The Central Limit Theorem and Berry--Esseen bound for logarithmic law of random determinants
AU - Song-Hao Liu
AU - Qi-Man Shao
AU - Jing-Yu Xu
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11688
UR - https://arxiv.org/abs/2609.11688
AB - Let A=(A_n)_n≥2 be a triangular array of random matrices, where A_n=(a_ij)_1≤ i,j≤ n is an n× n random matrix with independent real entries satisfying 𝔼 a_ij=0 and 𝔼a_ij^2=1, and put ℒ_n=log|det A_n| and W_n^d(A_n):=(ℒ_n − 1/2log(n−1)!)/(√(1/2log n)), W_n^e(A_n):= (ℒ_n−𝔼 ℒ_n)/(√(1/2log n)). We prove that W_n^d(A_n) ⇒ 𝒩(0,1), whenever the family {|a_ij|^4/(√(log(e+|a_ij|))) }_n≥ 2;1≤ i,j≤ n is uniformly integrable. If, in addition, the entries have uniformly bounded densities, then W_n^e(A_n) ⇒ 𝒩(0,1) whenever the family $\left\{\frac{|a_{ij}|^{4}}{\log(e+|a_{ij}|)}\right\}_{n\geq 2;1\leq i,j\leq n} $ is uniformly integrable. These two conditions are optimal at the level of universal moment assumptions. We further establish the corresponding Berry--Esseen bounds, and show that for 0<δ≤1/2, if sup_nmax_1≤ i,j≤ n𝔼 |a_ij|^4/({log(e+|a_ij|)}^1/2−δ)<∞, then d_K(W_n^d(A_n),𝒩(0,1))≤ C(log n)^−δ. For 0<γ≤1, if sup_nmax_1≤ i,j≤ n𝔼 |a_ij|^4/({log(e+|a_ij|)}^1−γ)<∞ and the entries have uniformly bounded densities, then d_K(W_n^e(A_n),𝒩(0,1))≤ C(log n)^−γ. When δ= 1/2 and γ= 1, the bounds (log n)^−1/2 and (log n)^−1 are optimal, respectively. Our results improve the earlier Central Limit Theorem by \cite{BaoPanZhou2015} and the Berry--Esseen bound by \cite{NguyenVu2014}.
ER -