Berry-Esseen bounds for products of random matrices under a 2+δ polynomial moment
Abstract
Let (gk)k ≥ 1 be a sequence of independent and identically distributed matrices with law μ on GL(d,ℝ), where d≥2 is an integer. Suppose that the closed semigroup generated by the support of μ is strongly irreducible and proximal and that the polynomial moment of order 2+δ is finite, where 0<δ≤ 1. We establish a Berry-Esseen bound of order n−δ/2 for log | gn ⋯ g1 v|, where v is a nonzero vector in ℝd and |·| is an arbitrary norm on ℝd. As an application, we also obtain a Berry-Esseen bound of order n−δ/2 for log | gn ⋯ g1|, where |·| is any matrix norm. The proof combines an m-dependent approximation with Stein's method. The argument uses identities for exchangeable pairs constructed by resampling individual matrices, together with a concentration inequality for sums of the approximating variables.
Related
- Berry--Esseen Bounds for Faces in Unimodular Triangulations of Random Polytopes
- Universality for the global spectrum of random inner-product kernel matrices in the polynomial regime
- Quantitative universality for products of i.i.d. random matrices
- Spectra of Random Polynomial Matrices: the Petaloid Law
- Characteristic polynomial of self-normalized random matrices
BibTeX
@misc{cai2026berryesseen,
title = {{Berry-Esseen bounds for products of random matrices under a $2+δ$ polynomial moment}},
author = {Zhi-Jun Cai and Qi-Man Shao and Hui Xiao and Zhuo-Song Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33342},
eprint = {2610.33342},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33342},
}
RIS
TY - UNPB
TI - Berry-Esseen bounds for products of random matrices under a 2+δ polynomial moment
AU - Zhi-Jun Cai
AU - Qi-Man Shao
AU - Hui Xiao
AU - Zhuo-Song Zhang
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.33342
UR - https://arxiv.org/abs/2610.33342
AB - Let (g_k)_k ≥ 1 be a sequence of independent and identically distributed matrices with law μ on GL(d,ℝ), where d≥2 is an integer. Suppose that the closed semigroup generated by the support of μ is strongly irreducible and proximal and that the polynomial moment of order 2+δ is finite, where 0<δ≤ 1. We establish a Berry-Esseen bound of order n^−δ/2 for log | g_n ⋯ g_1 v|, where v is a nonzero vector in ℝ^d and |·| is an arbitrary norm on ℝ^d. As an application, we also obtain a Berry-Esseen bound of order n^−δ/2 for log | g_n ⋯ g_1|, where |·| is any matrix norm. The proof combines an m-dependent approximation with Stein's method. The argument uses identities for exchangeable pairs constructed by resampling individual matrices, together with a concentration inequality for sums of the approximating variables.
ER -