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BERRY–ESSEEN BOUND AND LOCAL LIMIT THEOREM FOR THE COEFFICIENTS OF PRODUCTS OF RANDOM MATRICES
Published online by Cambridge University Press: 07 December 2022
Abstract
Let $\mu $ be a probability measure on
$\mathrm {GL}_d(\mathbb {R})$, and denote by
$S_n:= g_n \cdots g_1$ the associated random matrix product, where
$g_j$ are i.i.d. with law
$\mu $. Under the assumptions that
$\mu $ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we prove a Berry–Esseen bound with the optimal rate
$O(1/\sqrt n)$ for the coefficients of
$S_n$, settling a long-standing question considered since the fundamental work of Guivarc’h and Raugi. The local limit theorem for the coefficients is also obtained, complementing a recent partial result of Grama, Quint and Xiao.
MSC classification
- Type
- Research Article
- Information
- Journal of the Institute of Mathematics of Jussieu , Volume 23 , Issue 2 , March 2024 , pp. 705 - 735
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press
References
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