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BERRY–ESSEEN BOUND AND LOCAL LIMIT THEOREM FOR THE COEFFICIENTS OF PRODUCTS OF RANDOM MATRICES

Part of: Limit theorems Ergodic theory Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  07 December 2022

Tien-Cuong Dinh ,
Lucas Kaufmann Open the ORCID record for Lucas Kaufmann [Opens in a new window]  and
Hao Wu
Tien-Cuong Dinh
Affiliation:
Department of Mathematics, National University of Singapore, 10, Lower Kent Ridge Road, Singapore 119076 (matdtc@nus.edu.sg)
Lucas Kaufmann*
Affiliation:
Center for Complex Geometry, Institute for Basic Science (IBS), 55 Expo-ro Yuseong-gu, Daejeon 34126 South Korea; Institut Denis Poisson, CNRS, Université d’Orléans, Rue de Chartres B.P. 6759, 45067 Orléans Cedex 2 France
Hao Wu
Affiliation:
Department of Mathematics, National University of Singapore, 10, Lower Kent Ridge Road, Singapore 119076 (matwu@nus.edu.sg)
*
lucas.kaufmann@univ-orleans.fr
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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

Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see ) 60F99: None of the above, but in this section 37A30: Ergodic theorems, spectral theory, Markov operators

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 23 , Issue 2 , March 2024 , pp. 705 - 735
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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