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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.

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Berry–esseen Bound and Local Limit Theorem for the Coefficients of Products of Random Matrices

Dinh¹,

Kaufmann

Wu³

2022

J. Inst. Math. Jussieu

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Berry–Esseen type bounds for the left random walk on GLd(R) under polynomial moment conditions

et al. 2023

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with common distribution µ. In this paper, under standard assumptions on µ (strong irreducibility and proximality), we prove Berry-Esseen type theorems for log( A n ) when µ has a polynomial moment. More precisely, we get the rate ((log n)/n) q/2−1 when µ has a moment of order q ∈]2, 3] and the rate 1/ √ n when µ has a moment of order 4, which significantly improves earlier results in this setting.

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Berry–Esseen type bounds for the matrix coefficients and the spectral radius of the left random walk on GL d (ℝ)

Cuny¹,

Dedecker²,

Merlevède³

et al. 2022

Comptes Rendus. Mathématique

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Berry–Esseen bounds and moderate deviations for random walks on GLd(R)

Cited by 3 publications

References 21 publications

Berry–esseen Bound and Local Limit Theorem for the Coefficients of Products of Random Matrices

Berry–esseen Bound and Local Limit Theorem for the Coefficients of Products of Random Matrices

Berry–Esseen type bounds for the left random walk on GLd(R) under polynomial moment conditions

Berry–Esseen type bounds for the matrix coefficients and the spectral radius of the left random walk on GL d (ℝ)

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