Berry-Esseen bounds with targets and Local Limit Theorems for products of random matrices

2021/11/28 by Dinh, Tien-Cuong, Kaufmann, Lucas, Wu, Hao
#60B15 #60B20 #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2111.14109

Abstract

Let μ be a probability measure on GL_d(\mathbb R) and denote by S_n:= g_n ⋯ g₁ the associated random matrix product, where g_j's are i.i.d.'s with law μ. We study statistical properties of random variables of the form σ(S_n,x) + u(S_n x), where x ∈ \mathbb P^d-1, σ is the norm cocycle and u belongs to a class of admissible functions on \mathbb P^d-1 with values in \mathbb R ∪ \± ∞\. Assuming that μ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we obtain optimal Berry-Esseen bounds and the Local Limit Theorem for such variables using a large class of observables on \mathbb R and Hölder continuous target functions on \mathbb P^d-1. As particular cases, we obtain new limit theorems for σ(S_n,x) and for the coefficients of S_n.

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