Berry–Esseen Bounds with Targets and Local Limit Theorems for Products of Random Matrices

• Published in: The Journal of geometric analysis 2023-03, Vol.33 (3), Article 76
• Main Authors: Dinh, Tien-Cuong, Kaufmann, Lucas, Wu, Hao
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Continuity (mathematics); Convex and Discrete Geometry; Differential Geometry; Dynamical Systems and Ergodic Theory; Fourier Analysis; Geometry; Global Analysis and Analysis on Manifolds; Mathematics; Mathematics and Statistics; Nessim Sibony: In Memoriam; Random variables; Statistical analysis; Theorems
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-022-01127-3

Let μ be a probability measure on GL d ( R ) and denote by S n : = g n ⋯ g 1 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 ∈ P d - 1 , σ is the norm cocycle and u belongs to a class of admissible functions on P d - 1 with values in 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 R and Hölder continuous target functions on P d - 1 . As particular cases, we obtain new limit theorems for σ ( S n , x ) and for the coefficients of S n .