Some Limit Theorems Regarding Products of Random Matrices I: Directional Derivative of the Lyapunov Exponent

2022/09/16 by Wang, Fan
#Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2209.07750

Abstract

Given an i.i.d. sequence \A_n(ω)\_{n≥ 1} of invertible matrices and a random matrix B(ω), we consider the random matrix sequences inductively defined by S_n(ω) = A_n(ω)S_n-1(ω) and T_n(ω) = B(σ^n-1ω)S_n-1(ω)+A_n(ω)T_n-1(ω), and study several limit theorems involving T_n(ω) as well as the asymptotic behaviour of the action of T_n(ω) on the projective space and on the unit circle.

Related

• Limit laws for random matrix products
• A central limit theorem with explicit Lyapunov exponent and variance for products of 2×2 random non-invertible matrices
• Precise large deviation asymptotics for products of random matrices
• Analiticity of the Lyapunov exponents of random products of matrices
• Local limit theorem for the operator norm of products of random matrices