Limit theorems for inhomogeneous random walks on $GL(d,\mathbb R)$

TL;DR

This work develops a comprehensive framework for limit theorems of inhomogeneous random walks on GL(d,R) by focusing on the cocycle $S_n=\,\ln\|g_n\cdots g_1 x_0\|$ and assuming average logarithmic contraction in projective space. The authors establish Berry-Esseen rates, almost sure invariance principle rates, large deviations, and a moderate deviations principle without requiring exponential growth of norms, by reducing the problem to a non-stationary Bernoulli-shift setting through a time-reversal argument and a detailed perturbative analysis. Central contributions include a robust abstract Prop. ['Approx Prop'], a reversal-time transfer-operator approach with analytic perturbation theory, and perturbation results showing stability under small random perturbations; these yield concrete CLT-type results for a broad class of non-identically distributed matrix products. The paper also provides practical applicability via applications to small perturbations of contracting matrices and to inhomogeneous random walks on GL(d,R), including a specialized GL(2,R) treatment, expanding the scope of non-stationary limit theorems in random matrix products.

Abstract

We prove Berry-Esseen theorems, almost sure invariance principle rates and large deviations for products of independent but not identically distributed invertible matrices with some average (logarithmic) projective contraction and uniform boundedness assumptions. We also characterize the divergence of the variance of the logarithm of the norm of the product. Our approach is based on verifying the conditions of \cite{NewBE} after reversing time.

Theorems & Definitions (37)

• Proposition 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8
• Theorem 2.9
• Proposition 3.1
• proof
• proof : Proof of Proposition \ref{['Approx Prop']}