Berry-Esseen bound and precise moderate deviations for products of random matrices

TL;DR

This work advances the quantitative understanding of products of random matrices by establishing a Berry–Esseen bound and an Edgeworth expansion for the joint process $(X_n^x,\sigma(G_n,x))$, as well as Cramér-type moderate deviations and a local limit theorem with moderate deviations. The authors develop a robust complex-analytic smoothing framework and a detailed spectral-gap theory for perturbed transfer operators $P_z$, along with a change-of-measure $\mathbb{Q}_s^x$, to transfer limit theorems from the i.i.d. matrix setting to the projective Markov chain. Under moment, irreducibility/positivity, and non-arithmeticity conditions (with positivity for matrices in $\mathscr{G}_+^\circ$), they obtain uniform results for both invertible and positive matrices, including an operator-norm LLT and local limit results that refine classical CLTs. The techniques—smoothing on the complex plane, saddle-point methods, and refined spectral-gap perturbations—enable precise asymptotics (up to $o(n^{-1/2})$ terms) and robust moderate deviation principles with broad potential applications in branching processes, random recursions, and financial models. The results provide a near-complete picture of the rates of convergence and deviations for products of random matrices, complementing the LLN and CLT with detailed, practically applicable asymptotic expansions.

Abstract

Let $(g_{n})_{n\geq 1}$ be a sequence of independent and identically distributed (i.i.d.) $d\times d$ real random matrices. For $n\geq 1$ set $G_n = g_n \ldots g_1$. Given any starting point $x=\mathbb R v\in\mathbb{P}^{d-1}$, consider the Markov chain $X_n^x = \mathbb R G_n v $ on the projective space $\mathbb P^{d-1}$ and the norm cocycle $σ(G_n, x)= \log \frac{|G_n v|}{|v|}$, for an arbitrary norm $|\cdot|$ on $\mathbb R^{d}$. Under suitable conditions we prove a Berry-Esseen type theorem and an Edgeworth expansion for the couple $(X_n^x, σ(G_n, x))$. These results are established using a brand new smoothing inequality on complex plane, the saddle point method and additional spectral gap properties of the transfer operator related to the Markov chain $X_n^x$. Cramér type moderate deviation expansions as well as a local limit theorem with moderate deviations are proved for the couple $(X_n^x, σ(G_n, x))$ with a target function $\varphi$ on the Markov chain $X_n^x$.

Theorems & Definitions (60)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Proposition 3.1
• Remark 3.2
• proof : Proof of Proposition \ref{['transfer operator']}
• Lemma 3.3
• proof