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

Hui Xiao,Quansheng Liu,Ion Grama

doi:10.4171/jems/1142

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 define the norm cocycle by ${\\sigma(G_n, x)= \\log (|G_n v|/|v|)}$, for an arbitrary norm ${|\\cdot|}$ on $\\smash{\\mathbb R^{d}}$. Under suitable conditions we prove a Berry–Esseen-type theorem and an Edgeworth expansion for the couple ${(X_n^x, \\sigma(G_n, x))}$. 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 ${(X_n^x, \\sigma(G_n, x))}$ with a target function ${\\varphi}$ on the Markov chain ${X_n^x}$.

Highlights

The study of the asymptotic properties of the Markov chain (Xnx)n 1 and of the product (Gn)n 1 has attracted a good deal of attention since the groundwork of Furstenberg and Kesten [14], where the strong law of large numbers for log Gn has been established, which is a fundamental result for the products of random matrices
Our second objective, which is the key point in proving (1.4), is a Berry-Esseen bound for the couple (Xnx, log |Gnx|): for any Hölder continuous function φ on Pd−1, 1 sup E φ(Xnx) log |Gn√x|−nλ σn y
We complement the results in [14, 19, 21] by giving a Cramér type moderate deviation expansion and a local limit theorem with moderate deviations

Summary

Introduction

Our second objective, which is the key point in proving (1.4), is a Berry-Esseen bound for the couple (Xnx, log |Gnx|): for any Hölder continuous function φ on Pd−1,. We extend the BerryEsseen theorem of [21] to the couple (Xnx, log |Gnx|) with a target function φ on the Markov chain Xnx. We complement the results in [14, 19, 21] by giving a Cramér type moderate deviation expansion and a local limit theorem with moderate deviations. In order to prove (1.4) we have to rework the spectral gap theory for the transfer operators Pz and Rs,z, by considering the case when s can take values in the interval (−η, η) with η > 0 small, and z belongs to a small complex ball centered at the origin, see Section 3 This allows to define the change of measure Qxs and to extend the Berry-Esseen bound (1.5) for the changed measure Qxs , see Theorem 5.1. Under the non-arithmeticity condition, in Theorem 2.2 we obtain an Edgeworth expansion for (Xnx, |Gnx|) with the target function φ on Xnx, which is of independent interest

Main results

Spectral gap theory

Smoothing inequality on the complex plane

Proofs of Berry-Esseen bound and Edgeworth expansion

Proof of moderate deviation expansions

Proof of a local limit theorem