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

Abstract

Let $(g_n{)}_{n\ge 1}$ be a sequence of independent and identically distributed (i.i.d.) $d\times d$ real random matrices. For $n\ge 1$ set $G_n=g_n\dots g_1$. 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 ${\mathbb{R}}^d$. 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 $\phi$ on the Markov chain $X_n^x$.