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

### Hui Xiao

Universität Hildesheim, Germany### Ion Grama

Université de Bretagne-Sud, Vannes, France### Quansheng Liu

Université de Bretagne-Sud, Vannes, France

## 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 ${X_n^x}$.

## Cite this article

Hui Xiao, Ion Grama, Quansheng Liu, Berry–Esseen bound and precise moderate deviations for products of random matrices. J. Eur. Math. Soc. 24 (2022), no. 8, pp. 2691–2750

DOI 10.4171/JEMS/1142