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

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 $\sigma(G_n, x)= \log \frac{|G_n v|}{|v|}$, 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 $(X_n^x, \sigma(G_n, x))$. 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 $X_n^x$. Cram\'{e}r 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 $X_n^x$.