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

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 $G_n = g_n \ldots g_1$\mathrm{.}Givenanystartingpoint$x=\mathbb R v\in$\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 $\sigma(G_n, x)= \log \frac{|G_n v|}{|v|}$,foranarbitrarynorm$|\cdot|$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 $\phi$ on the Markov chain $X_n^x$.