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

Let (g(n))(n >= 1) (g be a sequence of independent and identically distributed (i.i.d.) d x d real .0 random matrices. For n >= 1 set G(n) = g(n) center dot center dot center dot g(1). Given any starting point x = Rv is an element of Pd-1, con- sider the Markov chain X-n(x) = RG(n)(v) on the projective space Pd-1 and define the norm cocycle by sigma(G(n), x) = log (vertical bar G(n)(v)vertical bar/vertical bar v vertical bar), for an arbitrary norm vertical bar center dot vertical bar on 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). Cramer-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).