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

Let (g₍) ₍ ₁ be a sequence of independent and identically distributed (i. i. d. ) d d real random matrices. For n 1 set Gₙ = gₙ g₁. Given any starting point x= R v^{d-1}, consider the Markov chain Xₙˣ = R Gₙ v on the projective space P^{d-1} and define the norm cocycle by (Gₙ, x) = (|Gₙ v|/|v|), for an arbitrary norm || on R^d. Under suitable conditions we prove a Berry–Esseen-type theorem and an Edgeworth expansion for the couple (Xₙˣ, (Gₙ, 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 Xₙˣ. Cramér-type moderate deviation expansions as well as a local limit theorem with moderate deviations are proved for the couple (Xₙˣ, (Gₙ, x) ) with a target function on the Markov chain Xₙˣ.