Moderate deviations and local limit theorems for the coefficients of random walks on the general linear group

Consider the random walk Gn:=gn…g1, n?1, where (gn)n?1 is a sequence of independent and identically distributed elements with law ? on general linear group GL(V) V=Rd. Under suitable conditions ?, we establish Cramér type moderate deviation expansions local limit theorems deviations for coefficients ?f,Gnv?, v?V f?V?. Our approach based Hölder regularity invariant measure Markov chain Gn?x=RGnv projective space V starting point x=Rv, under changed measure.