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

Abstract

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