On the rate of convergence in the kesten renewal theorem[1]

Abstract

We consider the stochastic recursion X n+1 = M n+1 X n Q n+1 on R d, where (M n, Q n) are i.i.d. random variables such that Q n are translations, M n are similarities of the Euclidean space R d. Under some standard assumptions the sequence X n converges to a random variable R and the law v of R is the unique stationary measure of the process. Moreover, the weak limit of properly dilated measure v exists, defining thus a homogeneous tail measure A. In this paper we study the rate of convergence of dilations of v to A In particular in the one dimensional setting, when (M n, Q n) e R + x R and X n e R, the Kesten renewal theorem says that t α P[|R| > t] converges to some strictly positive constant C +. Our main result says that \t α P[|R| > t - C + \ < C(logt)-σ, for some σ > 0 and large t. It generalizes the previous one by Goldie.