On the rate of convergence in the Kesten renewal theorem*

We consider the stochastic recursion Xn+1 = Mn+1Xn +Qn+1 on R , where (Mn, Qn) are i.i.d. random variables such that Qn are translations, Mn are similarities of the Euclidean space R. Under some standard assumptions the sequence Xn converges to a random variable R and the law ν of R is the unique stationary measure of the process. Moreover, the weak limit of properly dilated measure ν exists, defining thus a homogeneous tail measure Λ. In this paper we study the rate of convergence of dilations of ν to Λ In particular in the one dimensional setting, when (Mn, Qn) ∈ R×R and Xn ∈ R, the Kesten renewal theorem says that tP[|R| > t] converges to some strictly positive constant C+. Our main result says that ∣∣tαP[|R| > t]− C+∣∣ ≤ C(log t), for some σ > 0 and large t. It generalizes the previous one by Goldie.