A simple proof of heavy tail estimates for affine type Lipschitz recursions

We study the affine recursion Xn=AnXn−1+Bn where (An,Bn)∈R+×R is an i.i.d. sequence and recursions Xn=Φn(Xn−1) defined by Lipschitz transformations such that Φ(x)≥Ax+B. It is known that under appropriate hypotheses the stationary solution X has regularly varying tail, i.e. limt→∞tαP[X>t]=C. However positivity of C in general is either unknown or requires some additional involved arguments. In this paper we give a simple proof that C>0. This applies, in particular, to the case when Kesten-Goldie assumptions are satisfied.