Perpetuities with thin tails revisited

We consider the tail behavior of random variables R which are solutions of the distributional equation Rd=Q+MR, where (Q, M) is independent of R and |M|.1. Goldie and Grübel showed that the tails of R are no heavier than exponential and that if Q is bounded and M resembles near 1 the uniform distribution, then the tails of R are Poissonian. In this paper, we further investigate the connection between the tails of R and the behavior of M near 1. We focus on the special case when Q is constant and M is nonnegative.