Perpetuities with thin tails revisited

Details Perpetuities with thin tails revisited Perpetuities with thin tails revisited

2009-12-09

arxiv.org/abs/0912.1694v3

Annals of Applied Probability 2009, Vol. 19, No. 6, 2080-2101

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AAP603 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/09-AAP603

We consider the tail behavior of random variables $R$ which are solutions of the distributional equation $R\stackrel{d}{=}Q+MR$, where $(Q,M)$ is independent of $R$ and $|M|\le 1$. Goldie and Gr\"{u}bel 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.

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