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arXiv · 0912.1694

Perpetuities with thin tails revisited

Paweł Hitczenko·Jacek Wesołowski
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Abstract

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 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.

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Paweł Hitczenko, Jacek Wesołowski. 2010-02-08. Perpetuities with thin tails revisited. https://doi.org/10.1214/09-aap603

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