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Perpetuities with thin tails revisited. (English) Zbl 1185.60074
Ann. Appl. Probab. 19, No. 6, 2080-2101 (2009); erratum ibid. 20, No. 3, 1177 (2010).
Summary: We consider the tail behavior of random variables $$R$$ which are solutions of the distributional equation $$R \overset {d} = Q+MR$$, where $$(Q, M)$$ is independent of $$R$$ and $$|M|\leq 1$$. Ch. M. Goldie and R. Grübel [Adv. Appl. Probab. 28, No. 2, 463–480 (1996; Zbl 0862.60046)] 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.

##### MSC:
 60H25 Random operators and equations (aspects of stochastic analysis) 60E99 Distribution theory
##### Keywords:
perpetuity; stochastic difference equation; tail behavior
Full Text:
##### References:
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