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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
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