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arXiv · 1703.02330

On perpetuities with gamma-like tails

Dariusz Buraczewski·Piotr Dyszewski·Alexander Iksanov·Alexander Marynych
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Abstract

An infinite convergent sum of independent and identically distributed random variables discounted by a multiplicative random walk is called perpetuity, because of a possible actuarial application. We give three disjoint groups of sufficient conditions which ensure that the distribution right tail of a perpetuity $\mathbb{P}\{X>x\}$ is asymptotic to $ax^ce^{-bx}$ as $x\to\infty$ for some $a,b>0$ and $c\in\mathbb{R}$. Our results complement those of Denisov and Zwart [J. Appl. Probab. 44 (2007), 1031--1046]. As an auxiliary tool we provide criteria for the finiteness of the one-sided exponential moments of perpetuities. Several examples are given in which the distributions of perpetuities are explicitly identified.

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Dariusz Buraczewski, Piotr Dyszewski, Alexander Iksanov, Alexander Marynych. 2017-03-07. On perpetuities with gamma-like tails. https://doi.org/10.1017/jpr.2018.24

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