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On perpetuities with gamma-like tails

Part of: Stochastic analysis Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  26 July 2018

Dariusz Buraczewski ,
Piotr Dyszewski ,
Alexander Iksanov  and
Alexander Marynych
Dariusz Buraczewski*
Affiliation:
University of Wrocław
Piotr Dyszewski*
Affiliation:
University of Wrocław
Alexander Iksanov*
Affiliation:
Taras Shevchenko National University of Kyiv
Alexander Marynych*
Affiliation:
Taras Shevchenko National University of Kyiv
*
* Postal address: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
* Postal address: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
**** Postal address: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine.
**** Postal address: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine.
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Abstract

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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 provide three disjoint groups of sufficient conditions which ensure that the right tail of a perpetuity ℙ{X > x} is asymptotic to axce-bx as x → ∞ for some a, b > 0, and c ∈ ℝ. Our results complement those of Denisov and Zwart (2007). As an auxiliary tool we provide criteria for the finiteness of the one-sided exponential moments of perpetuities. We give several examples in which the distributions of perpetuities are explicitly identified.

Keywords

Distribution tailselfdecomposable distributionperpetuityexponential moment

MSC classification

Primary: 60H25: Random operators and equations
Secondary: 60G50: Sums of independent random variables; random walks 60E07: Infinitely divisible distributions; stable distributions
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 2 , June 2018 , pp. 368 - 389
Copyright
Copyright © Applied Probability Trust 2018 

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