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On perpetuities with gamma-like tails

Published online by Cambridge University Press:  26 July 2018

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.

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 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.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2018 

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