On perpetuities with gamma-like tails

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 $a{x}^c{e}^{-bx}$ 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.

To appear in Journal of Applied Probability, 55, no. 2, 2018

References in corpus (1)

• Inverse problems for regular variation of linear filters, a cancellation property for $\sigma$-finite measures and identification of stable laws

Cited by in corpus (3)

• On perpetuities with light tails
• Stochastic recursions: between Kesten's and Grincevičius-Grey's assumptions
• A result on power moments of Lévy-type perpetuities and its application to the $L_p$-convergence of Biggins' martingales in branching Lévy processes