# On perpetuities with gamma-like tails

## TL;DR

This paper establishes conditions under which perpetuities have gamma-like tail distributions, providing asymptotic formulas and criteria for exponential moments, with explicit examples.

## Contribution

It introduces three new sets of sufficient conditions for gamma-like tail asymptotics of perpetuities, extending previous results and offering explicit distribution examples.

## Key findings

- Tail distribution asymptotics $ o ax^ce^{-bx}$ as $x oig$
- Criteria for finiteness of exponential moments of perpetuities
- Explicit examples of perpetuity distributions

## 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.