  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.
