  We consider the tail behavior of random variables $R$ which are solutions of
the distributional equation $R\stackrel{d}{=}Q+MR$, where $(Q,M)$ is
independent of $R$ and $|M|\le 1$. Goldie and Gr\"{u}bel showed that the tails
of $R$ are no heavier than exponential and that if $Q$ is bounded and $M$
resembles near 1 the uniform distribution, then the tails of $R$ are
Poissonian. In this paper, we further investigate the connection between the
tails of $R$ and the behavior of $M$ near 1. We focus on the special case when
$Q$ is constant and $M$ is nonnegative.
