  We consider the random variables $R$ which are solutions of the
distributional equation $R\overset{\cL}{=}MR+Q$, where $(Q,M)$ is independent
of $R$ and $\ABS{M}\leq 1$. Goldie and Gr\"ubel showed that the tails of $R$
are no heavier than exponential. In this note we provide the exact lower and
upper bounds of the domain of the Laplace transform of $R$.
