Regular Variation in the Tail Behaviour of Solutions of Random Difference Equations

Let $Q$ and $M$ be random variables with given joint distribution. Under some conditions on this joint distribution, there will be exactly one distribution for another random variable $R$, independent of $(Q,M)$, with the property that $Q + MR$ has the same distribution as $R$. When $M$ is nonnegative and satisfies some moment conditions, we give an improved proof that if the upper tail of the distribution of $Q$ is regularly varying, then the upper tail of the distribution of $R$ behaves similarly; this proof also yields a converse. We also give an application to random environment branching processes, and consider extensions to cases where $Q + MR$ is replaced by $\Psi(R)$ for random but nonlinear $\Psi$ and where $M$ may be negative.