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 .(R) for random but nonlinear . and where M may be negative.