On a theorem of Breiman and a class of random difference equations

We consider the tail behavior of the product of two independent nonnegative random variables X and Y. Breiman (1965) has considered this problem, assuming that X is regularly varying with index a and that E{Ya+e} <8 for some e > 0. We investigate when the condition on Y can be weakened and apply our findings to analyze a class of random difference equations.