Random Difference Equations with Subexponential Innovations∗

In this paper we consider the random difference equations S =d (X + S)Y and T =d X+TY , where =d denotes equality in distribution, X and Y are two nonnegative random variables, and S and T on the right-hand side are independent of (X,Y ). Under the assumptions that X follows a subexponential distribution with a nonzero lower Karamata index, that Y takes values in [0, 1] and is not degenerate at 0 or 1, and that (X,Y ) fulfills a certain dependence structure via the conditional tail probability of X given Y , we derive some asymptotic formulas for the tail probabilities of the weak solutions S and T to these equations. In doing so we also obtain some by-products which are interesting in their own right.