On the behavior of the product of independent random variables

For two independent non-negative random variables X and Y, we treat X as the initial variable of major importance and Y as a modifier (such as the interest rate of a portfolio). Stability in the tail behaviors of the product compared with that of the original variable X is of practical interests. In this paper, we study the tail behaviors of the product XY when the distribution of X belongs to the classes L and S, respectively. Under appropriate conditions, we show that the distribution of the product XY is in the same class as X when X belongs to class L or S, in other words, classes L and S are stable under some mild conditions on the distribution of Y. We also show that if the distribution of X is in class L(γ) (γ>0) and continuous, then the product XY is in L if and only if Y is unbounded.