On convolution equivalence with applications

A distribution F on ( 1, 1) is said to belong to the class S(a) for some a > 0 if limx!1 F(x u)=F(x) 1⁄4 eau holds for all u and limx!1 F (x)=F(x) 1⁄4 2mF exists and is finite. Let X and Y be two independent random variables, where X has a distribution in the class S(a) and Y is non-negative with an endpoint ŷ 1⁄4 sup y : P(Y < y) , 1 f g 2 (0, 1). We prove that the product XY has a distribution in the class S(a= ŷ). We further apply this result to investigate the tail probabilities of Poisson shot noise processes and certain stochastic equations with random coefficients.