The product distribution of dependent random variables with applications to a discrete-time risk model

Let X be a real valued random variable with an unbounded distribution F and let Y be a nonnegative valued random variable with a distribution G. Suppose that X and Y satisfy that P(X>x|Y=y)∼h(y)P(X>x) holds uniformly for y≥0 as x→∞ , where h(·) is a positive measurable function. Under the condition that G¯(bx)=o(H¯(x)) holds for all constant b > 0, this paper proved that F∈ℓ(γ) for some γ≥0 implied H∈ℓ(γ/βG) and that F∈S(γ) for some γ≥0 implied H∈S(γ/βG) , where H is the distribution of the product XY, and βG>0 is the right endpoint of G, that is, βG=sup{y: G(y)