The Product Of Dependent Random Variables With Applications To A Discrete-time Risk Model

In this paper,we consider first the product distribution of random variables which follow an Asmit dependence structure.Let X be a real valued random variable with an unbounded distribution F and let Y be a nonnegative random variable with a distribution G.H is the distribution of the product Z = XY.Suppose that X and Y follows an Asmit dependence structure defined as follows:P(X>x|Y=y)～h(y)P(X>x)holds uniformly for y ≥ 0 as x →∞,where h(y)is a positive measurable function.Under the condition that G(bx)= o(H(x))holds for all constant&>0,we proved that F∈L(γ)for some γ ≥ 0 implied H∈L(γ/βG)and that F∈S(γ)for some>0 implied H∈S(γ/βG),where H is the distribution of the product XY,and βG is the right endpoint of G,that is,βG sup{y:G(y)<1} ∈(0,∞],and when βG=∞,γ/βG is understood as 0.In addition,we apply these results to a discrete-time insurance risk model,in which the net insurance loss and the stochastic discount factor are equipped with an Asmit dependence structure.When the net insurance loss has a subexponential tail a general asymptotic formula for the finite-time ruin probability is obtained;when the net insurance loss has a regularly varying tail a general asymptotic formula for the infinite-time ruin probability is obtained.