The product 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 unbounded distribution $G$, which satisfy that \begin{eqnarray*} P(X>x|Y=y)\sim h(y)P(X>x) \end{eqnarray*} holds uniformly for $y\geq0$ as $x\to \infty$. Under the condition that $\overline{G}(bx)=o(\overline H(x))$ holds for all constant $b>0$, we proved that $F\in\mathcal{L}(\gamma)$ for some $\gamma\geq 0$ implied $H\in \mathcal{L}(\gamma/\beta_G)$ and that $F\in\mathcal{S}(\gamma)$ for some $\gamma\geq 0$ implied $H\in \mathcal{S}(\gamma/\beta_G)$, where $H$ is the distribution of the product $XY$, and $\beta_G$ is the right endpoint of $G$, that is, $\beta_G=\sup\{y:~G(y)<1\}\in (0,\infty],$ and when $\beta_G=\infty$, $\gamma/\beta_G$ is understood as 0. Furthermore, in a discrete-time risk model in which the net insurance loss and the stochastic discount factor are equipped with a dependence structure, a general asymptotic formula for the finite-time ruin probability is obtained when the net insurance loss has a subexponential tail.