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

Abstract

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 $y\ge 0$ 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 $\gamma \ge 0$ implied $H\in \mathcal{L}(\gamma /{\beta }_G)$ and that $F\in \mathcal{S}(\gamma )$ for some $\gamma \ge 0$ 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.

13 pages