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

TL;DR

This paper investigates the tail behavior of the product of dependent random variables and applies the findings to derive asymptotic ruin probabilities in a discrete-time risk model with dependent components.

Contribution

It establishes conditions under which the tail properties of the product distribution are preserved from the individual distributions, and applies these results to risk modeling with dependence.

Findings

Proved that tail properties are preserved under certain dependence conditions.

Derived asymptotic formulas for finite-time ruin probabilities with subexponential tails.

Extended classical risk models to include dependence between loss and discount factors.

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 $y\geq0$ 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.