The product distribution 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 distribution G. Suppose that X and Y satisfy that $\mathbb{P}(X>x|Y=y)\sim h(y)\mathbb{P}(X>x)$ holds uniformly for $y\ge 0$ as $x\to \infty$, where $h(·)$ is a positive measurable function. Under the condition that $\overline{G}(\mathit{bx})=o(\overline{H}(x))$ holds for all constant b > 0, this paper proved that $F\in \ell (\gamma )$ for some $\gamma \ge 0$ implied $H\in \ell (\gamma /{\beta }_G)$ 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 ${\beta }_G>0$ 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 losses follow a common subexponential distribution.

Keywords:

• Long-tailed distribution
• dependent random variables
• finite-time ruin probability
• risk model
• subexponential distribution

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 60E05
• 62E10

Acknowledgements

The authors would like to express their deep gratitude to the two referees for their valuable comments and suggestions which help a lot in the improvement of the paper.

Additional information

Funding

Research supported by National Natural Science Foundation of China (No. 11401415), and the Priority Academic Program Development of Jiangsu Higher Education Institutions.