A Breiman's theorem for conditional dependent random vector and its applications to risk theory

In this paper, we give a Breiman's theorem for conditional dependent random vector, where one component has a regularly-varying-tailed distribution with the index $\alpha\ge0$ and its slowly varying function satisfies a relaxed condition, while the other component is non-negative and its tail distribution is lighter than the former. This result substantially extends and improves Theorem 2.1 of Yang and Wang (Extremes,\ 2013). %with a lower moment condition requirement for many occasions. We also provide some concrete examples and some interesting properties of conditional dependent random vector. Further, we apply the above Breiman's theorem to risk theory, and obtain two asymptotic estimates of the finite-time ruin probability and the infinite-time ruin probability of a discrete-time risk model, in which the corresponding net loss and random discount are conditionally dependent.