  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.
