Asymptotics for the joint tail probability of bidimensional randomly weighted sums with applications to insurance

This paper studies the joint tail behavior of two randomly weighted sums ∑i=1m ΘiXi and ∑j=1nθjYj for some m, n ∈ ℕ ∪{∞}, in which the primary random variables {Xi;i ∈ ℕ} and {Yi;i ∈ ℕ}, respectively, are real-valued, dependent and heavy-tailed, while the random weights {Θi, θi; i ∈ ℕ} are nonnegative and arbitrarily dependent, but the three sequences {Xi;i ∈ ℕ}, {Yi;i ∈ ℕ} and {Θi, θi;i ∈ ℕ} are mutually independent. Under two types of weak dependence assumptions on the heavy-tailed primary random variables and some mild moment conditions on the random weights, we establish some (uniformly) asymptotic formulas for the joint tail probability of the two randomly weighted sums, expressing the insensitivity with respect to the underlying weak dependence structures. As applications, we consider both discrete-time and continuous-time insurance risk models, and obtain some asymptotic results for ruin probabilities.