We consider the randomly weighted sums \({\sum }_{i=1}^{m}{\Theta }_{i}{X}_{i}\) and \({\sum }_{j=1}^{n}{\theta }_{j}{Y}_{j}\) for m, n ∈ℕ, where the real-valued random variables {Xi; i ∈ ℕ} and {Yj; j ∈ ℕ}, possibly dependent, have heavy-tailed distributions, and the random weights {Θi, θj; i, j ∈ ℕ} are nonnegative and arbitrarily dependent, but the sequences {Xi; i ∈ N}, {Yj; j ∈ ℕ}, and {Θi, θj; i, j ∈ℕ} are mutually independent. Under some mild conditions on the random weights, we derive the asymptotics of the joint tail probability of the two randomly weighted sums, in which the primary random variables satisfy some dependence assumptions.
