Randomly weighted sums of subexponential random variables with application to capital allocation

We are interested in the tail behavior of the randomly weighted sum \( \sum _{i=1}^{n}\theta _{i}X_{i}\) , in which the primary random variables X ₁, …, X _n are real valued, independent and subexponentially distributed, while the random weights ?? ₁, …, ?? _n are nonnegative and arbitrarily dependent, but independent of X ₁, …, X _n . For various important cases, we prove that the tail probability of \(\sum _{i=1}^{n}\theta _{i}X_{i}\) is asymptotically equivalent to the sum of the tail probabilities of ?? ₁ X ₁, …, ?? _n X _n , which complies with the principle of a single big jump. An application to capital allocation is proposed.