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

Abstract

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 𝜃 1, …, 𝜃 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 𝜃 1 X 1, …, 𝜃 n X n , which complies with the principle of a single big jump. An application to capital allocation is proposed.