Randomly Weighted Sums of Subexponential Random Variables with Application to Capital Allocation

We are interested in the tail behavior of the randomly weighted sum ∑n i=1 θiXi, in which the primary random variables X1, . . . , Xn are real valued, independent and subexponentially distributed, while the random weights θ1, . . . , θn are nonnegative and arbitrarily dependent, but independent of X1, . . . , Xn. For various important cases, we prove that the tail probability of ∑n i=1 θiXi is asymptotically equivalent to the sum of the tail probabilities of θ1X1, . . . , θnXn, which complies with the principle of a single big jump. An application to capital allocation is proposed.