On the joint tail behavior of randomly weighted sums of heavy-tailed random variables

Abstract

We focus on the joint tail behavior of randomly weighted sums Sn=U1X1+⋯+UnXn and Tm=V1Y1+⋯+VmYm. The vectors of primary random variables (X1,Y1), (X2,Y2),… are assumed to be independent with dominatedly varying marginal distributions, and the dependence within each pair (Xi,Yi) satisfies a condition called strong asymptotic independence. The random weights U1, V1,… are non-negative and arbitrarily dependent, but they are independent of the primary random variables. Under suitable conditions, we obtain asymptotic expansions for the joint tails of (Sn,Tm) with fixed positive integers n and m, and (SN,TM) with integer-valued random variables N and M that are independent of the primary random variables. When the marginal distributions of the primary random variables are extended regularly varying, the result is proved to hold uniformly for any n and m under stronger conditions. Our results rely critically on moment conditions that are generally easy to check.