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 S n = U 1 X 1 + ⋯ + U n X n and T m = V 1 Y 1 + ⋯ + V m Y m . The vectors of primary random variables ( X 1 , Y 1 ) , ( X 2 , Y 2 ) , … are assumed to be independent with dominatedly varying marginal distributions, and the dependence within each pair ( X i , Y i ) satisfies a condition called strong asymptotic independence. The random weights U 1 , V 1 , … 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 ( S n , T m ) with fixed positive integers n and m , and ( S N , T M ) 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.