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 $S_n=U_1{X}_1+\cdots +U_n{X}_n$ and $T_m=V_1{Y}_1+\cdots +V_m{Y}_m$. The vectors of primary random variables $(X_1,Y_1)$, $(X_2,Y_2),\dots$ 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 $U_1$, $V_1,\dots$ 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.