Closure property and maximum of randomly weighted sums with heavy-tailed increments

In this paper, we consider the randomly weighted sum S2Θ=Θ1X1+Θ2X2, where the two primary random summands X1 and X2 are real-valued and dependent with long or dominatedly varying tails, and the random weights Θ1 and Θ2 are positive, with values in [a,b], 0<a≤b<∞, and arbitrarily dependent, but independent of X1 and X2. Under some dependence structure between X1 and X2, we show that S2Θ has a long or dominatedly varying tail as well, and obtain the corresponding (weak) equivalence results between the tails of S2Θ and M2Θ=max{Θ1X1,Θ1X1+Θ2X2}. As corollaries, we establish the asymptotic (weak) equivalence formulas for the tail probabilities of randomly weighted sums of even number of long-tailed or dominatedly varying-tailed random variables.