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

In this paper, we consider the randomly weighted sum S-2(Theta) = Theta X-1(1) + Theta X-2(2), where the two primary random summands X-1 and X-2 are real-valued and dependent with long or dominatedly varying tails, and the random weights Theta(1) and Theta(2) are positive, with values in [a, b], 0 < a <= b < infinity, and arbitrarily dependent, but independent of X-1 and X-2. Under some dependence structure between X-1 and X-2, we show that S-2(Theta) has a long or dominatedly varying tail as well, and obtain the corresponding (weak) equivalence results between the tails of S-2(Theta) and M-2(Theta) = max {Theta X-1(1), Theta X-1(1) + Theta X-2(2)}. 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. (C) 2014 Elsevier B.V. All rights reserved.