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 X1X1 and X2X2 are real-valued and dependent with long or dominatedly varying tails, and the random weights Θ1Θ1 and Θ2Θ2 are positive, with values in [a,b][a,b], 0