Randomly weighted sums and their maxima with heavy-tailed increments and dependence structure

Consider the randomly weighted sums Sm(θ) = ∑mi = 1θiXi, 1 ⩽ m ⩽ n, and their maxima Mn(θ) = max 1 ⩽ m ⩽ nSm(θ), where Xi, 1 ⩽ i ⩽ n, are real-valued and dependent according to a wide type of dependence structure, and θi, 1 ⩽ i ⩽ n, are non negative and arbitrarily dependent, but independent of Xi, 1 ⩽ i ⩽ n. Under some mild conditions on the right tails of the weights θi, 1 ⩽ i ⩽ n, we establish some asymptotic equivalence formulas for the tail probabilities of Sn(θ) and Mn(θ) in the case where Xi, 1 ⩽ i ⩽ n, are dominatedly varying, long-tailed and subexponential distributions, respectively.