Randomly weighted sums of dependent random variables with dominated variation

For any fixed n≥1, consider the randomly weighted sum ∑k=1nθkXk and the maximum max1≤m≤n⁡∑k=1mθkXk, where Xk, 1≤k≤n, are n real-valued and not necessarily identically distributed random variables (r.v.s) with dominated variation, and θk, 1≤k≤n, are n nonnegative r.v.s without any dependence assumptions. Let Xk, 1≤k≤n, be independent of θk, 1≤k≤n. Under some relatively weaker conditions on the weights θk, 1≤k≤n (which are weaker than the moment conditions in the existing results), this paper derives asymptotically lower (upper) bounds for the tail probabilities of the randomly weighted sums and their maxima, where Xk, 1≤k≤n, are pairwise asymptotically independent or pairwise tail quasi-asymptotically independent. In particular, when the above-mentioned distributions are consistently-varying-tailed, an asymptotically equivalent result is derived.