Randomly weighted sums of dependent random variables with dominated variation

For any fixed n ≥ 1 , consider the randomly weighted sum ∑ k = 1 n θ k X k and the maximum max 1 ≤ m ≤ n ⁡ ∑ k = 1 m θ k X k , where X k , 1 ≤ k ≤ n , are n real-valued and not necessarily identically distributed random variables (r.v.s) with dominated variation, and θ k , 1 ≤ k ≤ n , are n nonnegative r.v.s without any dependence assumptions. Let X k , 1 ≤ k ≤ n , be independent of θ k , 1 ≤ k ≤ n . Under some relatively weaker conditions on the weights θ k , 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 X k , 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.