Uniform estimate for the tail probabilities of randomly weighted sums

Several authors have studied the uniform estimate for the tail probabilities of randomly weighted sums \(\sum\limits_{i = 1}^n {\theta _i X_i }\) and their maxima. In this paper, we generalize their work to the situation that {X _i , i ≥ 1} is a sequence of upper tail asymptotically independent random variables with common distribution from the class \(\mathcal{D} \cap \mathcal{L}\) , and {θ _i , i ≥ 1} is a sequence of nonnegative random variables, independent of {X _i , i ≥ 1} and satisfying some regular conditions. Moreover, no additional assumption is required on the dependence structure of {θ _i , i ≥ 1}.