Asymptotics for a Type of Randomly Weighted Sums and Its Application

This paper considers the asymptotics of randomly weighted sums and their maxima, where the increments X_i,i\geq1\ is a sequence of independent, identically distributed and real-valued random variables and the weights \theta_i,i\geq1\ form another sequence of non-negative and independent random variables, and the two sequences of random variables follow some dependence structures. When the common distribution F of the increments belongs to dominant variation class, we obtain some weakly asymptotic estimations for the tail probability of randomly weighted sums and their maxima. In particular, when the F belongs to consistent variation class, some asymptotic formulas is presented. Finally, these results are applied to the asymptotic estimation for the ruin probability.