Complete Convergence and Complete Moment Convergence for Maximum of Weighted Sums of ρ−-mixing Random Variables and Its Application

In this paper, complete convergence and complete moment convergence for maximal weighted sums of ρ⁻-mixing random variables are investigated, and some sufficient conditions for the convergence are provided. The relationships among the weights of the partial sums, boundary function and weight function are in a sense revealed. Additionally, a Marcinkiewicz–Zygmund type strong law of large numbers for maximal weighted sums of ρ⁻-mixing random variables is established. The results obtained extend the corresponding ones for random variables with independence structure and some dependence structures. As an application, the strong consistency for the tail-value-at-risk (TVaR) estimator in the financial and actuarial fields is established.