Complete convergence theorems for the maximum of weighted sums under sub-linear expectations

Let {X,Xn;n≥1} be a sequence of identically distributed random variables in a sub-linear expectation space (Ω,H,Ê). Suppose that Xk is independent of (Xk+1,…,Xn) for each k=1,…,n−1,n⩾1. We establish Baum-Katz-type complete and complete moment convergence theorems for the maximum of weighted sums under optimal moment condition in a sub-linear expectation space. As an application of our complete convergence theorem, strong laws for weighted sums is obtained. Our results generalize and improve the corresponding results of the probability space.