Marcinkiewicz strong laws for weighted sums under the sub-linear expectations

In this article, we establish complete convergence theorems for weighted sums in sub-linear expectations space. As corollaries, we obtain Marcinkiewicz strong laws for weighted sums under the sublinear expectations. Our results extend and improve the corresponding results of Bai and Cheng (Statist. Probab. Lett. 46: 105-112, 2000) and Cuzick (J. Theoret. Probab. 8: 625-641, 1995) from classical probability space to sub-linear expectation space.