Convergence of series and almost sure convergence for weighted random variables under sub-linear expectations

In this paper, convergence of series and almost sure convergence are established for weighted random variables under a sub-linear expectation space. Our results are very extensive versions which contain the related convergence of series and almost sure convergence for sequences of random variables and so on, and are extensions and improvements of classical convergence of series and almost sure convergence from the traditional probability space to the sub-linear expectation space.