Complete Convergence And Complete Integration Convergence For Negatively Dependent Random Variables Under Sub-linear Expectations

In order to solve some problems of uncertainty of risk measurement,theorems for sub-linear expec-tations were put forward,and have become the trend of the research about classical probability theory.In this paper,the limit theories in classical probability space were extended to the sub-linear expecta-tion space,we established the complete convergence and almost sure convergence for weighted sums of negatively dependent(ND)random variables under sub-linear expectations and the complete integration convergence for arrays of row-wise extend negatively dependent(END)random variables under sub-linear expectations.Firstly,we took the complete convergence theorem in probability space as the theoretical basis,by combining the negatively dependence for ND random variable sequences in sub-linear expectation space and the properties of sub-linear expectation and capacity,we obtained the complete convergence for weighted sums of ND random variables under sub-linear expectations.Since the almost sure convergence can be deduced from the complete convergence,we further obtain the almost sure convergence theorems for weighted sums of ND random variables in sub-linear expectation space.Secondly,in probability space,the complete moment convergence is more accurate than the com-plete convergence,the complete moment convergence includes the complete convergence,END random variables sequence is a wider sequence than ND random variable sequence.So,by using the methods of studying the theory of sub-linear expectation space,we obtained the complete integration convergence for arrays of row-wise END random variables under sub-linear expectations under a general moment condition.Our results further perfect the theoretical system of complete convergence and almost sure convergence for the ND random variables and complete integration convergence for the END random variables in the sub-linear expectation context.