The Complete Convergence And Strong Law Of Large Number For Weighted Sums Of END Sequence Under Sub-Linear Expectation

In order to solve the non-linear problems such as the calculation and analysis of various risk measures in the financial field,the theory of sub-linear expectation space was proposed,and the introduction of the concept of sub-linear expectation provided a brand-new research direction for the study of probability limit theory.Some of the very important or meaningful conclusions or theorems have been proved and generalized in the sub-linear expectation space,but the development is not perfect,and there are still many problems that need to be further studied.Therefore,this paper mainly studies the complete conver-gence and strong law of large numbers of weighted sum of extended negative dependent(END)sequences in sub-linear expectation space,and generalizes the existing conclusions in traditional probability space.First,we take the complete convergence theorem in the probability space as a reference,and combine the negative dependence of the END random variable sequence with the identically distribution in the sub-linear expectation space,and the properties of countable additivity and capacity of the sub-linear expectation,under the condition that the integral exists on the order of the random variable 2+r/α,and generalize the research object from the identically distribution NOD sequence in the probability space to the identically distribution under sub-linear expectation END sequences,we obtain the complete convergence of weighted and Stout-type identically distributed END sequences in a sub-linear expectation space,which enriches the content of complete convergence in a sub-linear expectation space.Secondly,the strong law of large numbers of weighted sums of END random variable sequences with the identically distribution under sub-linear expectations is studied.According to the subadditivity of the sub-linear expectations and the properties of the END sequences,using inequality processing techniques,sub-column methods and other methods to study the law of strong numbers of the weighted sum of Sung-type END sequences under sub-linear expectation and the law of strong numbers of the weighted sum of END sequences under general conditions,obtaining a very broad version of the strong number law of END sequences in sub-linear expectation spaces.The obtained results generalize and improve some existing conclusions in classical probability spaces,and provide a new type of research on such strong convergence problems research methods.