Law Of Large Numbers And Complete Convergence Theorem For Weighted Sums Under Sublinear Expectation

The sublinear expectation space,an extension of the classical probability space,is used in solving many nonlinear problems in the fields of statistics,quantum mechanics,and risk management.In the sublinear expectation space,some limit theories in the traditional probability space have been proved and generalized,but the related research on the law of large numbers and complete convergence theorem still needs further improvement.In the sublinear expectation space,it is more difficult to study because the capacity and sublinear expectation are not additive,and some capacity inequalities for partial sums are still not equivalent to the classical probability space,so the study of limit theory is very challenging.In this paper,after overcoming these difficulties,we extend the strong convergence theorem of i.i.d.sequence weighted sums,the full convergence theorem of ND sequence weighted sums and the large number law result of Pareto type sequence weighted sums studied under probability space to the sublinear expectation space in turn,and provide some methods for the study of such problems.First of all,comprehensive use of the partial sum of the capacity inequality,sub-sequence method,Kronecker lemma,Borel-Cantelli lemma and other methods.At the same time,the general moment con-dition in the probability space is changed to the Choquet integral condition in the sub-linear expectation space.This article explores the strong law of large numbers for weighted sums of i.i.d.random variable sequences and the complete convergence theorem for weighted sums of ND random variable sequences under sub-linear expectations.The research obtained two complete convergence theorems and two strong laws of large numbers in the sub-linear expectation space.The coefficients of the weighted sum are all general functions.The results obtained improve and generalize some of the existing conclusions in the classical probability space and the sub-linear expectation space at the same time.Secondly,based on the sub-linear expectation space limit theory,the Rosenthal inequality,Holder inequality,Borel-Cantelli lemma and other methods are used.We study the weak law of large numbers and the strong law of large numbers for the weighted sum of independent random variable sequences that satisfy the Pareto distribution of V {Xn>x)=1/x+cn(where cn is a non-decreasing constant sequence).The research results extend the relevant conclusions in the traditional probability space to the sub-linear expectation space,enriching the content of the law of large numbers of random variables that satisfy the Pareto distribution in the sub-linear expectation space.Finally,we summarize some issues that still need to be improved in the study of limit theory under sublinear expectations.Due to the complexity of the sublinear expectation theory,many common tech-niques and practical tools used in the original probability space are no longer applicable,which makes the study of limit theory under sublinear expectation space more difficult and complicated.In this paper,we give some effective methods to solve these problems.