Complete Convergence And Complete Integral Convergence Of Weighted Sums Under Sub-linear Expectations

The traditional probabilistic theory system is based on the linear expectation space,but many problems in real life such as statistics,quantum mechanics,and risk management are not additive,so to solve this problem,the academician of Shandong University,Shige Peng,proposed the concept of sub-linear expectation.In this paper,we study the complete convergence of the weighted sums of the WND(Widely negative dependent)sequence and the complete convergence and complete integral convergence of the weighted sums of the WA(Widely acceptable)sequence and the randomly weighted sums of WND arrays in the framework of the sub-linear expectation space,and extend the existing findings in the probability space.This paper studies the complete convergence of the weighted sums of the WND sequence in the sub-linear expectation space.Since both the sub-linear expectation and the capacity are no longer additive,the capacity inequality in the sub-linear expectation space is one-sided,so we can only study the result of one-sided complete convergence,and the bilateral complete convergence similar to that in the probability space can be obtained when the upper and lower expectations are equal.The complete convergence of weighted sums mainly reveals the relationship between the weight condition,the upper integration condition,the weight function,and the boundary function.In this paper,we prove the complete convergence of weighted sums by considering the relationship between each case in the upper integration condition,which not only extends the complete convergence of weighted sums of WOD(Widely orthant-dependent)sequence in probability space to sublinear expectation space,but also expands the scope of the conditions when the theorem holds,the weight function and boundary function are investigated which are different from those in the complete convergence of the existing sub-linear expectation space,and the upper integration condition is obtained for the complete convergence of the weighted sums of WND sequence in three different forms,which completes the content of the complete convergence of the weighted sums of WND random variable sequence in the sub-linear expectation space with different weight and boundary functions.The complete convergence and complete integral convergence of the weighted sums of the WA sequence and the randomly weighted sums of WND arrays in the sub-linear expectation space are studied.Based on the definition of WA random variables in the sub-linear expectation space,the complete convergence and complete integral convergence of the weighted sums of WA sequence with regular variation functions of control coefficients are obtained by extending the sequence of identically distributed END(Extended negatively dependent)random variables in the probability space to the sequence of WA random variables under random control.Therefore,this paper enriches the study of the complete integration convergence of WA sequences.Compared with the existing study of the complete integration convergence of WA sequence in the sub-linear expectation space,this paper extends the study object from identical distribution to random control,and the upper integration condition is weaker and the conclusion is stronger.In addition,under the condition of upper integration,the complete convergence and complete integration convergence of the randomly weighted sums of WND arrays when the weights are columns of random variables are investigated.Compared with the existing results,this paper extends the range of random variables from END sequence to WND random variable whose control coefficients are regular variation functions,and extends the random sequences to random arrays,so that the range of randomly weighted sums studied is wider.