Complete Convergence And Complete Integral Convergence Of Dependent Sequences Under Sub-linear Expectation

This paper mainly studies the complete convergence of the random weighted sum of END（Extended Negatively Dependent）sequence under the sub-linear expectation,the complete integral convergence of the random weighted sum of ND（Negatively Dependent）sequence,and the complete integral convergence of the linear process of the ND sequence with random coefficients.It generalizes the conclusions in the probability space to the sub-linear expectation.Firstly,we study the complete convergence of the random weighted sum of the END sequence under the sub-linear expectation.Let {An,n≥1} be the random variable sequence under the sub-linear expec-tation.{Xn,n≥1} is the sequence of END random variables in the sub-linear expectation that randomly controlled by the random variable X.{An,n≥1} is independent of {Xn,n≥1}.l（x）>0 is a slow vary-ing function.For α>0,p>1,αp>1,q>max（2,p）,we get sum from n=1 to ∞(n^αp-2l（n） )V（|sum from i=1 to n（A_iX_i|>εn^α））<∞ under the condition sum from i=1 to n（（?）|A_i|^q=O（n） ） and CV[|X|1/α)]<∞ using the tolerance inequality of partial sums,the independent nature,the nature of the END sequence,the nature of the slow varying function and the method of multiple truncation.That is,the complete convergence of the random weight-ed sum for the END sequence is obtained under the sub-linear expectation.The complete convergence with the random weighted sum of the END random variable sequence for the real number sequence in the probability space is generalized to the random variables sequence under the sub-linear expectation.Secondly,we study the complete integral convergence of the random weighted sum for the ND sequence under the sub-linear expectation.The probability and expectation are additive in the probability space,but are sub-additive under the sub-linear expectation.We construct the continuous functions belonging to Cl,Lip and if f≤I（A）≤g,f,g∈H then Ef≤V（A）≤Eg transforms expectation and capacity to overcome this difficulty.Using the integral inequality of the partial sum under the sub-linear expectation space,we obtain the complete integration convergence of the random weighted sum of the ND sequence under the sub-linear expectation,and extend the complete convergence of the random weighted sum of the ND random variable sequence in the probability space to the sub-linear expectation.Finally,we study the complete integral convergence of the linear process with the random coefficient ND sequence in the sub-linear expectation under the conditions of the linear process using the same method with the complete integration convergence of the random weighted sum for the ND sequence.The complete integral convergence of the linear process of the ND sequence in the probability space is extended to the sub-linear expectation.