Almost Sure Convergence And Complete Convergence Of Weighted Sums Under Sub-linear Expectations

The limit theorem for sub-linear expectations is a challenging field and has attracted a lot of attention in recent years.Some research achievements have been made under the sub-linear expectation,but the development is still not perfect.There are still many problems to be studied.Therefore,it is meaningful to study the limit theorem under the sub-linear expectations.In this paper,we establish the law of the iterated logarithm of the END sequence,the almost sure convergence of the weighted sums and the complete convergence law of the ND array under the sub-linear expectation,and generalize the some existing results in the probability space.Firstly,on the basis of Zhang s research,Let the law of the iterated logarithm for the moment condition of E(|X1|2+?)<? reduced to E(|X1|2 ln?|X1|)<?,by correlation properties of END random variables,we researched the law of the iterated logarithm for END random variables under the non-linear expectations,which deepen the Zhang result and expanded the scope of its application.Secondly,by using the moment inequalities of the END sequence,Cr inequality,exponential inequal-ity,Rosenthal inequality,and extend the range of p from 1?p<2 t0 0<p?2,we extend almost sure convergence of weighted sums from the traditional probability space to the sub-linear expectation space,and the almost sure convergence of the END sequence under the sub-linear expectation is obtained.Finally,we study the complete convergence theorem for ND arrays with sub-linear expectation,using the moment inequalities of ND sequences,some properties of sub-linear expectation,and Markov s inequalities.By establishing a continuous local Lipschitz function,the capacity and expectation have sub-additives.Using a new method that is different from the probability space,the complete convergence of the ND array under the sub-linear expectation is established,which generalize the corresponding conclusions.