Study On Complete Convergence Under Sub-linear Expectations

In this paper,we mainly study the complete convergence of identically distributed ND sequences and weighted sums of END sequences under sub-linear expectations.In order to further enrich and expand the theory and application of complete convergence,based on some basic research methods of probability probability limit theory and some latest theoretical researches under sub-linear expectations,we further study the complete convergence of sub linear expectation.Firstly,we use Rosenthal s inequality,Markov inequality,slowly varying function properties and the method of multiple truncated to discuss the complete convergence of ND sequence with the condition of Cv(|X1|pl(|X1|1/α))<∞ where αp>1,0<p<2,l(X)is a slowly varying function and E is countably sub-additive.We generalize the conclusion of complete convergence of NA sequences in prob-ability space and get the complete convergence of identically distributed ND sequences under sub-linear expectations,which enriches the complete convergence content under sub-linear expectations.Secondly,we combine Jansen inequality,Hold inequality,Markov inequality and the inequalities of capacities of END sequences under the sub-linear expectations to study the complete convergence for weighted sums of END sequences with the condition of that lim sup n→∞n-1 ∑i=1n aniβ<∞,β=max(α,γ),where 0<α≤2,γ>0 and E[|X|β]<∞.We get the complete convergence for weighted sums of the Marcinkiewicz type END sequences under sub-linear expectations,which extends and deepens the orig-inal result.At last,we through the special technique of truncation,combined with Markov inequalities and relevant basic properties of END sequences under sub-linear expectations to generalize the important lemma and prove the complete convergence for weighted sums of END sequences with the condition of that max1≤I≤n ani(n-1/p)=O,∑i=1n=O(ani2(logn)-1)and integral conditions of E[|Y|2p]≤Cv(|Y|2p)<∞.We obtain the complete convergence for weighted sums of Sung type END sequences under sub-linear expectations,and further perfect the complete convergence theory of END sequences in the sub-linear expectation context.Since the complexity of sub-linear expectation theory makes a lot of common research methods and practical tools in probability space is no longer applicable,the research on complete convergence of sub-linear expectations is more complex and difficult.In this paper,we present some effective methods to solve these problems.