The Convergence of the Sums of Independent Random Variables Under the Sub-linear Expectations  

作　　者：Li Xin ZHANG 

机构地区：[1]School of Mathematical Sciences,Zhejiang University,Hangzhou 310027,P.R.China

出　　处：《Acta Mathematica Sinica,English Series》2020年第3期224-244,共21页数学学报（英文版）

基　　金：Supported by grants from the NSF of China(Grant No.11731012);Ten Thousands Talents Plan of Zhejiang Province(Grant No.2018R52042);the 973 Program(Grant No.2015CB352302);the Fundamental Research Funds for the Central Universities。

摘　　要：Let {Xn;n≥1} be a sequence of independent random variables on a probability space(Ω,F,P) and Sn=∑k=1n Xk.It is well-known that the almost sure convergence,the convergence in probability and the convergence in distribution of Sn are equivalent.In this paper,we prove similar results for the independent random variables under the sub-linear expectations,and give a group of sufficient and necessary conditions for these convergence.For proving the results,the Levy and Kolmogorov maximal inequalities for independent random variables under the sub-linear expectation are established.As an application of the maximal inequalities,the sufficient and necessary conditions for the central limit theorem of independent and identically distributed random variables are also obtained.Let {Xn;n≥1} be a sequence of independent random variables on a probability space（Ω,F,P） and Sn=∑k=1n Xk.It is well-known that the almost sure convergence,the convergence in probability and the convergence in distribution of Sn are equivalent.In this paper,we prove similar results for the independent random variables under the sub-linear expectations,and give a group of sufficient and necessary conditions for these convergence.For proving the results,the Levy and Kolmogorov maximal inequalities for independent random variables under the sub-linear expectation are established.As an application of the maximal inequalities,the sufficient and necessary conditions for the central limit theorem of independent and identically distributed random variables are also obtained.

关 键 词：Sub-linear EXPECTATION capacity INDEPENDENCE LEVY MAXIMAL inequality central limit theorem 

分 类 号：O211.5[理学—概率论与数理统计]