Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm  

作　　者：ZHANG LiXin 

机构地区：[1]School of Mathematical Sciences, Zhejiang University

出　　处：《Science China Mathematics》2016年第12期2503-2526,共24页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant No. 11225104);the National Basic Research Program of China (Grant No. 2015CB352302);the Fundamental Research Funds for the Central Universities

摘　　要：Kolmogorov's exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper establishes the Kolmogorov type exponential inequalities of the partial sums of independent random variables as well as negatively dependent random variables under the sub-linear expectations. As applications of the exponential inequalities, the laws of the iterated logarithm in the sense of non-additive capacities are proved for independent or negatively dependent identically distributed random variables with finite second order moments.For deriving a lower bound of an exponential inequality, a central limit theorem is also proved under the sublinear expectation for random variables with only finite variances.Kolmogorov＇s exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper establishes the Kolmogorov type exponential inequalities of the partial sums of independent random variables as well as negatively dependent random variables under the sub-linear expectations. As applications of the exponential inequalities, the laws of the iterated logarithm in the sense of non-additive capacities are proved for independent or negatively dependent identically distributed random variables with finite second order moments.For deriving a lower bound of an exponential inequality, a central limit theorem is also proved under the sublinear expectation for random variables with only finite variances.

关 键 词：sub-linear expectation capacity Kolmogorov＇s exponential inequality negative dependence laws of the iterated logarithm central limit theorem 

分 类 号：O211.67[理学—概率论与数理统计]