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sTRONG LIMIT THEOREMs FOR EXTENDED INDEPENDENT RANDOM VARIABLEs AND EXTENDED NEGATIVELY DEPENDENT RANDOM VARIABLEs UNDER sUB-LINEAR EXPECTATIONs

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Acta Mathematica scientia 2022年第2期42卷 467-490页

作者：张立新 school of Mathematical sciences Zhejiang UniversityHangzhou 310027China

Limit theorems for non-additive probabilities or non-linear expectations are challenging issues which have attracted a lot of interest recently.The purpose of this paper is to study the strong law of large numbers and the law of the iterated logarithm for a sequence of random variables in a sub-linear expectation space under a concept of extended independence which is much weaker and easier to verify than the independence proposed by Peng[20].We introduce a concept of extended negative dependence which is an extension of the kind of weak independence and the extended negative independence relative to classical probability that has appeared in the recent literature.Powerful tools such as moment inequality and kolmogorov’s exponential inequality are established for these kinds of extended negatively independent random variables,and these tools improve a lot upon those of Chen,Chen and Ng[1].The strong law of large numbers and the law of iterated logarithm are also obtained by applying these inequalities.

关键词： sub-linear expectation capacity extended negative dependence kolmogorov’s exponential inequality laws of the iterated logarithm law of large numbers

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On the laws of the iterated logarithm under sub-linear expectations

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Probability, Uncertainty and Quantitative Risk 2021年第4期6卷 409-460页

作者： Li-Xin Zhang school of Mathematical sciences Zhejiang UniversityHangzhou 310027ZhejiangChina

In this paper,we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space,where the random variables are not necessarily identically distributed.exponential inequalities for the maximum sum of independent random variables and kolmogorov’s converse exponential inequalities are established as tools for showing the law of the iterated logarithm.As an application,the sufficient and necessary conditions of the law of the iterated logarithm for independent and identically distributed random variables under the sub-linear expectation are obtained.In the paper,it is also shown that if the sub-linear expectation space is rich enough,it will have no continuous capacity.The laws of the iterated logarithm are established without the assumption on the continuity of capacities.

关键词： sub-linear expectation Capacity kolmogorov’s exponential inequality Laws of the iterated logarithm

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