Related Research About The Law Of Large Numbers Under Sublinear Expectation

The law of large numbers is an important limit theorem in the probability theory.The law of large numbers has played a very important role in the whole development of probability theory.The existence of the law of large numbers makes probability theory form perfect theoretical system.It has extensive application background.The classic law of large numbers is based on the linear expectations and additive probabilities.However,there arc still many uncertainties existing in the fields of statistics,finance,economy and so forth.These uncertainties do not satisfy the linear and additive conditions.Therefore,those uncertainties can not be well explained and predicted by those theories due to the traditional linear set-up.In order to model these uncertainties,people tend to think more about nonlinear expectation or non-additive probabilities.In order to depict the uncertainties in statistics;finance and economy more reasonably.Peng(2007)gave the definition of sublinear expectation and further obtained the law of large numbers and the central limit theorem under sublincar expectation which formed the theoretical frameework of sublinear expectation.Subsequently,a number of researchers leave stuudied and extended the law of large numbers based on this theoretical framework.For example,Chen at al.(2013)demonstrated the strong law of large numbers for non-additive probabilities in the framework of sublinear expectation.It needed an assumption that random variables were ?D under sublinear expectation Hu et al.(2016)extended the strong law of large numbers in Chen et al.(2013)by only needing the assumption that random variables were independent:Zhang(2016)proved the strong law of large numbers for non-additive probabilities in the framework of sublincar expertation.It needed the assumption that.random variables were negatively dependent.and identical distributed in the sublinear expectation.Their work has enriched the limit theory of sublincar expertation to a large extent.In this paper,we further extend the law of large numbers based on the existing results.The whole paper is divided into four chapters:In chapter 1,we introduce the development history of the law of large numbers and the researching status.In chapter 2.we state the basic definition and existing results.In chapter 3.we introduce the main results of this paper.We prove the strong law of large numbers for negatively dependent and non-identical distributed random variables in the framework of sublinear expec tation.In chapter 4.we give an example to illustrate that the law of large numbers is meaningful in reality.