Some Limit Results Under Sublinear Expectations And Convex Expectations

Limit theorems such as the laws of large numbers,the central limit theorems and the laws of the iterated logarithm are important parts of probability theory.After a long period of study and development,they have formed quite perfect theoretical system and have been widely used in real life.The laws of large numbers are theorems that describe the stable long-term result of performing the same experiment a large number of times,the central limit theorems are theorems that describe the limit distribution of a sequence of random variables,and the laws of the iterated logarithm are theorems that describe the magnitude of the fluctuations of a random walk.In the framework of classical linear expectations,the above three kinds of limit theorems have been studied by many scholars and have formed quite rich classical results and methods.However,these limit theorems have been always considered under classical linear expectations and additive probabilities.In fact,the addi-tivity of expectations and probabilities has been abandoned in some areas because many uncertain phenomena cannot be well modelled using classical linear expectations and addi-tive probabilities.As an alternative to classical expectations or probabilities,nonlinear expectations or non-additive probabilities have been studied in many fields such as statistics,finance and.economics.Taking the stock as an instance,an important question is how to measure the investment risk brought by the fluctuations of the stock price.Because of the deficiency of linear expectations,many scholars have explored nonlinear expectation theory to describe the risk more accurately.Since the paper(Artzner et al.(1999))on coherent risk measures,people are more and more interested in sublinear expectations(or more generally,convex expectations).By Peng(2009),we know that a sublinear expectation E can be represented as the upper expectation of a subset of linear expectations {E?:???},i.e.,E[·]= sup E?[·].In most cases,this subset is often treated as an uncertain model of probabilities {P?:???}and the notion of sublinear expectation provides a robust way to measure a risk loss.In fact,the nonlinear expectation theory provides many rich,flexible and elegant tools.Motivated by modelling uncertainty in finance,Peng(2009,2010)generally defined the sublinear expectations and the convex expectations.He further established the new law of large numbers and central limit theorem under sublinear expectations.Many limit theorems such as the laws of large numbers,the central limit theorems,the laws of the iterated loga-rithm under sublinear expectations or convex expectations have been studied and extended.For example,Hu and Chen(2016)obtained a general strong law of large number for indepen-dent and non-identical distributed random variables under sublinear expectations;Li and Shi(2010)obtained a central limit theorem for independent and non-identical distributed random variables under sublinear expectations;Hu Mingshang(2010)established a central limit theorem for independent and identical distributed random variables under convex ex-pectations;Chen and Hu(2014)established a law of the iterated logarithm for bounded,independent and identical distributed random variables under sublinear expectations.In this paper,we further extend some limit theorems under sublinear expectations and convex expectations.This paper can be divided into four chapters:In chapter 1,we mainly introduce our research background and some related basic concepts and lemmas.In chapter 2,a law of large numbers under sublinear expectations is proved for independent and non-identical distributed random variables with finite first order moments and a central limit theorem under sublinear expectations is proved for independent and non-identical distributed random variables with finite second order moments.In chapter 3,a law of the iterated logarithm under sublinear expectations is proved for negatively dependent and non-identical distributed random variables with any finite order moments.In chapter 4,a central limit theorem under convex expectations is proved for independent and non-identical distributed random variables with finite second order moments and a law of large numbers under convex expectations is proved for independent and non-identical distributed random variables with finite first order moments.