Limiting Properties Of Several Dependent Sequences Under Sublinear Expectations

In 1933,Russian mathematician Kolmogorov published his book《The Foundations of the Theory of Probability》,which established the theoretical system of probability theory on the basis of measure theory for the first time.This classical probability space is denoted by(Ω,F,P),its probability measure P is linear,so the expectation E defined therein is a linear functional,considering that many situations in life have uncertainty,Academician Peng Shige in Peng(1997,2006,2007)proposed concepts of g-expect at ion,G-expectation,and subelinear expectation respectively to deal with problems that do not conform to the linear hypothesis,and established the framework of sublinear expectation space(Ω,H,E)by introducing the concept of Choquet expectation(integral).The main content of this paper is based on this framework.Firstly,we study the complete convergence and complete moment convergence of ND random variable sequences under sublinear expectations.We prove an inequality of sum of capacity.By using this inequality and the truncation method,we get the equivalent conditions of complete convergence and complete moment convergence of negatively dependent(ND)random variable sequences under the sublinear expectation space.Secondly,we study the upper bound of the moderate deviation principle for the linear process generated by ND sequence under sublinear expectation and the moderate deviation principle of the linear process generated by m-dependent random variables.Using the decomposition form of the Beveridge-Nelson linear process and the moderate deviation principle of sequence of m-dependent random variables under sublinear expectations,we proved the moderate deviation principle of the linear process generated by ND and m-dependent random variables.Next,we study the strong law of large numbers for m-Asymptotically Almost Negatively Associated(m-AANA)random variable sequence under sublinear expectation.Using the properties and inequalities of m-AANA random variable sequence under sublinear expectations and Kronecker lemma,the three series theorem of mAANA random variable sequence is obtained.Applying the three series theorem,the Marcinkiewicz’s strong law of large numbers for the identically distributed m-AANA random variables,the Kolmogorov’s strong law of large numbers for non-identically distributed m-AANA random variables and the strong law of large numbers for weighted sums of non-identically distributed m-AANA random variables are obtained.Then,we study the complete convergence and the complete moment convergence of the linear process of generating sequences of m-AANA random variables under sublinear expectation.On the basis of the previous research,the moment inequality of m-AANA random variable sequence is proved first,and then the complete convergence and complete moment convergence of pα=1 and pα>1 are proved respectively.Finally,we consider the precise large deviation of the random sum of the sequence of independent identically distributed heavy-tailed distribution random variables with Poisson INGARCH(1,1)process counting process under the classical space.Using the property of Poisson INGARCH(1,1)process,we prove the exponential inequality for the sum of counting process {Nt,t≥ 0},then the precise large deviation of the random sum is established,and the simulation study is carried out.