Almost Sure Central Limit Theorem For Mixing Sequences Of Random Variables

Theory of probability is a science of quantitative law of random phenome-na, which is extensively applied in natural science,managerial science,economic and finance etc. The significance of the probability theory is to describe the regularity manifested by a large number of random factors,so the limit proper-ty for random variables plays an extremely important role to reveal the nature of random phenomena.Hence, Limit theory is not only one of the main branch-es of probability, but also an essential theoretical foundation of other fields of probability theory and mathematics statistics. As stated in the classical book 《Limit Distributions for Sums of Independent Random Variables》(1954) by Gnedenko and Kolmogrov,"The epistemological value of the theory of prob-ability is revealed only by limit theorems. Moreover, without limit theorems it is impossible to understand the real content of the primary concept of all our sciences-the concept of probability."The classical limit theorems of probability theory for independent ran-dom variables had been developed successfully in1930’s and1940’s, and they were the significant achievements in the progress of Probability. In many practical problems, the sample in most cases is not independent or the func-tion of independent sample is not independent.In addition, requirement of dependency is made in the theoretical research and other branches.In1950’s and1960s,following the classical limit theory of independent random vari-ables fully developed,the limit theorems of probability theory for martingale and dependent random variables had been developed.In recent decades, many scholars maintain interested in some new fields of probability limit theory for example,the almost sure central limit theorem,the theory of products of partial sums, which become the popular direction in the study of modern probability theory.In this paper, we will make further research in almost sure central limit theorems for kinds of mixing random variables and the products of sums of partial sums.In Chapter1, we introduce some dependent sequences of random variables in Section1:NA(PA), pairwise NQD,LNQD,φ-mixing,ρ-mixing,α-mixing ψ-mixing,β-mixing, λ-mixing and φ-mixing,ρ*-mixing,ρ--ixing. The author introduce their concepts,some simple properties and the rela-tionship between them.It makes good prepare for discussing the problem in Chapter2and3.In Section2the author recalls some results of central lim-it theorem of i.i.d. and φ-mixing,ρ-mixing,α-mixing sequences,and then introduce the concept of almost sure central limit theorem and the fruits of previous research.In Chapter2,we discuss almost sure central limit theorems for weighted sums of-mixing,-mixing and-mixing sequences of random variables respec-tively. In Section1, the research background is explored. In the past decades, the limit properties for weighted sums random variables were given by many scholars,but most of them were focus on the stationary sequences. Rare are almost sure central limit theorem for weighted sums of non-stationary mixing sequences.In this chapter, we prove some almost sure central limit theorems for weighted sums of mixing sequences of random variables without stationary assumption.Theorem2.1.1Let{Xn,n≥1} be a sequence of random variables with EXn=0and {ani,1≤i≤n,n≥1} be an array of real numbers satisfying the following(1) sup for some γ>0;(2) Var(Sn)�'1as n�'∞o and {Xn2} is a uniformly integrable family, where(3) For a certain δ>0,{Xn} is strongly mixing,{|Xn|2+δ} is a uniformly integrable family, infn Var(Xn)>0and Let Then for any x E R,Theorem2.1.2Let {Xn,n≥1} be a sequence of random variables with EXn=0and {ani,1≤i≤n,n≥1} be an array of real numbers satisfying the following(1) for some γ>0;(2) Var(Sn)�'1as n�'∞and {Xn2} is a uniformly integrable family, where(3){Xn} is φ-mixing and Let Then for any x E R,Theorem2.1.3Let {Xn,n≥1} be a sequence of random variables with EXn=0and {ani,1≤i≤n,n≥1} be an array of real numbers satisfying the following(1) for some γ>0;(2) Var(Sn)�'1as n�'∞and{Xn2} is a uniformly integrable family, where(3){Xn}is ρ-mixing and Let Then for any x∈R,In Section2,we list some lemmas used in the proof of the theorems.In Section3,the author gives the proof of the theorem.In Chapter3,the author discusses an almost sure central limit theorem for the products of sums of partial sums for ρ--mixing sequences.In Section l,we introduce the related background of the products of sums of partial sums for random variables sequences,and recall some almost sure central limit theorems for the products of partial sums and the products of sums of partial sums for some mixing sequences.Based on these,the author discusses the almost sure central limit theorem for the products of sums of partial sums for ρ-mixing sequences in more general case.Theorem3.1.1be a stationary ρ-sequence of positive random variables with EX1=μ>0, Var(X1)=σ2, E|X1|r<∞, for somer>2. Denote Assume that(al)(a2) (a3) ρ (n)=O(logδn), for some δ>2.(a4) thenTheorem3.1.2Under the conditions of Theorem3.1.1,we have where F(·)is the distribution function of random variable eN. In Section2,we list and prove some lemmas used in the proof of the theorems.In Section3,the author give the proof of the theorem.In Chapter4,the author discusses some issues yet to be resolved or to be further explored, preparing for the further study of the problem.