Some Results On Limit Theory For Sequences Of Random Variables

Theory of Probability is a science of quantitatively studying regularity of random phe-nomena, which is extensively applied in natural science, technological science, and managerialscience etc. Hence, it has been developing rapidly since 1930 s and many new branches haveemerged from time to time. Limit Theory is one of the important branches and also anessential theoretical basis of science of Probability and Statistics. As stated in the classicalbook"Limit Distributions for Sums of Independent Random Variables"(1954) by Gendenkoand Kolmogrov,"The epistemological value of the theory of probability is revealed only bylimit theorems. Without limit theorems it is impossible to understand the real content of theprimary concept of all our sciences-the concept of probability."The classical limit theoremsof probability theory for independent random variables had been developed successfully in1930 s and 1940 s, and they are the significant achievements in the progress of Probability.The basic results were summed up in Gendenko and Kolmogrov s monograph《Limit Distri-butions for Sums of Independent Random Variables》(1954) and Petrov s monograph《Sumsof Independent Random Variables》(1975). The strong limit theorems of probability theoryfor mixing random variables, dependent random variables and martingale had been developedin 1950 s and 1960 s. The basic results were summed up in Lu Chuanrong and Lin Zhengyanmonograph《Limit theory for mixing dependent random variables》(1997) and Hall andHeyde《Martingale limit theory and its applications》(1980). Limit theory has become themost important and popular orientations of the current study of Probability Theory. Somesignificant results have been reached through deep research in this dissertation.In Chapter one, the author deals with precise asymptotics of random variables. Firstof all, in Section 2 the author discuss the precise asymptotics in the law of the iteratedlogarithm and the complete convergence for uniform empirical process. Let {ξ1,ξ2,···,ξn}be a sequence of independent and identically distributed U[0,1]-distributed random variables. Define the uniform empirical process In Section 3, the author deals with the precise asymptotics of linear process generatedby dependent random variables and obtains the same results as the Theorems 1.2.1-1.2.4.In Section 4, the author obtains a general law of precise asymptotics for a new kind ofcomplete moment convergence of i.i.d. random variables.Theorem 1.4.1 Let g(x) be a positive and differentiable function defined on [n0,∞),and the following conditions are satisfied: In Section 5, the author obtains a general law of precise asymptotics for products ofsums under dependence.We need the following assumptions:(A1) Let g(x) be a positive and di?erentiable function defined on [n0,∞), which isstrictly increasing to∞;(A2)ρ(x) = ggt ((xx)) is monotone for t < 1, and ifρ(x) is monotone nondecreasing, weassume limx(A3) ?(x) = gg ((xx)) is monotone, and if ?(x) is monotone nondecreasing, we assume In Chapter 2, the author deals with the almost sure central limit theorem of randomvariables. At first, in Section 2, we prove an almost sure central limit theorem for weightedsums under association. Theorem 2.3.3 Assume that (2.3.1) and (2.3.2) are satisfied and X is in the domainof attraction of the normal law, thenIn Section 4, the author deals with an almost sure central limit theorem for products ofsums of partial sums under association.Theorem 2.4.1 Let {Xn;n≥1} be a strictly stationary NA (PA,LNQD,LPQD)sequence of positive random variables with EX1 =μ> 0, and VarX1 =σ2 <∞. DenoteSn = nXi, Tn = nSi andγ=σ/μthe coe?cient of variation. Assume thatWhere F(·) is the distribution function of the random variables e10/3N .In Chapter 3, the author deals with the important model of time series analysis-autoregressionmodels of order one. We discuss the tests for unit root and strong consistency of the ordinaryleast squares estimator under dependence. The main results are as follows: In Chapter 4, the author deals with upper large deviations for mixing random sequence.In Section 2, we discuss upper large deviations for empirical measure generated by mixingrandom sequence.