随机变量序列的极限理论的若干结果
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摘要
概率极限理论是概率论的主要分支之一,也是概率论的其它分支和数理统计的重要基础.前苏联著名概率论学家Gnedenko和Kolmogrov曾说过“:概率论的认识论的价值只有通过极限定理才能被揭示,没有极限定理就不可能去理解概率论的基本概念的真正含义.”本文也就此对概率极限理论的若干问题进行了初步的研究.
本文利用概率极限理论的相关工具,首先,依次讨论了均匀经验过程完全收敛性及重对数律的精确渐近性、由相依序列生成的线性过程的精确渐近性、独立同分布随机变量序列矩完全收敛性的精确渐近性的一般形式以及相依序列部分和乘积的精确渐近性的一般形式.其次,讨论了非平稳相依序列加权和的几乎处处中心极限定理、独立随机变量序列自正则加权和的几乎处处中心极限定理以及相依序列部分和之和的乘积的几乎处处中心极限定理.再次,利用弱收敛定理讨论了误差项为相依情形下的一阶自回归模型的单位根检验,还研究了误差项为NA情形下的一阶自回归模型中最小二乘估计的强相合性.最后,给出了混合序列的大偏差上界以及由混合序列产生的经验测度序列的大偏差上界.
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.
本文利用概率极限理论的相关工具,首先,依次讨论了均匀经验过程完全收敛性及重对数律的精确渐近性、由相依序列生成的线性过程的精确渐近性、独立同分布随机变量序列矩完全收敛性的精确渐近性的一般形式以及相依序列部分和乘积的精确渐近性的一般形式.其次,讨论了非平稳相依序列加权和的几乎处处中心极限定理、独立随机变量序列自正则加权和的几乎处处中心极限定理以及相依序列部分和之和的乘积的几乎处处中心极限定理.再次,利用弱收敛定理讨论了误差项为相依情形下的一阶自回归模型的单位根检验,还研究了误差项为NA情形下的一阶自回归模型中最小二乘估计的强相合性.最后,给出了混合序列的大偏差上界以及由混合序列产生的经验测度序列的大偏差上界.
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.
引文
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