ρ~--混合序列加权和及部分和乘积的几乎处处中心极限定理

发布时间：2018-05-14 14:32

  本文选题：几乎处处中心极限定理 + ρ~--混合序列　； 参考：混合序列加权和及部分和乘积的几乎处处中心极限定理

【摘要】：极限理论在概率论中占据着重要的地位,而几乎处处中心极限定理又一直是概率论研究的中心课题,很多有关于随机样本的线性统计量都可以看作是随机变量加权和的形式,因而研究加权和的极限理论在概率论与数理统计应用中占据着重要的地位.几乎处处中心极限定理是概率论中讨论随机变量和的极限分布为正态分布的一组定理,这组定理是数理统计与误差分析的理论基础,指出了在一定条件下,大量的随机变量和的分布都可近似看作服从正态分布.本文在前人研究NA序列加权和的几乎处处中心极限定理及ρ~--混合序列部分和乘积的几乎处处中心极限定理的基础上,研究了一般权重下ρ~--混合序列加权和及部分和乘积的几乎处处中心极限定理,得到以下结论:定理1假设{Xn,n≥1}是零均值严平稳的ρ~--混合序列,并且当r2时,0E|X1|r∞,{ani,1≤i≤n,n≥1}为一实值非负三角阵列,令Sn=(?)aniXi,假设下列条件成立(1)sup(?)ani2∞,|ani|≤Cn1/2/l∨logη(n/i),1≤i≤n,n≥1,对某个η0;(2)(?)|Cov(X1,Xj)|∞;(3)ρ-(n)= O(log-δn),δ1;(4)Var(S_n)= 1,当n→∞.那么对任意的x∈R,有(?)dkI{Sk≤x}=Φ(x).a.s.(1)其中,D_n=(?)dk,dk=exp(logαk)/k,α∈[0,1/2).定理2当D_n=∑dk,dk= logrk/k,r-1,并满足定理1的条件时,定理1的结论仍然成立.定理3假设{Xn,n ≥ 1}是严平稳正值的ρ~--混合序列,满足E|X1|=μ0,VarX1=σ2∞,且0E|X1|∞,当r2时,记Sn=(?)Xi,变异系数γ = σ/μ,假设下列条件成立(1)0σ12= EX12+ 2(?)Cov(X1,Xj)∞;(2)(?)|Cov(X1,Xj)|∞;(3)ρ-(n)=O(log-δn),δ12(4)(?)0.那么对的x ∈R,有其中,F(x)表示随机变量e(?)的分布函数,N表示标准正态分布D_n=(?)dk,dk=k/exp(logαk),α i,∈[0,1/2),σn2= Var(Sn,n),Sn,n=(?)bk,nYk,Yk=Xk-μσ,k≥1,bk,n =(?)1/i,k≤n,bk,n=0,kn.定理4当D_n=(?)dk,dk=kogrk/k,r-1,并满足定理3的条件时,定理3的结论仍然成立.
[Abstract]:Limit theory plays an important role in probability theory, and almost everywhere central limit theorem is the central subject of probability theory. Many linear statistics about random samples can be regarded as the weighted sum of random variables. Therefore, the study of the limit theory of weighted sum plays an important role in the application of probability theory and mathematical statistics. Almost everywhere central limit theorem is a set of theorems that discuss that the limit distribution of sum of random variables is normal distribution in probability theory. This set of theorems is the theoretical basis of mathematical statistics and error analysis, and points out that under certain conditions, The distribution of the sum of a large number of random variables can be approximately regarded as a normal distribution. This paper is based on the previous studies of almost everywhere central limit theorems for weighted sums of na sequences and almost everywhere central limit theorems for the product of partial sums of 蟻 -mixed sequences. In this paper, we study the almost everywhere central limit theorem of weighted sum and partial sum product of 蟻 -mixed sequence under general weight. The following conclusions are obtained: theorem 1 assumes that {Xnn 鈮，

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