本文選題：幾乎處處中心極限定理 + ρ~--混合序列��； 參考：混合序列加權(quán)和及部分和乘積的幾乎處處中心極限定理

【摘要】：極限理論在概率論中占據(jù)著重要的地位,而幾乎處處中心極限定理又一直是概率論研究的中心課題,很多有關于隨機樣本的線性統(tǒng)計量都可以看作是隨機變量加權(quán)和的形式,因而研究加權(quán)和的極限理論在概率論與數(shù)理統(tǒng)計應用中占據(jù)著重要的地位.幾乎處處中心極限定理是概率論中討論隨機變量和的極限分布為正態(tài)分布的一組定理,這組定理是數(shù)理統(tǒng)計與誤差分析的理論基礎,指出了在一定條件下,大量的隨機變量和的分布都可近似看作服從正態(tài)分布.本文在前人研究NA序列加權(quán)和的幾乎處處中心極限定理及ρ~--混合序列部分和乘積的幾乎處處中心極限定理的基礎上,研究了一般權(quán)重下ρ~--混合序列加權(quán)和及部分和乘積的幾乎處處中心極限定理,得到以下結(jié)論:定理1假設{Xn,n≥1}是零均值嚴平穩(wěn)的ρ~--混合序列,并且當r2時,0E|X1|r∞,{ani,1≤i≤n,n≥1}為一實值非負三角陣列,令Sn=(?)aniXi,假設下列條件成立(1)sup(?)ani2∞,|ani|≤Cn1/2/l∨logη(n/i),1≤i≤n,n≥1,對某個η0;(2)(?)|Cov(X1,Xj)|∞;(3)ρ-(n)= O(log-δn),δ1;(4)Var(S_n)= 1,當n→∞.那么對任意的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當D_n=∑dk,dk= logrk/k,r-1,并滿足定理1的條件時,定理1的結(jié)論仍然成立.定理3假設{Xn,n ≥ 1}是嚴平穩(wěn)正值的ρ~--混合序列,滿足E|X1|=μ0,VarX1=σ2∞,且0E|X1|∞,當r2時,記Sn=(?)Xi,變異系數(shù)γ = σ/μ,假設下列條件成立(1)0σ12= EX12+ 2(?)Cov(X1,Xj)∞;(2)(?)|Cov(X1,Xj)|∞;(3)ρ-(n)=O(log-δn),δ12(4)(?)0.那么對的x ∈R,有其中,F(x)表示隨機變量e(?)的分布函數(shù),N表示標準正態(tài)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當D_n=(?)dk,dk=kogrk/k,r-1,并滿足定理3的條件時,定理3的結(jié)論仍然成立.
[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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