NA序列部分和之和乘积的极限定理

Limit Theorems of Products of Sumsof Partial Sums of NA Sequence

设{X_(n),n≥1}是一严平稳正值负相关(NA)随机变量序列,满足EX_(1)=μ>0,Var X_(1)=σ^(2)<∞,σ^(2)_(1)=1+2/σ^(2)∞∑j=2Cov(X_(1),X_(j))>0,Cov(X_(1),X_(n+1))=O(n^(-1)(log n)^(-2-■)),对某个■>0.记S_(n)=n∑i=1X_(i),T_(n)=n∑i=1S_(i),γ=σ/μ.首先利用NA序列加权和的中心强极限定理和矩不等式证明(n∏k=12T_(k)/k(k+1)μ)1/γσ_(1)√3/10nd→e^(N),n→∞,其中N为标准正态随机变量;其次,对于边界函数和拟权函数给出NA序列部分和之和乘积的完全收敛性中精确渐近性的一般结果.

Let{X_(n),n≥1}be a strictly stationary negatively associated(NA)sequence of positive random variables with EX_(1)=μ>0,Var X_(1)=σ^(2)<∞,σ^(2)_(1)=1+2/σ^(2)∞∑j=2Cov(X_(1),X_(j))>0,Cov(X_(1),X_(n+1))=O(n^(-1)(log n)^(-2-■)),for some ■>0.Denote S_(n)=n∑i=1X_(i),T_(n)=n∑i=1S_(i) and γ=σ/μ.Firstly,by using the central strong limit theorems and moment inequalities of weighted sums of NA sequence,we prove that(n∏k=12T_(k)/k(k+1)μ)1/γσ_(1)√3/10nd→e^(N),n→∞,where N is a standard normal random variable.Secondly,we give a general result of precise asymptotics in complete convergence for products of sums of partial sums of NA sequence for the boundary function and quasi weight function.