An almost sure central limit theorem for products of sums of partial sums under association

作者：Zhang Yong^*; Yang Xiao Yun; Dong Zhi Shan

来源：Journal of Mathematical Analysis and Applications, 2009, 355(2): 708-716.

DOI：10.1016/j.jmaa.2009.01.071

• Almost sure central limit theorem
• Products of sums of partial sums
• Negatively (positively) associated
• LNQD (LPQD)

Let (X(n), n >= 1) be a strictly stationary positively or negatively associated sequence of positive random variables with EX(1) = mu > 0, and Var X(1) = sigma(2) < infinity. Denote S(n) = Sigma(n)(i=1) Xi, T(n) = Sigma(n)(i=1) S(i) and gamma = sigma/mu the coefficient of variation. Under suitable conditions, we show that
for all x lim(n ->infinity) Sigma(n)(k=1) 1/k I {(2(k)Pi(k)(j=1) T(j)/k!(k + 1)!mu(k))(1/(gamma sigma 1 root k)) <= x} = F (x) a.s.,
where sigma(2)(1) = 1 + 2/sigma(2) Sigma(infinity)(j=2) Cov(X(I), X(j)), F(.) is the distribution function of the random variables e(root 10/3N) and N is a standard normal random variable.

• 出版日期2009-7-15
• 单位吉林大学