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Precise asymptotics in the complete moment convergence of NA sequence. (Chinese. English summary) Zbl 1082.60507

Summary: If \(\{X,X_n;n\geq 1\}\) is a stationary sequence of negatively associated random variables (NA), based on some means about convergence, precise asymptotics in the complete moment convergence of NA sequence is obtained. Suppose that \(EX_1=0,E|X_1|^3<\infty\), some conditions are satisfied. Set \(S_n= X_1+X_2+\cdots+X_n\), \(n\geq 1\), \(\sigma^2=EX_1+2\sum^\infty_{i=2} E(X_1X_j)>0\). If \(E|X|^r< \infty\), \(1<p<2\), \(r>1+p/2\), then \[ \lim_{\varepsilon \searrow 0} \varepsilon^{2(r-p)/(2-p)-1}\sum^\infty_{n=1}n^{r/(p-2)-1/p}E\{ |S_n|-\sigma \varepsilon n^{1/p}\}_+=\frac{p(2-p)\sigma}{(r-p)(2r-p-2)}E|N|^{2(r-p)/(2-p)}, \] where \(N\) is a standard normal random variable.

MSC:

60F15 Strong limit theorems
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