关键词: 线性过程, φ -混合, 负相伴, Baum-Katz 律, 完全收敛性

Abstract: In this paper, the author discusses moving-average process
$X_k=\sum\limits_{i=-\infty}^\infty a_{i+k}\varepsilon_i$,
where $\{\varepsilon_i; -\infty $\varphi$-mixing or negatively associated random variables with mean zeros and finite variances,
$\{a_i;-\infty $S_n=\sum\limits_{k=1}^nX_k, n\geq 1$, the author proves that, if
$E|\varepsilon_1|^r<\infty$, then, for $1\leq p<2$ and $r>p$
$$\lim_{\epsilon\searrow 0}\epsilon^{2(r-p)/(2-p)}\sum\limits_{n=1}^\infty n^{r/p-2}
P\{|S_n|\geq \epsilon
n^{1/p}\}=\frac{p}{r-p}E|Z|^{2(r-p)/(2-p)},$$
where $Z$ has a normal distribution with mean 0 and variance
$\tau^2=\sigma^2\cdot(\sum\limits_{i=-\infty}^\infty a_i)^2.

Key words: Moving-average process, φ -mixing, Negative association, Baum-Katz law, Complete convergence.