线性过程关于大数律的精确渐近性

该文主要讨论的是滑线性过程 $X_k=\sum\limits_{i=-\infty}^\infty a_{i+k}\varepsilon_i$,其中 $\{\varepsilon_i; -\infty$\varphi$ -混合或负相伴随机变量序列,$\{a_i;-\inftyp$, 若 $E|\varepsilon_1|^r<\infty$$\lim_{\epsilon\searrow 0}\epsilon^{2(r-p)/(2-p)}\sum\limits_{n=1}^\infty n^{r/p-2}P\{|S_n|\geq \epsilonn^{1/p}\}=\frac{p}{r-p}E|Z|^{2(r-p)/(2-p)},$ 其中 $Z$ 是服从均值为零,方差为 $\tau^2=\sigma^2\cdot(\sum\limits_{i=-\infty}^\infty a_i)^2$的正态分布.

Precise Asymptotics in the Law of Large Numbers of Moving-average Processes

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$$$\lim_{\epsilon\searrow 0}\epsilon^{2(r-p)/(2-p)}\sum\limits_{n=1}^\infty n^{r/p-2}P\{|S_n|\geq \epsilonn^{1/p}\}=\frac{p}{r-p}E|Z|^{2(r-p)/(2-p)},$$ where $Z$ has a normal distribution with mean 0 and variance $\tau^2=\sigma^2\cdot(\sum\limits_{i=-\infty}^\infty a_i)^2.