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

Li Yunxia

Zhejiang University of Finance and Economics, Hangzhou 310018

Received:2004-08-30 Revised:2006-04-11 Online:2006-10-25 Published:2006-10-25
Contact: Li Yunxia

Abstract

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.

CLC Number:

60F15

Li Yunxia.

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

[J].Acta mathematica scientia,Series A, 2006, 26(5): 675-687.

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

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