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高校应用数学学报
Applied Mathematics A Journal of Chinese Universities  2014, Vol. 29 Issue (3): 261-268    DOI:
    
A general strong approximation theorem for the long memory process generated by $\varphi$-mixing sequences
LI Hui-jie, FU Ke-ang
School of Stat. and Math., Zhejiang Gongshang Univ., Hangzhou 310018, China
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Abstract  Let $\{X_k;k\ge1\}$ be a moving average process defined by $X_k=\sum_{i=0}^\infty a_i\varepsilon_{k-i},$ where $\{\varepsilon_i;-\infty<i<\infty\}$ is a doubly infinite sequence of identically distributed $\varphi$-mixing random variables, $a_i\sim i^{-\alpha}l(i)$ and $l(i)$ is a slowly varying function. When $1/2<\alpha<1,$ $\{X_k;k\ge1\}$ is a long memory process. Under the assumption that $\text{E}\varepsilon_0^2$ may be infinite, a general strong approximation theorem for partial sums of the long memory process is derived.

Key words： long memory process      mixing dependence      strong approximation     
Received: 25 December 2013      Published: 10 June 2018
CLC:  O211.4  
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LI Hui-jie, FU Ke-ang. A general strong approximation theorem for the long memory process generated by $\varphi$-mixing sequences. Applied Mathematics A Journal of Chinese Universities, 2014, 29(3): 261-268.

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http://www.zjujournals.com/amjcua/     OR     http://www.zjujournals.com/amjcua/Y2014/V29/I3/261


由$\varphi-$混合序列产生的长程相依过程的广义强逼近定理

设$\{X_k; k\geq1\}$是由$X_k=\sum_{i=0}^\infty a_i\varepsilon_{k-i}$所定义的滑动平均过程, 其中$\{\varepsilon_i;-\infty<i<\infty\}$是一同分布的$\varphi$-混合相依变量序列, $\{a_i;i\ge0\}$为满足条件$a_i\sim i^{-\alpha}l(i)$的实数序列, $l(i)$为一缓变函数. 当$1/2<\alpha<1$时, $\{X_k; k\geq1\}$为一长程相依过程. 在$\text{E}\varepsilon_0^2$可能为无穷的条件下, 对长程相依过程$\{X_k; k\geq1\}$的部分和建立了一个更为一般性的强逼近定理.

关键词： 长程相依过程,  混合相依,  强逼近 
[1] LI Yun-xia. A general law of precise asymptotics for long memory processes[J]. Applied Mathematics A Journal of Chinese Universities, 2015, 30(2): 150-156.