线性过程的广义强逼近及其应用

A GENERAL STRONG APPROXIMATION THEOREM FOR LINEAR PROCESSES AND ITS APPLICATIONS

设{Xt；t≥1}是由Xt=∑^∞i=0 aiεt-i所定义的线性过程，其中{ai；i≥0}是一实系数序列，{εi；-∞〈i〈∞}是一双边无穷的独立同分布随机变量序列．在髓Eε0^2可能为无穷的情形下，证明了{Xt；t≥1}的一个广义强逼近定理，作为应用，得到了线性过程部分和与部分和乘积的广义重对数律，以及具有相依重尾扰动项的AR（1）模型的渐近性质．

Let {Xt;t≥1}be a linear process given by Xt=∑^∞i=0 aiεt-i, where {ai;i≥0} is a sequence of real numbers and {εi;-∞〈i〈∞} is a doubly infinite sequence of i.i.d, random variables. In this article, a general strong approximation for {Xt;t≥1} is proved, when Eε0^2 may be infinite. As applications, the general laws of the iterated logarithm for the partial sums and the product of partial sums, and the limit property of the AR（1） model with dependent and heavy tailed innovations, are derived.