Limit Theorems For A Linear Process

From nearly two centuries,the laws of large numbers and the central limit theorems of random variable sequence (X_n, n≥1} have been the central issue of probability. When {X_n, n≥1} is an independent random variable sequence, these issues have been perfect conclusion. In recent years,the work on the limit theorems has mainly two categories.One is to study the limit theorems of some random variable sequences under some dependent conditions, such as the study of limit theorems under various mixed conditions, the study of limit theory about martingale (semimartingale)and some statistics and so on. The other is the study of limit theorems of random processes.In this paper, we study the strong law of large numbers and the central limit theorem for a linear process. Usually there are two basic ways to prove a strong law of large numbers. The first method is firstly to prove that some subsequences of S_n/B_n(B_n＞0, B_n↑∞) obey the strong laws of large numbers, then to promote the conclusions to the sequences(methods of subsequences). In this method, it is necessary to use some maximum inequalities of the sequences. The second method is to use Hajek-Renyi maximum inequality to prove it.As the Hajek-Renyi maximum inequality is not easy to obtain, the first method is more popular. But once attained the Hajek-Renyi maximum inequality, the proof of the strong laws of large numbers is obvious. In this paper, on the basis of the paper of Phillips and solo(1992), we use the B-N decomposition of the linear filter to obtain the Hajek-Renyi inequality of a linear process, and then obtain the law of large numbers and central limit theorem for the linear process. So it promotes and improves the corresponding results of the paper of Phillips and solo(1992).This paper is divided into three chapters:Chapter 1 prior knowledge. Chapter 2 Hajek-Renyi Inequality of linear process.Chapter 3 Limit Theorems of linear process.