线性平稳过程的协方差和均值估计的大偏差结果

LARGE DEVIATION RESULTS FOR THE ESTIMATIONS OF COVARIANCE AND MEAN OF LINEAR STATIONARY PROCESSES

摘要: 本文讨论了具有有界输入的线性平稳过程的参数估计的大偏差的上界和下界。

Abstract: We consider the discrete time linear stationary process: $X(t)=\sum_{j=0}^{\infty} a_j \varepsilon(t-j), \quad t=0, \pm 1, \pm 2, \cdots$ Where {ε（t）} is a {$\mathscr{F}_t$}-stationary ergodic martingale difference sequences. We assume a₀=1, $\sum_{j=0}^{\infty}\left|a_j\right|<\infty$ and $E\left[\varepsilon^2(t) \mid \mathscr{F}_{t-1}\right]=\sigma^2>0$ a. s. Let the sample covariance function and the sample mean function of the prooess X （t） be respoectively: $\hat{r}_N(k)=\frac{1}{N} \sum_{t=1}^{N-k} X(t) X(t+k)=\hat{r}_N(-k), \quad k=0,1,2, \cdots$ and $\hat{W}_N=\frac{1}{N} \sum_{t=0}^N X(t)$ where N is an arbitrary positive integer. The aim of this paper is to derive largo deviation for the sample eovarianee function and thesample mean fuuetions.