论文标题
Rosenblatt关于确定性固定序列的预测误差渐近行为的结果的扩展
Extensions of Rosenblatt's results on the asymptotic behavior of the prediction error for deterministic stationary sequences
论文作者
论文摘要
离散时间二阶固定过程$ x（t）$的预测理论的主要问题之一是将最佳线性平均值预测错误的渐近行为描述在预测$ x（0）$给定$ x（t），$ x（t），$ -n \ le t \ le-t \ le-1 $中，为$ n $。此行为取决于该过程$ x（t）$的规律性（确定性或非确定性）。在他的开创性论文{\ IT“一些纯粹确定性的过程”（J. of Math。和Mech。，} 6（6），801-810，1957）中，M。Rosenblatt描述了在以下两种情况下离散时间确定性过程的预测错误的渐近行为：频谱密度$ f（λ）$与零具有很高的接触。他表明，在情况下（a）预测误差差异呈指数式行为，而在（b）的情况下，它的行为为$ n \ to \ infty $。在本文中，使用一种新方法，我们将Rosenblatt结果扩展到更广泛的光谱密度类别。示例说明了获得的结果。
One of the main problem in prediction theory of discrete-time second-order stationary processes $X(t)$ is to describe the asymptotic behavior of the best linear mean squared prediction error in predicting $X(0)$ given $ X(t),$ $-n\le t\le-1$, as $n$ goes to infinity. This behavior depends on the regularity (deterministic or non-deterministic) of the process $X(t)$. In his seminal paper {\it "Some purely deterministic processes" (J. of Math. and Mech.,} 6(6), 801-810, 1957), M. Rosenblatt has described the asymptotic behavior of the prediction error for discrete-time deterministic processes in the following two cases: (a) the spectral density $f(λ)$ of $X(t)$ is continuous and vanishes on an interval, (b) the spectral density $f(λ)$ has a very high order contact with zero. He showed that in the case (a) the prediction error variance behaves exponentially, while in the case (b), it behaves hyperbolically as $n\to\infty$. In this paper, using a new approach, we describe extensions of Rosenblatt's results to broader classes of spectral densities. Examples illustrate the obtained results.
