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 ≤ t ≤ −1, as n goes to infinity. This behavior depends on the regularity (deterministic or non-deterministic) of the process X(t). In his seminal article ‘Some purely deterministic processes’ (J. of Math. and Mech., 6(6), 801–10, 1957), Rosenblatt has described the asymptotic behavior of the prediction error for deterministic processes in the following two cases: (i) the spectral density f of X(t) is continuous and vanishes on an interval, (ii) the spectral density f has a very high order contact with zero. He showed that in the case (i) the prediction error behaves exponentially, while in the case (ii), it behaves like a power as $n\to \infty$. In this article, using an approach different from the one applied in Rosenblatt's article, we describe extensions of Rosenblatt's results to broader classes of spectral densities. Examples illustrate the obtained results.

中文翻译：

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Rosenblatt 关于确定性平稳序列预测误差渐近行为的结果的扩展

离散时间二阶平稳过程X ( t )预测理论的主要问题之一是在给定X ( t ) 的情况下，在预测X (0) 时描述最佳线性均方预测误差的渐近行为， − n  ≤  t  ≤ -1，因为n趋于无穷大。这种行为取决于过程X ( t )的规律性（确定性或非确定性）。在他的开创性文章“一些纯粹的确定性过程”（J. of Math. and Mech.，6（6），801-10，1957），布拉特已经描述了用于确定性过程的预测误差的渐近行为在以下两种情况：（i）所述频谱密度˚F的X（吨）是连续的并且消失上的间隔，（ii）谱密度f与零具有非常高的阶次接触。他表明，在情况 (i) 中，预测误差呈指数分布，而在情况 (ii) 中，它的行为类似于幂$n\to \infty$. 在本文中，我们使用与 Rosenblatt 文章中应用的方法不同的方法，将 Rosenblatt 的结果扩展到更广泛的光谱密度类别。示例说明了所获得的结果。