One of the main problem in prediction theory of 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.,} {\bf 6}(6), 801-810, 1957), for a specific spectral density that has a very high order contact with zero M. Rosenblatt showed that the prediction error behaves like a power as $n\to\f$. In the paper Babayan et al. {\it 'Extensions of Rosenblatt's results on the asymptotic behavior of the prediction error for deterministic stationary sequences' (J. Time Ser. Anal.} {\bf 42}, 622-652, 2021), Rosenblatt's result was extended to the class of spectral densities of the form $f=f_dg$, where $f_d$ is the spectral density of a deterministic process that has a very high order contact with zero, while $g$ is a function that can have polynomial type singularities. In this paper, we describe new extensions of the above quoted results in the case where the function $g$ can have {\it arbitrary power type singularities}. Examples illustrate the obtained results.
翻译:固定过程预测理论的主要问题之一 $X(t) 是描述在预测X(0)$X(t),$-n\le t\le-1美元(美元到无限度)方面, 最佳线性平均平方预测错误在预测美元X(0美元), 美元美元- n\le t\le-1美元(美元到无限度) 时的无症状行为。 该行为取决于该过程的规律性( 确定性或非确定性) $X(t) 。 在他的原始论文中, “ 某些纯粹的确定性过程”(数学和梅奇的J.} $bf(6) ; 6, 801-810, 1957) 最佳线性平均平方预测错误的行为。 对于特定光谱密度与0 M(t), 美元(t), 美元-n=lexblatt(美元) 美元- 美元(xl) 美元(m) 直径直径直径(xxxxxxxxx) 直径(ral) 直径(ral) ral_x) rexal_xxxxx(ral_xx) ral_xx) ral_xx) ral_xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx。