2020/05/30 by Nikolay M. Babayan, Babayan, Nikolay M., Mamikon S. Ginovyan +3
Engineering · Decision Sciences · #Control Systems and Identification #Probabilistic and Robust Engineering Design #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2006.00430
One of the main problem in prediction theory of discrete-time second-order\nstationary processes X(t) is to describe the asymptotic behavior of the best\nlinear mean squared prediction error in predicting X(0) given X(t), -n\≤\nt\≤-1, as n goes to infinity. This behavior depends on the regularity\n(deterministic or non-deterministic) of the process X(t). In his seminal\npaper it "Some purely deterministic processes" (J. of Math. and Mech., 6(6),\n801-810, 1957), M. Rosenblatt has described the asymptotic behavior of the\nprediction error for discrete-time deterministic processes in the following two\ncases: (a) the spectral density f(\λ) of X(t) is continuous and\nvanishes on an interval, (b) the spectral density f(\λ) has a very high\norder contact with zero. He showed that in the case (a) the prediction error\nvariance behaves exponentially, while in the case (b), it behaves\nhyperbolically as n\→\∞. In this paper, using a new approach, we\ndescribe extensions of Rosenblatt's results to broader classes of spectral\ndensities. Examples illustrate the obtained results.\n