Extensions of Rosenblatt's results on the asymptotic behavior of prediction error variance for deterministic stationary sequences
Ginovyan, Mamikon; Babayan, Nikolay; Taqqu, Murad
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 paper "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 deterministic processes in the
following two cases: (a) the spectral density f of X(t) is continuous and vanishes on an interval,
(b) the spectral density f has a very high order contact with zero. He showed that in the case
(a) the prediction error behaves exponentially, while in the case (b), it behaves like a power as
n ⟶ ∞. In this paper, using an approach different from the one applied in Rosenblatt's paper,
we describe extensions of Rosenblatt's results to broader classes of spectral densities. Examples
illustrate the obtained results.
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