Extensions of Rosenblatt's results on the asymptotic behavior of the prediction error for deterministic stationary sequences

Abstract

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→∞. 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.

On Asymptotic Behavior of the Prediction Error for a Class of Deterministic Stationary Sequences

“…Using this result, Rosenblatt's Theorem 5.2 was extended in Babayan and Ginovyan [5] and in Babayan et al [7] to a class of spectral densities of the form f = f a g, where f a is as in (5.3) and g is a nonnegative function that can have arbitrary power type singularities (see Corollary 5.5 in Section 5.5.2).…”

“…Thus, the function f a in (5.3) has a very high order of contact with zero at points λ = 0, ±π, due to which the process with spectral density f a is deterministic and the prediction error σ 2 n (f a ) in (5.4) decreases to zero like a power as n → ∞. In the papers Babayan [2, 3], Babayan and Ginovyan [4,5,6], and Babayan et al [7], the above stated Theorems 5.1 and 5.2 were extended to broader classes of spectral densities.…”

“…The next result of Rakhmanov [54,55] shows that for (3.16) (or equivalently, for lim n→∞ σ n+1 (µ)/σ n (µ) = 1) it is enough only to have a.e. positiveness on T of the spectral density f (see also Babayan et al [7] and Simon [65], p. 5).…”

“…More precisely, the result that follows extends the asymptotic relation (5.2) to the case of several arcs, without having to stipulate continuity of the spectral density f . Theorem 5.3 (Babayan et al [7]). Let the support E f of the spectral density f of the process X(t) consists of a finite number of closed arcs of the unit circle T, and let f > 0 a.e.…”

“…Later the problems (a) and (b) were studied by Babayan [2,3], Babayan and Ginovyan [4,5,6], and Babayan et al [7] (see also Davisson [15] and Fortus [20]), where some generalizations and extensions of Rosenblatt's results have been obtained.…”