Regularity Conditions for Stationary Processes

Abstract

In this chapter we characterize different regularity conditions introduced in the previous chapter in spectral terms. In §1 we characterize the minimal stationary processes and find the spectral density of the interpolation error process in terms of the spectral density of the initial process. In §2 we consider the angles between the past and the future of a stationary process. We characterize the processes with nonzero angles between the past and the future. In the next section we consider various regularity conditions for stationary processes (such as complete regularity, complete regularity of order a, p-regularity, etc.) and we characterize such regularity conditions in spectral terms. Note that the original proofs of these results were quite different for different regularity conditions; some proofs were quite complicated. In Peller and Khrushch6v [1] a single approach to all regularity conditions was found. This approach is based on Hankel operators and the results on best approximation given in Chapter 7 and it simplifies the original proofs. Finally, in §4 we consider several stronger regularity conditions and we also characterize them in spectral terms.