×

Induced stationary process and structure of locally square integrable periodically correlated processes. (English) Zbl 0947.60032

The author studies periodically correlated (PC) processes with period \(T>0\) defined as functions \(x : {\mathbb R} \to {\mathcal H}\), where \({\mathcal H}\) is a complex Hilbert space, which are Bochner square integrable over each compact interval and such that for every \(t \in {\mathbb R}\) \[ (x(t+u),x(u)) = (x(t+T+u),x(T+u)) \qquad du\text{-a.e.} \] The main result is the correspondence theorem between PC processes and a certain class of infinite-dimensional stationary processes; a constructive procedure to recover a PC process from an associated infinite-dimensional process is provided. A representation of a PC process as a unitary deformation of a periodic function is derived and is connected with the correspondence theorem. These results clarify and complement those of E. G. Gladyshev [Sov. Math., Dokl. 2, 385-388 (1961); translation from Dokl. Akad. Nauk SSSR 137, 1026-1029 (1961; Zbl 0212.21401) and Theory Probab. Appl. 8, 173-177 (1963); translation from Teor. Veroyatn. Primen. 8, 184-189 (1963; Zbl 0138.11003)]. A link between the correspondence theorem and Mackey’s imprimitivity theorem is discussed.

MSC:

60G12 General second-order stochastic processes
60G25 Prediction theory (aspects of stochastic processes)
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
60G10 Stationary stochastic processes
PDFBibTeX XMLCite
Full Text: DOI EuDML