Makagon, Andrzej Induced stationary process and structure of locally square integrable periodically correlated processes. (English) Zbl 0947.60032 Stud. Math. 136, No. 1, 71-86 (1999). 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. Reviewer: Alexander Gushchin (Moskva) Cited in 2 ReviewsCited in 5 Documents 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 Keywords:periodically correlated process; stationary process; imprimitivity theorem Citations:Zbl 0212.21401; Zbl 0138.11003 PDFBibTeX XMLCite \textit{A. Makagon}, Stud. Math. 136, No. 1, 71--86 (1999; Zbl 0947.60032) Full Text: DOI EuDML