Induced stationary process and structure of locally square integrable periodically correlated processes
A one-to-one correspondence between locally square integrable periodically correlated (PC) processes and a certain class of infinite-dimensional stationary processes is obtained. The correspondence complements and clarifies Gladyshev's known result  describing the correlation function of a continuous periodically correlated process. In contrast to Gladyshev's paper, the procedure for explicit reconstruction of one process from the other is provided. A representation of a PC process as a unitary deformation of a periodic function is derived and is related to the correspondence mentioned above. Some consequences of this representation are discussed.
-  W. A. Gardner and L. E. Frank, Characterization of cyclostationary signal processes, IEEE Trans. Inform. Theory IT-21 (1975), 4-14.
-  E. G. Gladyshev, Periodically correlated random sequences, Soviet Math. Dokl. 2 (1961), 385-388.
-  E. G. Gladyshev, Periodically and almost periodically correlated random processes with continuous time parameter, Theory Probab. Appl. 8 (1963), 173-177.
-  J. Górniak, A. Makagon and A. Weron, An explicit form of dilation theorems for semispectral measures, in: Prediction Theory and Harmonic Analysis, The Pesi Masani Volume, V. Mandrekar and H. Salehi (eds.), North-Holland, 1983, 85-111.
-  H. L. Hurd, Periodically correlated processes with discontinuous correlation functions, Theory Probab. Appl. 19 (1974), 804-808.
-  H. L. Hurd, Stationarizing properties of random shift, SIAM J. Appl. Math. 26 (1974), 203-211.
-  H. L. Hurd, Representation of strongly harmonizable periodically correlated processes and their covariances, J. Multivariate Anal 29 (1989), 53-67.
-  H. L. Hurd and G. Kallianpur, Periodically correlated processes and their relationship to $L_1 (0,T)$-valued stationary processes, in: Nonstationary Stochastic Processes and Their Applications, A. G. Miamee (ed.), World Sci., 1991, 256-284.
-  A. A. Kirillov, Elements of the Theory of Representations, Springer, 1976.
-  A. Makagon, A. G. Miamee and H. Salehi, Periodically correlated processes and their spectrum, in: Nonstationary Stochastic Processes and Their Applications, A. G. Miamee (ed.), World Sci., 1991, 147-164.
-  A. Makagon, A. G. Miamee and H. Salehi, Continuous time periodically correlated processes; spectrum and prediction, Stochastic Process. Appl. 49 (1994), 277-295.
-  A. Makagon and H. Salehi, Structure of periodically distributed stochastic sequences, in: Stochastic Processes, A Festschrift in Honour of Gopinath Kallianpur, S. Cambanis et al. (eds.), Springer, 1993, 245-251.
-  A. G. Miamee, Explicit formula for the best linear predictor of periodically correlated sequences, SIAM J. Math. Anal. 24 (1993), 703-711.