# Induced stationary process and structure of locally square integrable periodically correlated processes

Studia Mathematica (1999)

- Volume: 136, Issue: 1, page 71-86
- ISSN: 0039-3223

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topMakagon, Andrzej. "Induced stationary process and structure of locally square integrable periodically correlated processes." Studia Mathematica 136.1 (1999): 71-86. <http://eudml.org/doc/216661>.

@article{Makagon1999,

abstract = {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 [3] 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.},

author = {Makagon, Andrzej},

journal = {Studia Mathematica},

keywords = {periodically correlated process; stationary process; imprimitivity theorem},

language = {eng},

number = {1},

pages = {71-86},

title = {Induced stationary process and structure of locally square integrable periodically correlated processes},

url = {http://eudml.org/doc/216661},

volume = {136},

year = {1999},

}

TY - JOUR

AU - Makagon, Andrzej

TI - Induced stationary process and structure of locally square integrable periodically correlated processes

JO - Studia Mathematica

PY - 1999

VL - 136

IS - 1

SP - 71

EP - 86

AB - 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 [3] 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.

LA - eng

KW - periodically correlated process; stationary process; imprimitivity theorem

UR - http://eudml.org/doc/216661

ER -

## References

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- [3] E. G. Gladyshev, Periodically and almost periodically correlated random processes with continuous time parameter, Theory Probab. Appl. 8 (1963), 173-177. Zbl0138.11003
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- [5] H. L. Hurd, Periodically correlated processes with discontinuous correlation functions, Theory Probab. Appl. 19 (1974), 804-808. Zbl0326.60064
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- [8] 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.
- [9] A. A. Kirillov, Elements of the Theory of Representations, Springer, 1976. Zbl0342.22001
- [10] 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. Zbl0791.60031
- [11] A. Makagon, A. G. Miamee and H. Salehi, Continuous time periodically correlated processes; spectrum and prediction, Stochastic Process. Appl. 49 (1994), 277-295. Zbl0791.60031
- [12] 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. Zbl0783.60039
- [13] A. G. Miamee, Explicit formula for the best linear predictor of periodically correlated sequences, SIAM J. Math. Anal. 24 (1993), 703-711. Zbl0776.60056

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