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

Andrzej Makagon

Studia Mathematica (1999)

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

Abstract

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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.

How to cite

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Makagon, 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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  2. [2] E. G. Gladyshev, Periodically correlated random sequences, Soviet Math. Dokl. 2 (1961), 385-388. Zbl0212.21401
  3. [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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  8. [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. [9] A. A. Kirillov, Elements of the Theory of Representations, Springer, 1976. Zbl0342.22001
  10. [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. [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. [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. [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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