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Arranging a periodically correlated sequence (PC) with period T into blocks of length T generates a T-dimensional stationary sequence. In this paper we discuss two other transformations that map PC sequences into T-dimensional stationary sequences and study their properties. We also indicate possible applications of these mappings in the theory of PC processes and, in particular, for study of PARMA systems. The presented construction is both a simplification and enhancement of the construction given in [20].
Słowa kluczowe
Czasopismo
Rocznik
Tom
Strony
263--283
Opis fizyczny
Bibliogr. 38 poz.
Twórcy
autor
- Department of Mathematics, Hampton University, Queen and Tyler Street, Hampton, VA 23668, USA
Bibliografia
- [1] P. L. Anderson and M. M. Meerschaert, Parameter estimation for periodically stationary time series, J. Time Ser. Anal. 26 (4) (2005), pp. 489-518.
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- [7] D. Dehay and D. H. L. Hurd, Spectral estimation for strongly periodically correlated random fields defined on R2, Math. Methods Statist. 11 (2) (2002), pp. 135-151.
- [8] E. G. Gladyshev, Periodically correlated random sequences, Soviet Math. 2 (1961), pp. 385-388.
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- [10] E. J. Hannan, Multiple Time Series, Wiley, 1970.
- [11] H. L. Hurd, Stationarizing properties of random shifts, SIAM J. Appl. Math. 26 (1) (1974), pp. 203-211.
- [12] H. L. Hurd, Representation of strongly harmonizable periodically correlated processes and their covariance, J. Multivariate Anal. 29 (1989), pp. 53-67.
- [13] H. L. Hurd, Almost periodically unitary stochastic processes, Stochastic Process. Appl. 43 (1) (1992), pp. 99-113.
- [14] H. L. Hurd, G. Kallianpur and J. Farshidi, Correlation and spectral theory for periodically correlated random fields indexed on Z2, J. Multivariate Anal. 90 (2) (2004), pp. 359-383.
- [15] H. L. Hurd, A. Makagon and A. G. Miamee, On AR(1) models with periodic and almost periodic coefficients, Stochastic Process. Appl. 100 (2002), pp. 167-185.
- [16] H. L. Hurd and A. Miamee, Periodically Correlated Random Sequences. Spectral Theory and Practice, Wiley Ser. Probab. Stat., New York 2007.
- [17] J. Leśkow and A. Weron, Ergodic behavior and estimation for periodically correlated processes, Statist. Probab. Lett. 15 (1992), pp. 299-304.
- [18] R. B. Lund and I. V. Basawa, Recursive prediction and likelihood evaluation for periodic ARMA models, J. Time Ser. Anal. 21 (1) (2000), pp. 75-93.
- [19] A. Makagon, Induced stationary process and structure of locally square integrable periodically correlated processes, Studia Math. 136 (1) (1999), pp. 71-86.
- [20] A. Makagon, Theoretical prediction of periodically correlated sequences, Probab. Math. Statist. 19 (2) (1999), pp. 287-322.
- [21] A. Makagon, Characterization of the spectra of periodically correlated processes, J. Multivariate Anal. 78 (1) (2001), pp. 1-10.
- [22] A. Makagon, On a stationary process induced by an almost periodically correlated process, Demonstratio Math. 34 (2) (2001), pp. 321-326.
- [23] A. Makagon, An alternative approach to analysis of PARMA models, in: Proceedings of the 3rd Iranian Seminar on Probability and Stochasic Processes, Isfahan-Khansar, August 2001, pp. 26-37.
- [24] 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.), Word Scientific, 1991, pp. 147-164.
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- [29] A. G. Miamee, Explicit formula for the best linear predictor of periodically correlated sequences, SIAM J. Math. Anal. 24 (1993), pp. 703-711.
- [30] A. G. Miamee and H. Salehi, On the prediction of periodically correlated stochastic processes, in: Multivariate Analysis V, R. Krishnaiah (Ed.), North Holland, Amsterdam 1980, pp. 167-179.
- [31] A. G. Miamee and G. Shahkar, Shift operator for periodically correlated processes, Indian J. Pure Appl. Math. 33 (5) (2002), pp. 705-712.
- [32] M. Pagano, On periodic and multiple autoregression, Ann. Statist. 6 (1978), pp. 1310-1317.
- [33] H. Sakai, On the spectral density matrix of a periodic ARMA process, J. Time Ser. Anal. 12 (1991), pp. 72-82.
- [34] A. R. Soltani and A. Parvardeh, Decomposition of discrete time periodically correlated and multivariate stationary symmetric stable processes, Stochastic Process. Appl. 115 (11) (2005), pp. 1838-1859.
- [35] A. R. Soltani and Z. Shishebor, On infinite dimensional discrete time periodically correlated processes, Rocky Mountain J. Math. 37 (3) (2007), pp. 1043-1058.
- [36] A. R. Soltani, Z. Shishebor and A. Zamani, Inference on periodograms of infinite dimensional discrete time periodically correlated processes, J. Multivariate Anal. 101 (2) (2010), pp. 368-373.
- [37] A. V. Vecchia, Periodic Autoregressive Moving Average (PARMA) modeling with applications to water resources, Water Resources Bulletin 21 (5) (1985), pp. 721-730.
- [38] A. Weron and A. Wyłomańska, On ARMA(1; q) models with bounded and periodically correlated solutions, Probab. Math. Statist. 24 (1) (2004), pp. 165-172.
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Bibliografia
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