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Theory of Probability and Mathematical Statistics
(Teoriya Imovirnostei ta Matematychna Statystyka)



The Wold decomposition of Hilbertian periodically correlated processes

A. Zamani, Z. Sajjadnia, M. Hashemi

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Abstract: l{L}}$-Closed Subspaces, Moving Average Representation,

Keywords: $H$-Valued Random Variables,

Bibliography:
Correlated Processes, Wold Decomposition.
Abstract: The Wold decomposition of stationary processes is widely applied in
time series prediction and provides interesting
insights into the structure of stationary stochastic processes. In
1971, Kallianpur and Mandrekar, using the notion of resolution of
identity and unitary operators, presented the Wold decomposition
for weakly stationary stochastic processes with values in infinite
dimensional separable Hilbert spaces. This paper aims to
expand the idea of Wold decomposition to Hilbertian periodically
correlated processes, applying the concept of ${\mathcal{L}}$-closed
subspaces.
Bibliography: 1. A. T. Bharucha-Reid, Random integral equations, Academic Press, Inc., 1972.
2. D. Bosq, Linear Processes in Function Spaces: Theory and Applications, Springer, Berlin, 2000.
3. D. Bosq, General linear processes in Hilbert spaces and prediction, Journal of Statistical Planning and Inference, 137 (2007), no. 3, 879–894.
4. P. J. Brockwell, R. A. Davis, Time Series: Theory and Methods, Springer Science & Business Media, New York, 1991.
5. E. G. Gladyshev, Periodically correlated random sequences, Sow. Math., 2 (1961), 385–388.
6. H. L. Hurd, A. Miamee, Periodically Correlated Random Sequences Spectral Theory and Practice, John Wiley & Sons, Inc., 2007.
7. G. Kallianpur, V. Mandrekar, Spectral theory of stationary H-valued processes, Journal of Multivariate Analysis, 1 (1971), no. 1, 1–16.
8. A. N. Kolomogorov, Stationary sequences in Hilbert space, Bull. Moscow State Univ., 2 (1941), 1–40.
9. A. Makagon, Theoretical prediction of periodically correlated sequences, Probability and Mathematical Statistics – Wroclaw University, 19 (1999), 287–322.
10. A. G. Miamee, H. Salehi, On the prediction of periodically correlated stochastic processes, In Multivariate Analysis V (P. R. Krishnaiah, Ed.), pp. 167–179. North-Holland, Amsterdam, 1980.
11. M. Pagano, On periodic and multiple autoregressions, The Annals of Statistics, 6 (1978), no. 6, 1310–1317.
12. M. Pourahmadi, Foundations of Time Series Analysis and Prediction Theory, John Wiley & Sons, 2001.
13. M. Pourahmadi, H. Salehi, On subordination and linear transformation of harmonizable and periodically correlated processes. In Probability Theory on Vector Spaces III (pp. 195–213). Springer, Berlin, Heidelberg, 1984.
14. P. Rothman (Ed.), Nonlinear Time Series Analysis of Economic and Financial Data (Vol. 1). Springer Science & Business Media, New York, 2012.
15. Y. A. Rozanov, Stationary random processes, Holden Day, 1967.
16. R. Schatten, Norm ideals of completely continuous operators, Springer–Verlag, 2013.
17. A. R. Soltani, M. Hashemi, Periodically correlated autoregressive Hilbertian processes, Statistical inference for stochastic processes, 14 (2011), no. 2, 177–188.
18. N. Vakhania, V. Tarieladze, S. Chobanyan, Probability distributions on Banach spaces, Springer Science & Business Media, 1987.
19. H. Wold, Study in the analysis of stationary time series, Almqvist and Wiksell, 1954.