Volume 19, Issue 2 (12-2020)                   JIRSS 2020, 19(2): 1-13 | Back to browse issues page

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Hashemi M, Zamani A. Convergence Rate of Empirical Autocovariance Operators in H-Valued Periodically Correlated Processes. JIRSS. 2020; 19 (2) :1-13
URL: http://jirss.irstat.ir/article-1-619-en.html
Department of Statistics, Faculty of Science, Shiraz University, Shiraz, IRAN. , zamania@shirazu.ac.ir
Abstract:   (1260 Views)

This paper focuses on the empirical autocovariance operator of H-valued periodically correlated processes. It will be demonstrated that the empirical estimator converges to a limit with the same periodicity as the main process. Moreover, the rate of convergence of the empirical autocovariance operator in Hilbert-Schmidt norm is derived.

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Type of Study: Original Paper | Subject: 60Gxx: Stochastic processes
Received: 2019/10/23 | Accepted: 2021/01/28 | Published: 2020/12/11

1. Bennett, W. R. (1958), Statistics of regenerative digital transmission. Bell System Technical Journal, 37, 1501-1542. [DOI:10.1002/j.1538-7305.1958.tb01560.x]
2. Chow, Y. S., and Teicher, H. (2012), Probability theory: independence, interchangeability, martingales. Springer Science & Business Media.
3. Conway, J. B. (2000), A course in operator theory. American Mathematical Soc. [DOI:10.1090/gsm/021]
4. Gawarecki, L., and Mandrekar, V. (2010), Stochastic differential equations in infinite dimensions: with applications to stochastic partial differential equations. Springer Science & Business Media. [DOI:10.1007/978-3-642-16194-0]
5. Gladyshev, E. G. (1961), Periodically correlated random sequences. Soviet Math. Dokl., 2, 385-388.
6. Gladyshev, E. G. (1963). Periodically and almost periodically correlated random processes with continuous time parameter. Theory prob. appl., 8, 173-177. [DOI:10.1137/1108016]
7. Gudzenko, L. I. (1959), On periodically nonstationary processes. Radiotekhnika elektronika, 4 (6), 1062-1064.
8. Haghbin, H., Zamani, A., and Shishebor, Z. (2017), Inference on the asymptotic behavior of covariance operator of first-order periodically correlated autoregressive Hilbertian processes. Communications in Statistics-Theory and Methods, 46 (2), 761-769. [DOI:10.1080/03610926.2015.1005099]
9. Hashemi, M., Zamani, A., and Haghbin, H. (2019), Rates of convergence of autocorrelation estimates for periodically correlated autoregressive Hilbertian processes. Statistics, 53 (2), 283-300. [DOI:10.1080/02331888.2018.1547907]
10. Hsing, T., and Eubank, R. (2015), Theoretical foundations of functional data analysis, with an introduction to linear operators (Vol. 997). John Wiley & Sons. [DOI:10.1002/9781118762547]
11. Kannan, D. (1972), An operator-valued stochastic integral (II). Ann. Inst. Poincare, Sect. B. 8, 9-32.
12. Makagon, A. (1999), Theoretical prediction of periodically correlated sequences. Probability and Mathematical Statistics, 19 (2), 287-322.
13. Makagon, A. (2011), Stationary sequences associated with a periodically correlated sequence. Probability and Mathematical Statistics, 31 (2), 263-283.
14. Makagon, A., and Miamee, A. G. (2013), Spectral representation of periodically correlated sequences. Probability and Mathematical Statistics, 33 (1), 175-188.
15. Moricz, F. (1976), Moment inequalities and the strong laws of large numbers. Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete, 35 (4), 299-314. [DOI:10.1007/BF00532956]
16. Schatten, R. (2013), Norm ideals of completely continuous operators (Vol. 27). Springer-Verlag.
17. Serpedin, E., Panduru, F. and Giannakis, G. (2005), Bibilography on cyclostationary. Signal processing, 85, 2232- 2303. [DOI:10.1016/j.sigpro.2005.05.002]
18. Shishebor, Z., Soltani, A. R., and Zamani, A. (2011), Asymptotic distribution for periodograms of infinite dimensional discrete time periodically correlated processes. Journal of multivariate analysis, 102 (7), 1118-1125. [DOI:10.1016/j.jmva.2011.03.005]
19. Soltani, A. R., and Hashemi, M. (2011), Periodically correlated autoregressive Hilbertian processes. Statistical inference for stochastic processes, 14 (2), 177-188. [DOI:10.1007/s11203-011-9056-0]
20. Soltani, A. R., and Shishebor, Z. (2007), On infinite dimensional discrete time periodically correlated processes. The Rocky Mountain Journal of Mathematics, 37 (3), 1043-1058. [DOI:10.1216/rmjm/1182536176]
21. Soltani, A. R., Shishebor, Z., Zamani, A. (2010), Inference on periodograms of infinite dimensional discrete time periodically correlated processes. Journal of multivariate analysis, 101 (2), 368-373. [DOI:10.1016/j.jmva.2009.01.004]
22. Zamani, A., Sajjadnia, Z., Hashemi, M. (2020). The Wold decomposition of Hilbertian periodically correlated processes. Theory of Probability and Mathematical Statistics, 101, 119-127. [DOI:10.1090/tpms/1116]
23. Tian, C. J. (1988), A limiting property of sample autocovariances of periodically correlated processes with application to period determination. Journal of Time Series Analysis, 9 (4), 411-417. [DOI:10.1111/j.1467-9892.1988.tb00480.x]

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