Journal of the Iranian Statistical Society

Convergence Rate of Empirical Autocovariance Operators in H-Valued Periodically Correlated Processes

Authors

1 Department of Statistics, University of Khansar, Khansar, IRAN.

2 Department of Statistics, Faculty of Science, Shiraz University, Shiraz, IRAN.

10.52547/jirss.19.2.1

Abstract

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.

Keywords

  1. Bennett, W.R. (1958), Statistics of regenerative digital transmission. Bell System Technical Journal, 37, 1501-1542. Chow, Y. S., and Teicher, H. (2012), Probability theory: independence, interchangeability, martingales. Springer Science & Business Media. Conway, J. B. (2000), A course in operator theory. American Mathematical Soc.. Gawarecki, L., Mandrekar, V. (2010), Stochastic di_erential equations in infinite dimensions: with applications to stochastic partial di_erential equations. Springer Science & Business Media. Gladyshev, E.G. (1961), Periodically correlated random sequences. Soviet Math. Dokl., 2, 385-388. Gladyshev, E.G. (1963). Periodically and almost periodically correlated random processes with continuous time parameter. Theory prob. appl., 8, 173-177. Gudzenko, L.I. (1959), Onperiodically nonstationary processes. Radiotekhnika elektronika, 4 (6), 1062-1064. Haghbin, H., Zamani, A., 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. Hashemi, M., Zamani, A., Haghbin, H. (2019), Rates of convergence of autocorrelation estimates for periodically correlated autoregressive Hilbertian processes. Statistics, 53 (2), 283-300. Hsing, T., Eubank, R. (2015), Theoretical foundations of functional data analysis, with an introduction to linear operators (Vol. 997). John Wiley & Sons. Kannan, D. (1972), An operator-valued stochastic integral (II). Ann. Inst. Poincare, Sect. B.8 9-32. Makagon, A. (1999), Theoretical prediction of periodically correlated sequences. Probability and Mathematical Statistics, 19 (2), 287-322. Makagon, A. (2011), Stationary sequences associated with a periodically correlated sequence. Probability and Mathematical Statistics,31 (2), 263-283. Makagon, A., and Miamee, A. G. (2013), Spectral representation of periodically correlated sequences. Probability and Mathematical Statistics, 33 (1), 175-188. Móricz, F. (1976), Moment inequalities and the strong laws of large numbers. Zeitschrift fúr Wahrscheinlichkeitstheorie und verwandte Gebiete, 35 (4), 299-314. Schatten, R. (2013), Norm ideals of completely continuous operators (Vol. 27). Springer-Verlag. Serpedin, E., Panduru, F. and Giannakis, G. (2005), Bibilography on cyclostationary. Signal processing, 85, 2232- 2303. 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. Soltani, A. R., Hashemi, M. (2011),Periodically correlated autoregressive Hilbertian processes. Statistical inference for stochastic processes, 14 (2), 177-188. 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. 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. Zamani, A., Sajjadnia, Z., Hashemi, M. (2019), The wold decomposition of hilbertian periodically correlated processes. Teoriya ˘Imovirnoste˘i ta Matematychna Statystyka, 101, 106-114. 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.
  2. 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]
  3. 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]
  4. Chow, Y. S., and Teicher, H. (2012), Probability theory: independence, interchangeability, martingales. Springer Science & Business Media.
  5. Chow, Y. S., and Teicher, H. (2012), Probability theory: independence, interchangeability, martingales. Springer Science & Business Media.
  6. Conway, J. B. (2000), A course in operator theory. American Mathematical Soc. [DOI:10.1090/gsm/021]
  7. Conway, J. B. (2000), A course in operator theory. American Mathematical Soc. [DOI:10.1090/gsm/021]
  8. 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]
  9. 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]
  10. Gladyshev, E. G. (1961), Periodically correlated random sequences. Soviet Math. Dokl., 2, 385-388.
  11. Gladyshev, E. G. (1961), Periodically correlated random sequences. Soviet Math. Dokl., 2, 385-388.
  12. 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]
  13. 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]
  14. Gudzenko, L. I. (1959), On periodically nonstationary processes. Radiotekhnika elektronika, 4 (6), 1062-1064.
  15. Gudzenko, L. I. (1959), On periodically nonstationary processes. Radiotekhnika elektronika, 4 (6), 1062-1064.
  16. 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]
  17. 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]
  18. 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]
  19. 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]
  20. 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]
  21. 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]
  22. Kannan, D. (1972), An operator-valued stochastic integral (II). Ann. Inst. Poincare, Sect. B. 8, 9-32.
  23. Kannan, D. (1972), An operator-valued stochastic integral (II). Ann. Inst. Poincare, Sect. B. 8, 9-32.
  24. Makagon, A. (1999), Theoretical prediction of periodically correlated sequences. Probability and Mathematical Statistics, 19 (2), 287-322.
  25. Makagon, A. (1999), Theoretical prediction of periodically correlated sequences. Probability and Mathematical Statistics, 19 (2), 287-322.
  26. Makagon, A. (2011), Stationary sequences associated with a periodically correlated sequence. Probability and Mathematical Statistics, 31 (2), 263-283.
  27. Makagon, A. (2011), Stationary sequences associated with a periodically correlated sequence. Probability and Mathematical Statistics, 31 (2), 263-283.
  28. Makagon, A., and Miamee, A. G. (2013), Spectral representation of periodically correlated sequences. Probability and Mathematical Statistics, 33 (1), 175-188.
  29. Makagon, A., and Miamee, A. G. (2013), Spectral representation of periodically correlated sequences. Probability and Mathematical Statistics, 33 (1), 175-188.
  30. 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]
  31. 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]
  32. Schatten, R. (2013), Norm ideals of completely continuous operators (Vol. 27). Springer-Verlag.
  33. Schatten, R. (2013), Norm ideals of completely continuous operators (Vol. 27). Springer-Verlag.
  34. Serpedin, E., Panduru, F. and Giannakis, G. (2005), Bibilography on cyclostationary. Signal processing, 85, 2232- 2303. [DOI:10.1016/j.sigpro.2005.05.002]
  35. Serpedin, E., Panduru, F. and Giannakis, G. (2005), Bibilography on cyclostationary. Signal processing, 85, 2232- 2303. [DOI:10.1016/j.sigpro.2005.05.002]
  36. 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]
  37. 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]
  38. Soltani, A. R., and Hashemi, M. (2011), Periodically correlated autoregressive Hilbertian processes. Statistical inference for stochastic processes, 14 (2), 177-188. 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. https://doi.org/10.1216/rmjm/1182536176 [DOI:10.1007/s11203-011-9056-0]
  39. 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]
  40. 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]
  41. 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]
  42. 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]
  43. 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]
  44. 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]
  45. 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]
  46. 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]
Journal of the Iranian Statistical Society
Volume 19, Issue 2
December 2020
Pages 1-13
  • Receive Date: 23 July 2022
  • Revise Date: 10 May 2024
  • Accept Date: 23 July 2022