Structure of PC Sequences and the 3rd Prediction Problem

  • Andrzej MakagonEmail author
  • Abolghassem Miamee
Part of the Lecture Notes in Mechanical Engineering book series (LNME)


Founders of prediction theory formulated three prediction problems: extrapolation problem, interpolation problem, and the problem of positivity of the angle between the past and the future. The third one is strictly related to the question of representing the predictor as a series of past values of the process. All three have been solved in the case of stationary sequences, however, as far as we know in the case of periodically correlated sequences only the first prediction problem has been studied. The purpose of this chapter is to overview the third prediction problems.



This chapter deals with the theory of PC sequences, which is a very small part of a broad and multidisciplinary area of analysis of periodically correlated processes and cyclostationary signals. No paper on PCs would be complete without mentioning names as Hurd, Yavorskij, Leśkow, Dehay, Neapolitano, Weron, Wylomanska, who have shaped the theory and practice of periodically correlated processes (cf. Broszkiewicz-Suwaj et al. 2004; Cambanis et al. 1994; Dehay 1994; Dragan and Yavorskii 1985; Hurd 1974, 1989, 1991; Hurd and Leskow 1992a, b; Hurd et al. 2002; Hurd and Miamee 2007; Javors’kyj et al. 2003, 2007, 2010; Lenart et al. 2008; Leskow and Weron 1992; Neapolitano 2012; Weron and Wylomanska 2004; Wylomanska 2008) and provided a motivation for our study. We are grateful to Professor Leśkow for organizing annual workshops which give this diverse community an opportunity to share their research, experience, and ideas.


  1. Babenko KI (1948) On conjugate functions. Dokl Akad Nauk SSSR 62:157–160zbMATHMathSciNetGoogle Scholar
  2. Broszkiewicz-Suwaj E, Makagon A, Weron R, Wylomanska A (2004) On detecting and modeling periodic correlation in financial data. Phys A 336(1–2):196–205CrossRefMathSciNetGoogle Scholar
  3. Cambanis S, Houdr C, Hurd H, Leskow J (1994) Laws of large numbers for periodically and almost periodically correlated processes. Stochastic Process Appl 53(1):37–54CrossRefzbMATHMathSciNetGoogle Scholar
  4. Dehay D (1994) Spectral analysis of the covariance of the almost periodically correlated processes. Stochastic Process Appl 50(2):315–330CrossRefzbMATHMathSciNetGoogle Scholar
  5. Dragan Ya P, Yavorskii IN (1985) Statistical analysis of periodic random processes (Russian). Otbor i Peredacha Informatsii 71:20–29zbMATHMathSciNetGoogle Scholar
  6. Folland GB (1995) A course in abstract harmonic analysis. Studies in advanced mathematics. CRC Press, Boca RatonGoogle Scholar
  7. Gladyshev EG (1961) Periodically correlated random sequences. Soviet Math 2:385–388zbMATHGoogle Scholar
  8. Heil C (2011) A basis theory primer. Birkhäuser, BostonGoogle Scholar
  9. Helson H (1964) Lectures on invariant subspaces. Academic Press, New YorkzbMATHGoogle Scholar
  10. Helson H, Szegö G (1960) A problem in prediction theory. Ann Mat Pura Appl 51(4):107–138CrossRefzbMATHMathSciNetGoogle Scholar
  11. Hurd HL (1974) Stationarizing properties of random shifts. SIAM J Appl Math 26:203–212CrossRefzbMATHMathSciNetGoogle Scholar
  12. Hurd HL (1989) Representation of strongly harmonizable periodically correlated processes and their covariances. J Multivariate Anal 29(1):53–67CrossRefzbMATHMathSciNetGoogle Scholar
  13. Hurd HL (1991) Correlation theory of almost periodically correlated processes. J Multivariate Anal 37(1):24–45Google Scholar
  14. Hurd HL, Leskow J (1992) Strongly consistent and asymptotically normal estimation of the covariance for almost periodically correlated processes. Statist Decisions 10(3):201–225zbMATHMathSciNetGoogle Scholar
  15. Hurd HL, Leskow J (1992) Estimation of the Fourier coefficient functions and their spectral densities for \(\alpha \)-mixing almost periodically correlated processes. Statist Probab Lett 14(4):299–306CrossRefzbMATHMathSciNetGoogle Scholar
  16. Hurd H, Makagon A, Miamee AG (2002) On AR(1) models with periodic and almost periodic coefficients. Stochastic Process Appl 100:167–185CrossRefzbMATHMathSciNetGoogle Scholar
  17. Hurd HL, Miamee AG (2007) Periodically correlated random sequences. In: Spectral theory and practice. Wiley Series in Probability and Statistics. Wiley-Interscience, New JerseyGoogle Scholar
  18. Javors’kyj I, Mykhailyshyn V, Zabolotnyj O (2003) Least squares method for statistical analysis of polyrhythmics. (English summary). Appl Math Lett 16(8):1217–1222CrossRefzbMATHMathSciNetGoogle Scholar
  19. Javors’kyj I, Isayev I, Zakrzewski Z, Brooks SP (2007) Coherent covariance analysis of periodically correlated random processes. Signal Process 87:13–32CrossRefzbMATHGoogle Scholar
  20. Javors’kyj I, Isayev I, Majewski J, Yuzefovych R (2010) Component covariance analysis for periodically correlated random processes. Signal Process 90:1083–1102CrossRefGoogle Scholar
  21. Lenart L, Leskow J, Synowiecki R (2008) Subsampling in testing autocovariance for periodically correlated time series. J Time Ser Anal 29(6):995–1018CrossRefzbMATHMathSciNetGoogle Scholar
  22. Leskow J, Weron A (1992) Ergodic behavior and estimation for periodically correlated processes. Statist Probab Lett 15(4):299–304CrossRefzbMATHMathSciNetGoogle Scholar
  23. Lindenstrauss J, Tzafiri L (1997) Classical Banach spaces I. Springer, BerlinGoogle Scholar
  24. Makagon A (1984) Interpolation error operator for Hilbert space valued stationary stochastic processes. Probab Math Statist 4(1):57–65zbMATHMathSciNetGoogle Scholar
  25. Makagon A (2011) Stationary sequences associated with a periodically correlated sequence. Probab Math Stat 31(2):263–283zbMATHMathSciNetGoogle Scholar
  26. Makagon A, Miamee AG (2013) Spectral representation of periodically correlated sequences. Probab Math Stat 33(1):175–188Google Scholar
  27. Makagon A, Salehi H (1989) Notes on infinite-dimensional stationary sequences. In: Probability theory on vector spaces, IV, Lecture Notes in Mathematics 1391, Springer, Berlin, pp 200–238Google Scholar
  28. Makagon A, Weron A (1976) q -variate minimal stationary processes. Studia Math 59(1):41–52Google Scholar
  29. Masani P (1960) The prediction theory of multivariate stochastic processes III. Unbounded spectral densities. Acta Math 104:141–162CrossRefzbMATHMathSciNetGoogle Scholar
  30. Masani P (1966) Recent trends in multivariate prediction theory. In: Krishnaiah PR (ed) Multivariate analysis, Proceedings of the International Symposium Dayton, Ohio 1965, Academic Press, New York, pp 351–382Google Scholar
  31. Miamee AG (1993) On basicity of exponentials in \(L^p (\mu )\) and general prediction problems. Period Math Hungar 26(2):115–124Google Scholar
  32. Miamee AG (1991) The inclusion \(L^p (\mu ) \subseteq L^q (\nu )\). Amer Math Monthly 98(4):342–345Google Scholar
  33. Neapolitano A (2012) Generalizations of cyclostationary signal processing. Spectral analysis and applications. IEEE Press, ChichesterGoogle Scholar
  34. Pourahmadi M (2001) Foundation of time series analysis and prediction theory. Wiley, New YorkGoogle Scholar
  35. Weron A, Wylomanska A (2004) On ARMA(1, q) models with bounded and periodically correlated solutions. Probab Math Statist 24(1), Acta Univ. Wratislav. No. 2646, 165–172Google Scholar
  36. Wiener N, Masani P (1957) The prediction theory of multivariate stochastic processes I. The regularity condition. Acta Math 98:111–150CrossRefzbMATHMathSciNetGoogle Scholar
  37. Wiener N, Masani P (1958) The prediction theory of multivariate stochastic processes II. The linear predictor. Acta Math 99:93–137CrossRefMathSciNetGoogle Scholar
  38. Wylomanska A (2008) Spectral measures of PARMA sequences. J Time Ser Anal 29(1):1–13zbMATHMathSciNetGoogle Scholar

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsHampton UniversityHamptonUSA

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