Advances in Mathematical Economics Volume 20 pp 151-180

Part of the Advances in Mathematical Economics book series (MATHECON, volume 20) | Cite as

Fourier Analysis of Periodic Weakly Stationary Processes: A Note on Slutsky’s Observation

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Abstract

The periodic behavior of a specific weakly stationary stochastic process (w.s.p.) is examined from a viewpoint of classical Fourier analysis.(1) A w.s.p. has a spectral measure which is absolutely continuous with respect to the Lebesgue measure if and only if it is a moving average of a white noise. (2) A periodic or almost periodic w.s.p. must have a “discrete” spectral measure. Combining these two, we can conclude that any moving average of a white noise can neither be periodic nor almost periodic.However any w.s.p. can be approximated by a sequence of almost periodic w.s.p.’s in some specific sense.

Keywords

Weakly stationary process Periodicity Almost periodicity Spectral measure 

References

  1. 1.
    Billingsley P (1968) Convergence of probability measures. Wiley, New YorkMATHGoogle Scholar
  2. 2.
    Bochner S (1932) Vorlesungen über Fouriersche Integrale. Akademische Verlagsgesellschaft, LeipzigMATHGoogle Scholar
  3. 3.
    Bochner S (1933) Monotone Funktionen, Stieltjessche Integrale und harmonische Analyse. Math Ann 108:378–410MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bohr H (1932) Fastperiodische Funktionen. Springer, BerlinCrossRefMATHGoogle Scholar
  5. 5.
    Brémaud P (2014) Fourier analysis and stochastic processes. Springer, ChamMATHGoogle Scholar
  6. 6.
    Carleson L (1966) On convergence and growth of partial sums of Fourier series. Acta Math 116:135–157MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cramér H (1940) On the theory of stationary random processes. Ann Math 41:215–230MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Crum MM (1956) On positive-definite functions. Proc Lond Math Soc (3) 6:548–560Google Scholar
  9. 9.
    Doob JL (1953) Stochastic processes. Wiley, New YorkMATHGoogle Scholar
  10. 10.
    Doob JL (1949) Time series and harmonic analysis. In: Proceedings of Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley, pp 303–393Google Scholar
  11. 11.
    Frisch R (1933) Propagation problems and inpulse problems in dynamic economics. In: Economic essays in Honor of Gustav Cassel. Allen and Unwin, LondonGoogle Scholar
  12. 12.
    Granger CWJ, Newbold P (1986) Forecasting economic time series, 2nd edn. Academic, OrlandoMATHGoogle Scholar
  13. 13.
    Hamilton JD (1994) Time series analysis. Princeton University Press, PrincetonMATHGoogle Scholar
  14. 14.
    Herglotz G (1911) Über Potenzreihen mit positiven reellen Teil in Einheitskreis. Berichte Verh Säcks. Akad Wiss Leibzig Math Phys Kl 63:501–511Google Scholar
  15. 15.
    Itô K (1953) Probability theory. Iwanami, Tokyo (in Japanese)Google Scholar
  16. 16.
    Katznelson Y (2004) An introduction to harmonic analysis, 3rd edn. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  17. 17.
    Kawata T (1966) On the Fourier series of a stationary stochastic process I. Z Wahrsch Verw Gebiete 6:224–245MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kawata T (1969) On the Fourier series of a stationary stochastic process II. Z Wahrsch Verw Gebiete 13:25–38MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kawata T (1985) Stationary stochastic processes. Kyoritsu, Tokyo (in Japanese)MATHGoogle Scholar
  20. 20.
    Lax PD (2002) Functional analysis. Wiley, New YorkMATHGoogle Scholar
  21. 21.
    Loomis LH (1960) The spectral characterization of a class of almost periodic functions. Ann Math 72:362–368MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Malliavin P (1995) Integration and probability. Springer, New YorkCrossRefMATHGoogle Scholar
  23. 23.
    Maruyama G (1949) The harmonic analysis of stationary stochastic processes. Mem Fac Sci Kyushu Univ Ser A 4:45–106MathSciNetMATHGoogle Scholar
  24. 24.
    Naimark MA (1972) Normed algebras. Wolters Noordhoff, GroningenGoogle Scholar
  25. 25.
    Rudin W (1962) Fourier analysis on groups. Wiley, New YorkMATHGoogle Scholar
  26. 26.
    Sargent TJ (1979) Macroeconomic theory. Academic, New YorkMATHGoogle Scholar
  27. 27.
    Schwartz L (1966) Théorie des distributions. Hermann, ParisMATHGoogle Scholar
  28. 28.
    Slutsky E (1937) Alcune proposizioni sulla teoria delle funzioni aleatorie. Giorn Inst Ital degli Attuari 8:193–199Google Scholar
  29. 29.
    Slutsky E (1938) Sur les fonctions aléatoires presque périodiques et sur la décomposition des fonctions aléatoires stationnaires en composantes. Actualités Sci Ind 738:38–55MATHGoogle Scholar
  30. 30.
    Slutsky E (1937) The summation of random causes as the source of cyclic processes. Econometrica 5:105–146CrossRefGoogle Scholar
  31. 31.
    Wold H (1953) A study in the analysis of stationary time series, 2nd edn. Almquist and Wicksell, UppsalaMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Keio UniversityTokyoJapan

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