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The periodogram at the Fourier frequencies

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  • Kokoszka, Piotr
  • Mikosch, Thomas

Abstract

In the time series literature one can often find the claim that the periodogram ordinates of an iid sequence at the Fourier frequencies behave like an iid standard exponential sequence. We review some results about functions of these periodogram ordinates, including the convergence of extremes, point processes, the empirical distribution function and the empirical process. We show when the analogy with an iid exponential sequence is valid and study situations when it fails. Periodogram ordinates of an infinite variance iid sequence are also considered.

Suggested Citation

  • Kokoszka, Piotr & Mikosch, Thomas, 2000. "The periodogram at the Fourier frequencies," Stochastic Processes and their Applications, Elsevier, vol. 86(1), pages 49-79, March.
    • Handle: RePEc:eee:spapps:v:86:y:2000:i:1:p:49-79
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    References listed on IDEAS

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    1. Dahlhaus, Rainer, 1988. "Empirical spectral processes and their applications to time series analysis," Stochastic Processes and their Applications, Elsevier, vol. 30(1), pages 69-83, November.
    2. Knight, Keith, 1991. "On the empirical measure of the Fourier coefficients with infinite variance data," Statistics & Probability Letters, Elsevier, vol. 12(2), pages 109-117, August.
    3. Chen Zhao‐Guo & E. J. Hannan, 1980. "The Distribution Of Periodogram Ordinates," Journal of Time Series Analysis, Wiley Blackwell, vol. 1(1), pages 73-82, January.
    4. Kokoszka, P. & Mikosch, T., 1997. "The integrated periodogram for long-memory processes with finite or infinite variance," Stochastic Processes and their Applications, Elsevier, vol. 66(1), pages 55-78, February.
    5. Mikosch, T. & Norvaisa, R., 1997. "Uniform convergence of the empirical spectral distribution function," Stochastic Processes and their Applications, Elsevier, vol. 70(1), pages 85-114, October.
    6. Klüppelberg, Claudia & Mikosch, Thomas, 1993. "Spectral estimates and stable processes," Stochastic Processes and their Applications, Elsevier, vol. 47(2), pages 323-344, September.
    7. An, Hong-Zhi & Chen, Zhao-Guo & Hannan, E. J., 1983. "The maximum of the periodogram," Journal of Multivariate Analysis, Elsevier, vol. 13(3), pages 383-400, September.
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    Cited by:

    1. Laura Coroneo & Fabrizio Iacone, 2015. "Comparing predictive accuracy in small samples," Discussion Papers 15/15, Department of Economics, University of York.
    2. Cerovecki, Clément & Hörmann, Siegfried, 2017. "On the CLT for discrete Fourier transforms of functional time series," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 282-295.
    3. Fasen-Hartmann, Vicky & Mayer, Celeste, 2023. "Empirical spectral processes for stationary state space models," Stochastic Processes and their Applications, Elsevier, vol. 155(C), pages 319-354.
    4. Fay, Gilles & Soulier, Philippe, 2001. "The periodogram of an i.i.d. sequence," Stochastic Processes and their Applications, Elsevier, vol. 92(2), pages 315-343, April.

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