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Correlation theory of almost periodically correlated processes

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  • Hurd, Harry L.

Abstract

A second-order stochastic process X is called almost periodically correlated (PC) in the sense of Gladyshev if its mean function m(t) and covariance R(t + [tau], t) are uniformly continuous with respect to t, [tau] and are almosst periodic functions of t for every [tau]. We show that the mean uniformly almost periodic processes discussed by Kawata are also almost PC in the sense of Gladyshev. If X is almost PC, then for each fixed [tau] the function R(t + [tau], t) has the Fourier series and exists for every [lambda] and [tau], independently of the constant c. Assuming only that a([lambda], [tau]) exists in this sense for every [lambda] and [tau], we show a([lambda], [tau]) is a Fourier transform a([lambda] [tau]) = [integral operator]Rexp(iy[tau]) rlambda;(dy) if and only if a(0, [tau]) is continuous at [tau] = 0; under this same condition, the set [Lambda] = {[lambda]: a([lambda], [tau]) [not equal to] 0 for some [tau]} is countable. We show that a strongly harmonizable process is almost PC if and only if its spectral measure is concentrated on a countable set of diagonal lines S[lambda] = {([gamma]1, [gamma]2): [gamma]2 = [gamma]1 - [lambda]}; further, one may identify the spectral measure on the k th line with the measure r[lambda]k appearing in (iii). Finally we observe that almost PC processes are asymptotically stationary and give conditions under which strongly harmonizable almost PC processes may be made stationary by an independent random time shift.

Suggested Citation

  • Hurd, Harry L., 1991. "Correlation theory of almost periodically correlated processes," Journal of Multivariate Analysis, Elsevier, vol. 37(1), pages 24-45, April.
  • Handle: RePEc:eee:jmvana:v:37:y:1991:i:1:p:24-45
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    Citations

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    Cited by:

    1. Łukasz Lenart & Błażej Mazur, 2016. "On Bayesian Inference for Almost Periodic in Mean Autoregressive Models," FindEcon Chapters: Forecasting Financial Markets and Economic Decision-Making, in: Magdalena Osińska (ed.), Statistical Review, vol. 63, 2016, 3, edition 1, volume 63, chapter 1, pages 255-272, University of Lodz.
    2. Makagon, A. & Miamee, A. G., 1997. "On the spectrum of correlation autoregressive sequences," Stochastic Processes and their Applications, Elsevier, vol. 69(2), pages 179-193, September.
    3. Łukasz Lenart & Mateusz Pipień, 2017. "Non-Parametric Test for the Existence of the Common Deterministic Cycle: The Case of the Selected European Countries," Central European Journal of Economic Modelling and Econometrics, Central European Journal of Economic Modelling and Econometrics, vol. 9(3), pages 201-241, September.
    4. Averkamp, Roland, 1997. "Conditions for the completeness of the spectral domain of a harmonizable process," Stochastic Processes and their Applications, Elsevier, vol. 72(1), pages 1-9, December.
    5. Łukasz Lenart, 2016. "Generalized Resampling Scheme With Application to Spectral Density Matrix in Almost Periodically Correlated Class of Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(3), pages 369-404, May.
    6. Giuseppe Cavaliere & Dimitris N. Politis & Anders Rahbek & Dominique Dehay & Anna E. Dudek, 2015. "Recent developments in bootstrap methods for dependent data," Journal of Time Series Analysis, Wiley Blackwell, vol. 36(3), pages 327-351, May.
    7. Leonid G. Hanin & Bertram M. Schreiber, 1998. "Discrete Spectrum of Nonstationary Stochastic Processes on Groups," Journal of Theoretical Probability, Springer, vol. 11(4), pages 1111-1133, October.
    8. Lenart, Łukasz, 2013. "Non-parametric frequency identification and estimation in mean function for almost periodically correlated time series," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 252-269.
    9. ŁUkasz Lenart & Jacek Leśkow & Rafał Synowiecki, 2008. "Subsampling in testing autocovariance for periodically correlated time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(6), pages 995-1018, November.
    10. Łukasz Lenart & Mateusz Pipień, 2015. "Empirical Properties of the Credit and Equity Cycle within Almost Periodically Correlated Stochastic Processes - the Case of Poland, UK and USA," Central European Journal of Economic Modelling and Econometrics, Central European Journal of Economic Modelling and Econometrics, vol. 7(3), pages 169-186, September.

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