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An Extension Problem For Discrete‐Time Periodically Correlated Stochastic Processes

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  • D. Alpay
  • A. Chevreuil
  • Ph. Loubaton

Abstract

In the context of wide‐sense stationary processes, the so‐called Caratheodory–Fejer problem of extending a finite non‐negative sequence of matrices has been much studied. We here investigate a similar extension problem in the setting of wide‐sense periodically correlated processes: given the first N coefficients of T scalar‐valued sequences, we study under which condition(s) it is possible to find T extensions which are the cyclocorrelaion sequences of a periodically correlated process with period T. Using a result of Gladysev, the problem is shifted to a Caratheodory–Fejer problem with symmetry constraints. The existence of extensions is proved. In nondegenerate cases, the set of all solutions is given in terms of a homographic transformation of some Schur function G. The choice G=0 leads to the maximum entropy solution. The associated Gaussian processes are then proved to have a periodic autoregressive structure.

Suggested Citation

  • D. Alpay & A. Chevreuil & Ph. Loubaton, 2001. "An Extension Problem For Discrete‐Time Periodically Correlated Stochastic Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 22(1), pages 1-11, January.
    • Handle: RePEc:bla:jtsera:v:22:y:2001:i:1:p:1-11
    DOI: 10.1111/1467-9892.00209
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    Cited by:

    1. Sophie Lambert‐Lacroix, 2005. "Extension of Autocovariance Coefficients Sequence for Periodically Correlated Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(3), pages 423-435, May.

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