An extension problem for discrete-time periodically correlated stochastic processes

D. Alpay, A. Chevreuil, P. Loubaton

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


In the context of wide-sense stationary processes, the so-called Carathéodory-Fejér 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 Gladyšev, the problem is shifted to a Carathéodory-Fejér 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.

Original languageEnglish
Pages (from-to)1-11
Number of pages11
JournalJournal of Time Series Analysis
Issue number1
StatePublished - 1 Jan 2001


  • Extension of a non-negative sequence
  • Matrix-valued Szegö polynomials
  • Periodically correlated processes

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Applied Mathematics


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