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.
- 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