An extension problem for discrete-time periodically correlated stochastic processes

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 Gladygev, 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.