An Extension Problem For Discrete‐Time Periodically Correlated Stochastic Processes

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.