An extension problem for discrete-time periodically correlated stochastic processes (Q2722249)

A centered discrete-time stochastic process \(\{y(n)\}_{n\in {\mathbb Z}}\) is said to be periodically correlated (PC) with period \(T\in {\mathbb N}\) if for each \(n\) the sequence \(m\to E(y(n+m)y(m)^{*})\) is periodic with period \(T.\) In the context of wide-sense stationary processes the so-called Caratheodory-Fejer problem of extending a finite nonnegative sequence of matrices has been much studied. In this paper a similar extension problem for the wide-sense PC processes is investigated. Given the first \(N\) coefficients of \(T\) scalar-valued sequences it is studied under which condition(s) it is possible to find \(T\) extensions which are the cyclocorrelation sequences of a PC process with period \(T.\) Using a result of \textit{E. G. Gladyshev} [Sov. Math., Dokl. 2, 385-388 (1961); translation from Dokl. Akad. Nauk SSSR 137, 1026-1029 (1961; Zbl 0212.21401)] 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. \textit{H. Zhang} [IEEE Trans. Inf. Theory 43, No. 6, 2033-2035 (1997; Zbl 0952.94005)] solved a more general extension problem. But when applied to the present paper in particular context, the results are weaker than those shown in the paper. H.~Zhang supposed that the data of the extension problem are the \(T\) matrices \(\{E(y(m+k)y(m+l)^{*})\}_{(k,l)=(0,N_{m})}\) for \(m=0,\dots,T-1\) where, in contrast with the present paper, the indices \(N_{0},\dots,N_{T-1}\) do not necessarily coincide. H.~Zhang showed that if these matrices are strictly positive, then there exist PC extensions. In this case H.~Zhang also showed that the maximum entropy extension is a periodic autoregressive stochastic process. The approach is direct and elementary and does not provide any characterization of the set of all PC extensions. Therefore the results of the present paper are in some sense more complete.