Maximum of entropy and extension of covariance matrices for periodically correlated and multivariate processes

A one-to-one relationship exists between scalar periodically correlated nonstationary processes and multivariate stationary processes. This fact allows us to transfer results proven for ones to the others. We are interested in a probabilistic approach of results sometimes already known in a different (analytical or numerical) context, in order to simplify, generalize and unify them. We use a probabilistic approach of generalized reflection coefficients to give a constructive condition of extension of partial covariance sequences, achieved by an autoregressive model. We develop a Trench-Zohar recursion for the nonstationary case which leads to an economical algorithm to solve the associated Yule-Walker equations. Shannon and Burg entropies are linked through a Szëgo type theorem. A numerical example is given.