The problem of obtaining all stochastic realizations of a discrete-time non stalionary stochastic process defined on the set of integers is considered. The structure of the set of all covariances of the states of such realizations is studied. Special attention is paid to the minimum and maximum elements of this set. Finally, a non-Riccati-type algorithm—which does not require solution of a matrix Riccati equation—to generate (wide-sense) stochastic realizations is presented.