REGULAR MULTIDIMENSIONALSTATIONARY TIME SERIES (vol 43, pg 263, 2022)

In Theorem 2.1 which was the main result of the article it was implicitly assumed that for any regular d-dimensional weakly stationary time series {X-t} of rank r, 1 <= r <= d, there exists an analytic spectral factor Phi(z) of the form Phi(e(-i omega)) = root 2 pi U(omega) Lambda(1/2)(r) (omega), where Lambda(r)(omega) is the r x r diagonal matrix of the positive eigenvalues of the spectral density matrix f(omega) and U(omega) is the d x r sub-unitary matrix of the corresponding eigenvectors. In fact, to the best of my knowledge, it is not known if this assumption is true or false.