Corrigendum to the article “Regular multidimensional stationary time series”

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 $\left\{X_t\right\}$ of rank $r$, $1\le r\le d$, there exists an analytic spectral factor $\Phi \left(z\right)$ of the form $\Phi \left(e^{-i\omega }\right)=\sqrt{2\pi }\tilde{U}\left(\omega \right){\Lambda }_r^{1/2}\left(\omega \right)$, where ${\Lambda }_r\left(\omega \right)$ is the $r\times r$ diagonal matrix of the positive eigenvalues of the spectral density matrix $f\left(\omega \right)$ and $\tilde{U}\left(\omega \right)$ is the $d\times r$ 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.