System identification for stationary Gaussian processes includes an approximation problem. Currently, the subspace algorithm for this problem enjoys much attention. This algorithm is based on a transformation of a finite time series to canonical variable form followed by a truncation. There is no proof that this algorithm is the optimal solution to an approximation problem with a specific criterion. In this paper it is shown that the optimal solution to an approximation problem for Gaussian random variables with the divergence criterion is identical to the main step of the subspace algorithm. An approximation problem for stationary Gaussian processes with the divergence criterion is formulated.
Stoorvogel, A. A., & Schuppen, van, J. H. (1998). Approximation problems with the divergence criterion for Gaussian variables and Guassian processes. Systems and Control Letters, 35(4), 207-218. https://doi.org/10.1016/S0167-6911(98)00053-X