A comment to the paper of Approximate predictor and filter for partially observed vector ARMA processes
In 1985 a paper was presented (in the Computers & Mathematics with Applications, Volume 11, Issue 9, September 1985, Pages 943-947., by István, Bencsik) titled Approximate predictor and filter for partially observed vector ARMA processes, https://www.sciencedirect.com/science/article/pii/0898122185900987. The paper was reviewed by György Michaletzky.
Comment: Kalman-filters and ARMA predictors can be extended from the case of Gaussian noises with zero expected values to case of idendependent and identically distributed noises (https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables) with zero expected values
since the covariances can be any finite real values greater than zero and are conditionally independent of the expected values.
A guess: the extension may be possible for some other zero mean independent noises.
The main point of the paper was: An approximate L2-optimal predictor and filter was derived for partially observed vector autoregressive moving average processes of ARMA(2, 2) driven by white Gaussian noise. Some properties of the considered system—e.g. observability, controllability- were discussed.
The paper was about the state-space representations and vector difference equations relating to the input and output, widely used in time series modeling under the name ARMA models. In the case of k = 1 one can derive the Kalman-filter which is L2-optimal. The paper discussed an extension of the well known Kalman filter. In the higher order cases the structure of the L2 -optimal filter and predictor filter are the same, only the computation of exact parameters in the L2 -optimal case are complicated. Because of this fact the parameters are approximated, neglecting some terms connected with the problem of interpolation. In the case of ARMA (1, 1) processes the interpolation does not needed because of the order of the system.