We provide a characterisation of Gaussian time series which optimise the one-step prediction error subject to the covariance sequence being completely monotone with the first $m$ covariances specified.
"Optimising prediction error among completely monotone covariance sequences." Electron. Commun. Probab. 13 113 - 120, 2008. https://doi.org/10.1214/ECP.v13-1355