Prediction of stationary Gaussian random fields with incomplete quarterplane past

Let {Xm,n:(m,n)∈Z2} be a stationary Gaussian random field. Consider the problem of predicting X0,0X0,0 based on the quarterplane Q={(m,n):m≥0,n≥0}∖{(0,0)}Q={(m,n):m≥0,n≥0}∖{(0,0)}, but with finitely many observations missing. Two solutions are presented. The first solution expresses the best predictor in terms of the moving average coefficients of {Xm,n}{Xm,n}, under the assumption that the spectral density function has a strongly outer factorization. The second solution expresses the prediction error variance in terms of the autoregressive coefficients of {Xm,n}{Xm,n}; it requires the reciprocal of the density function to have a strongly outer factorization, and relies on a modified duality argument. These solutions are extended by allowing the quarterplane past to be replaced with a much broader class of parameter sets. This enables the solution, for example, of the quarterplane interpolation problem.