Some prediction problems for stationary random fields with quarter-plane past

We study several nonstandard prediction problems where a number of observations are added to the quarter-plane past of a stationary random field. The goal is to provide informative and explicit prediction error variance formulas in terms of either the autoregressive or moving average parameters of the random fields. However, unlike the time series situation, the prediction error variances for random fields seem to be expressible only in terms of the moving average parameters, and attempts to express them formally in terms of the autoregressive parameters lead to a new and mysterious projection operator which captures the nature of the “edge-effects” encountered in the estimation of the spectral density function of stationary random fields. The approach leads to a number of technical issues and open problems.