Extrapolation and moving average representation for stationary random fields and Beurling's theorem (Q792696)

This article deals with the extrapolation problems for a discrete parameter stationary process X(t), \(t\in Z^ n\). The main aim in the article is to employ the function theory in polydiscs in order to develop the theory of extrapolation for random fields. Different notions of regularity are introduced and necessary and sufficient conditions for strong regularity are given. An extension of the classical Beurling's theorem to functions of several variables is given. With the help of this extension necessary and sufficient conditions for the moving average representation of stationary random fields are obtained. Prediction problems are also discussed and a recipe formula for the best linear extrapolator is also given.