Prediction of Stationary Processes

Abstract

In this chapter we investigate the problem of predicting the values {X _t , t ≥ n + 1} of astationary process in terms of {X ₁,..., X _n }. Theideais to utilize observations taken at or before time n to forecast the subsequent behaviour of {X _t }. Given any closed subspace M of L ²(Ω, F, P), the best predictor in M of X _n+h is defined to be the element of M with minimum mean-square distance from X _n+h This of course is not the only possible definition of “best”, but for processes with finite second moments it leads to a theory of prediction which is simple, elegant and useful in practice. (In Chapter 12 we shall introduce alternative criteria which are needed for the prediction of processes with infinite second-order moments.)