Prediction of Stationary Processes

Abstract

In this chapter we investigate the problem of predicting the values {X _t , t ≥ n + 1} of a stationary process in terms of {X ₁,..., X _n }. The idea is 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 13 we shall introduce alternative criteria which are needed for the prediction of processes with infinite second-order moments.) In Section 2.7, we showed that the projections are respectively the best function of X ₁,..., X _n and the best linear combination of 1, X ₁,..., X _n for predicting X _n+h . For the reasons given in Section 2.7 we shall concentrate almost exclusively on predictors of the latter type (best linear predictors) instead of attempting to work with conditional expectations.