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Prediction of Stationary Processes

  • Peter J. Brockwell
  • Richard A. Davis
Part of the Springer Series in Statistics book series (SSS)

Abstract

In this chapter we investigate the problem of predicting the values {X t , tn + 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 of L 2(Ω, , P), the best predictor in of X n +h is defined to be the element of 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.

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Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Peter J. Brockwell
    • 1
  • Richard A. Davis
  1. 1.Department of StatisticsColorado State UniversityFort CollinsUSA

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