Pourahmadi, Mohsen

**Foundations of time series analysis and prediction theory.**
(English. English summary)

Wiley Series in Probability and Statistics: Applied Probability and Statistics. Wiley-Interscience, New York, 2001. xviii+414 pp. ISBN 0-471-39434-3, 62M10

The book presents a mathematical foundation for time series analysis and prediction theory. The author's approach is based on regression ideas and on the Hilbert space approach. The material is divided into ten chapters. After a short introduction the author gives an overview of time series analysis. About eight key mathematical results are presented and explained and their role in statistical time series analysis is illustrated. Then the models for "many short series" arising in longitudinal studies are described. This part is based mainly on papers published recently and it is really nice that information about new methods in this field is collected in a chapter of the book. Stationary ARMA models are introduced via solutions of stochastic difference equations without using the traditional backward shift operator. Prediction of ARMA models is studied and recursive formulas for predictors are derived. Then discrete-time covariance stationary processes are discussed. The Wold decomposition theorem is proved, and used in computing predictors and the corresponding error variances. Most of the results given here belong to the standard theory of stationary processes. In the next chapter, a regression lemma is introduced. Its use leads to several decompositions of the process. Each decomposition generates a corresponding set of parameters; this is a non-traditional approach to AR and MA representations. The method has the advantage that it can be used for processes with infinite variance where the classical procedures cannot be immediately applied. The chapter on finite prediction and partial correlations deals with the situation where the prediction is based only on a finite past. It is shown that with the increasing length of the history of the process the predictors converge to their infinite-past counterparts. Then the problem with missing values is treated. It concerns not only the problem of interpolation but also impacts on interventions and outliers. Further, the maximal correlation between past and future is calculated. The last two chapters contain descriptions of stationary sequences in Hilbert spaces and applications of Hardy spaces. The exercises at the end of each chapter help the reader to understand better and with more depth the results presented in the text. Some topics are not discussed in the book, for example nonlinear models, nonstationary models, and long-memory models.

The book is written very precisely and rigorously. At the same time, the presentation is interesting and understandable. The reader is informed about historical sources as well as about recent developments. The book is not a manual for time series analysis and forecasting. Instead, the mathematical, probabilistic, statistical, and computational concepts are emphasized along with some novel results from research papers. The book is suitable for researchers and advanced students. It can be recommended as an excellent textbook (one of the best which I have seen).

Reviewed by Jiri' Andel