Gaussian and Non-Gaussian Linear Time Series and Random Fields

1. Front Matter

   Pages i-xiii

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2. Reversibility and Identifiability

   • Murray Rosenblatt

   Pages 1-13

3. Minimum Phase Estimation

   • Murray Rosenblatt

   Pages 15-26

4. Homogeneous Gaussian Random Fields

   • Murray Rosenblatt

   Pages 27-39

5. Cumulants, Mixing and Estimation for Gaussian Fields

   • Murray Rosenblatt

   Pages 41-81

6. Prediction for Minimum and Nonminimum Phase Models

   • Murray Rosenblatt

   Pages 83-115

7. The Fluctuation of the Quasi-Gaussian Likelihood

   • Murray Rosenblatt

   Pages 117-139

8. Random Fields

   • Murray Rosenblatt

   Pages 141-154

9. Estimation for Possibly Nonminimum Phase Schemes

   • Murray Rosenblatt

   Pages 155-210

10. Back Matter

    Pages 211-246

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Much of this book is concerned with autoregressive and moving av­ erage linear stationary sequences and random fields. These models are part of the classical literature in time series analysis, particularly in the Gaussian case. There is a large literature on probabilistic and statistical aspects of these models-to a great extent in the Gaussian context. In the Gaussian case best predictors are linear and there is an extensive study of the asymptotics of asymptotically optimal esti­ mators. Some discussion of these classical results is given to provide a contrast with what may occur in the non-Gaussian case. There the prediction problem may be nonlinear and problems of estima­ tion can have a certain complexity due to the richer structure that non-Gaussian models may have. Gaussian stationary sequences have a reversible probability struc­ ture, that is, the probability structure with time increasing in the usual manner is the same as that with time reversed. Chapter 1 considers the question of reversibility for linear stationary sequences and gives necessary and sufficient conditions for the reversibility. A neat result of Breidt and Davis on reversibility is presented. A sim­ ple but elegant result of Cheng is also given that specifies conditions for the identifiability of the filter coefficients that specify a linear non-Gaussian random field.

• Covariance matrix
• Gaussian Linear Time Series
• Likelihood
• Linear Time Series
• Probability theory
• Time series
• Variance

From the reviews:

SHORT BOOK REVIEWS

"...will make this book useful as a reference source to the more theoretical among time series specialists."

ZENTRALBLATT MATH

"This publication can be recommended to readers familiar with the basic concepts of time series who are interested in estimation problems in nonminimum phase processes."

• Department of Mathematics, University of California, San Diego La Jolla, USA

  Murray Rosenblatt

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