Gaussian and Non-Gaussian Linear Time Series and Random Fields

1 Reversibility and Identifiability.- 1.1 Linear Sequences and the Gaussian Property.- 1.2 Reversibility.- 1.3 Identifiability.- 1.4 Minimum and Nonminimum Phase Sequences.- 2 Minimum Phase Estimation.- 2.1 The Minimum Phase Case and the Quasi-Gaussian Likelihood.- 2.2 Consistency.- 2.3 The Asymptotic Distribution.- 3 Homogeneous Gaussian Random Fields.- 3.1 Regular and Singular Fields.- 3.2 An Isometry.- 3.3 L-Fields and L-Markov Fields.- 4 Cumulants, Mixing and Estimation for Gaussian Fields.- 4.1 Moments and Cumulants.- 4.2 Higher Order Spectra.- 4.3 Some Simple Inequalities and Strong Mixing.- 4.4 Strong Mixing for Two-Sided Linear Processes.- 4.5 Mixing and a Central Limit Theorem for Random Fields.- 4.6 Estimation for Stationary Random Fields.- 4.7 Cumulants of Finite Fourier Transforms.- 4.8 Appendix: Two Inequalities.- 5 Prediction for Minimum and Nonminimum Phase Models.- 5.1 Introduction.- 5.2 A First Order Autoregressive Model.- 5.3 Nonminimum Phase Autoregressive Models.- 5.4 A Functional Equation.- 5.5 Entropy.- 5.6 Continuous Time Parameter Processes.- 6 The Fluctuation of the Quasi-Gaussian Likelihood.- 6.1 Initial Remarks.- 6.2 Derivation.- 6.3 The Limiting Process.- 7 Random Fields.- 7.1 Introduction.- 7.2 Markov Fields and Chains.- 7.3 Entropy and a Limit Theorem.- 7.4 Some Illustrations.- 8 Estimation for Possibly Nonminimum Phase Schemes.- 8.1 The Likelihood for Possibly Non-Gaussian Autoregressive Schemes.- 8.2 Asymptotic Normality.- 8.3 Preliminary Comments: Approximate Maximum Likelihood Estimates for Non-Gaussian Nonminimum Phase ARMA Sequences.- 8.4 The Likelihood Function.- 8.5 The Covariance Matrix.- 8.6 Solution of the Approximate Likelihood Equations.- 8.7 Cumulants and Estimation for Autoregressive Schemes.- 8.8 Superefficiency.- Bibliographic Notes.- References.- Notation.- Author Index.