Series of Irregular Observations: Forecasting and Model Building

I Discrete Time Random Processes.- 1. Random Variables and Probability Spaces.- 2. Random Vectors.- 3. Random Processes.- 4. Second-Order Process.- II Gaussian Processes.- 1. The Use (and Misuse) of Gaussian Models.- 2. Fourier Transform: A Few Basic Facts.- 3. Gaussian Random Vectors.- 4. Gaussian Processes.- III Stationary Processes.- 1. Stationarity and Model Building.- 2. Strict Stationarity and Second-Order Stationarity.- 3. Construction of Strictly Stationary Processes.- 4. Ergodicity.- 5. Second-Order Stationarity: Processes with Countable Spectrum.- IV Forecasting and Stationarity.- 1. Linear and Nonlinear Forecasting.- 2. Regular Processes and Singular Processes.- 3. Regular Stationary Processes and Innovation.- 4. Prediction Based on a Finite Number of Observations.- 5. Complements on Isometries.- V Random Fields and Stochastic Integrals.- 1. Random Measures with Finite Support.- 2. Uncorrected Random Fields.- 3. Stochastic Integrals.- VI Spectral Representation of Stationary Processes.- 1. Processes with Finite Spectrum.- 2. Spectral Measures.- 3. Spectral Decomposition.- VII Linear Filters.- 1. Often Used Linear Filters.- 2. Multiplication of a Random Field by a Function.- 3. Response Functions and Linear Filters.- 4. Applications to Linear Representations.- 5. Characterization of Linear Filters as Operators.- VIII ARMA Processes and Processes with Rational Spectrum.- 1. ARMA Processes.- 2. Regular and Singular Parts of an ARMA Process.- 3. Construction of ARMA Processes.- 4. Processes with Rational Spectrum.- 5. Innovation for Processes with Rational Spectrum.- IX Nonstationary ARMA Processes and Forecasting.- 1. Nonstationary ARMA Models.- 2. Linear Forecasting and Processes with Rational Spectrum.- 3. Time Inversion and Estimation of Past Observations.- 4. Forecasting and Nonstationary ARMA Processes.- X Empirical Estimators and Periodograms.- 1. Empirical Estimation.- 2. Periodograms.- 3. Asymptotic Normality and Periodogram.- 4. Asymptotic Normality of Empirical Estimators.- 5. The Toeplitz Asymptotic Homomorphism.- XI Empirical Estimation of the Parameters for ARMA Processes with Rational Spectrum.- 1. Empirical Estimation and Efficient Estimation.- 2. Computation of the ak and Yule-Walker Equations.- 3. Computation of the bl and of ?2.- 4. Empirical Estimation of the Parameters When p, q are Known.- 5. Characterization of p and q.- 6. Empirical Estimation of d for an ARIMA (p,d,q) Model.- 7. Empirical Estimation of (p,q).- 8. Complement: A Direct Method of Computation for the bk.- 9. The ARMA Models with Seasonal Effects.- 10. A Technical Result: Characterization of Minimal Recursive Identities.- 11. Empirical Estimation and Identification.- XII Effecient Estimation for the Parameters of a Process with Rational Spectrum.- 1. Maximum Likelihood.- 2. The Box-Jenkins Method to Compute (a, b).- 3. Computation of the Information Matrix.- 4. Convergence of the Backforecasting Algorithm.- XIII Asymptotic Maximum Likelihood.- 1. Approximate Log-Likelihood.- 2. Kullback Information.- 3. Convergence of Maximum Likelihood Estimators.- 4. Asymptotic Normality and Efficiency.- XIV Identification and Compensated Likelihood.- 1. Identification.- 2. Parametrization.- 3. Compensated Likelihood.- 4. Mathematical Study of Compensated Likelihood.- 5. Noninjective Parametrization.- 6. Almost Sure Bounds for the Maximal Log-Likelihood.- 7. Law of the Interated Logarithm for the Periodogram.- XV A Few Problems not Studied Here.- 1. Tests of Fit for ARMA Models.- 2. Nonlinearity.