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 (ã, 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.
At the university level, in probability and statistics departments or electrical engineering departments, this book contains enough material for a graduate course, or even for an upper-level undergraduate course if the asymptotic studies are reduced to a minimum. The prerequisites for most of the chapters (l - 12) are fairly limited: the elements of Hilbert space theory, and the basics of axiomatic probability theory including L 2-spaces, the notions of distributions, random variables and bounded measures. The standards of precision, conciseness, and mathematical rigour which we have maintained in this text are in clearcut contrast with the majority of similar texts on the subject. The main advantage of this choice should be a considerable gain of time for the noninitiated reader, provided he or she has a taste for mathematical language. On the other hand, being fully aware of the usefulness of ARMA models for applications, we present carefully and in full detail the essential algorithms for practical modelling and identification of ARMA processes. The experience gained from several graduate courses on these themes (Universities of Paris-Sud and of Paris-7) has shown that the mathematical material included here is sufficient to build reasonable computer programs of data analysis by ARMA modelling. To facilitate the reading, we have inserted a bibliographical guide at the end of each chapter and, indicated by stars (* ... *), a few intricate mathematical points which may be skipped over by nonspecialists.
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