Stationary Processes and Prediction Theory. (AM-44), Volume 44 (Annals of Mathematics Studies, 44)

The description for this book, Stationary Processes and Prediction Theory. (AM-44), Volume 44, will be forthcoming. ACKNOWLEDGEMENTS CONTENTS INTRODUCTION CHAPTER 1. STOCHASTIC PROCESSES AND STOCHASTIC SEQUENCES §1. Preliminaries: Commutative C*-algebras 1.1. C*- algebras 1.2. E - algebras §2. Preliminaries: Stationary Stochastic Processes 2.1. One-sides and Two-sided Processes 2.2. The Sample Paths of a Process 2.3. Subprocesses 2.4. Ergodicity §3. Stochastic Sequences 3.1. Definitions 3.2 . Distribution Functions of Stochastic Sequences 3.3. Examples §4.The Process Associated With a Stochastic Sequence 4.1. The Abstract Process of a Stochastic Sequence 4.2. The Process X(ξ) 4.3. Examples §5. Regular Sequences 5.1. Definitions 5.2. Existence of Regular Sequences 5.3. Generic Points §6. Examples 6.1. The Almost Periodic Case 6.2. The Random Case 6.3. Operations on Regular Sequences §7. Ergodic Properties of Regular Sequences 7.1. Criteria for Ergodicity 7.2. Examples CHAPTER 2. THE PREDICTION PROBLEMS FOR SEQUENCES §8. The Support of the Prediction Measure 8.1. L-sequences 8.2. Littlewood's Theorem §9. Deterministic Sequences 9.1. Definition 9.2. Construction of Deterministic Sequences §10. Prediction Measures and Continuous Predictability 10.1. Prediction Measures 10.2 . Preliminary Lemmas 10.3. A Criterion for Continuous Predictability 10.4. Continuous Predictability at Every Point 10.5 . Composition of Prediction Measures 10.6. Realization of Prediction Measures CHAPTER 3. EXAMPLES AND COUNTEREXAMPLES §11. Random and Markoff Sequences 11.1. Preliminaries 11.2. Continuous Predictability of Random, Markoff, and M-Markoff Sequences §12. A Non-Continuously Predictable Sequence 12.1. Construction of the Sequence 12.2. Application of the Prediction Procedure 12.3. Another Form of the Sequence §13. A Class of Deterministic Sequences 13.1. D-Sequences CHAPTER 4. SUBPROCESSES OF MARKOFF PROCESSES §14. Automorphism Groups 14.1. Definitions 14.2. An Illustration §15. Linear Transformations of Cones 15.1. The Projective Metric 15.2. Projectively Bounded Transformations §16. A Sufficient Condition for Continuous Predictability 16.1. Automorphisms and Their Adjoints 16.2. Construction of the Prediction Measures §17. Normality and Continuous Predictability 17.1. Reduction to Markoff Processes 17.2. Consequences of Normality 17.3. An Example CHAPTER 5. STOCHASTIC SEMIGROUPS AND CONTINUOUS PREDICTABILITY §18. Stochastic Semigroups 18.1. Definitions 18.2. Ergodicity of X and ς(X) 18.3. Application to Symmetric Processes §19. Realization of Stochastic Semigroups 19.1. Linear Semigroups 19.2 . Examples 19.3. Further Examples §20. Continuous Predictability for the Processes of Stochastic Semigroups 20.1. A Criterion for Continuous Predictability. 20.2. The Linear Case 20.3. Applications CHAPTER 6. STATISTICAL PREDICTABILITY §21. The Continuously Predictable Cover of a Process 21.1. Definitions §22. Statistical Predictability 22.1. Statistical Determination 22.2. Properties of Statistical Determination 22.3. Statistical Predictability §23. The Continuously Predictable Cover of a Finitely Valued Process 23.1. An Identity for Prediction Measures 23.2. Construction of §24. Applications to Finite Dimensional Processes 24.1. The Canonical Semigroup 24.2. The Sample Space CHAPTER 7. INDUCTIVE FUNCTIONS §25. Inductive Functions; An Example 25.1. Preliminaries 25.2. The Equation zn+1 = xn+1 + zn. Examples 25.3. A Condition for Regularity 25.4. Existence of Representative Sequences §26. G-roup-Valued Inductive Functions 26.1. The Equation zn+1 = xn+1zn 26.2. Applications to Equidistribution §27. Periodic Subsequences of Regular Sequences 27.1. Connection with Inductive Functions 27.2. Regularity of σ-Sequences 27.3. Existence of Fourier Coefficients CHAPTER 8. INDUCTIVE FUNCTIONS AND MARKOFF PROCESSES. §28. Inductive Functions of Markoff Processes 28.1. Preliminaries 28.2. A Counterexample 28.3. A Law of Large Numbers for Markoff Processes 28.4. Compact Inductive Functions of Random Processes 28.5. Compact Inductive Functions of Markoff Processes §29. Compact Inductive Functions of Markoff Sequences 29.1. Uniqueness of Inductive Functions 29.2. Application to Markoff Sequences CHAPTER 9. PROJECTIVE INDUCTIVE FUNCTIONS AND PREDICTION §30. Projective Inductive Functions 30.1. Notation 30.2. Preliminary Lemmas 30.3. Projective Inductive Functions of a Markoff Process 30.4. The Range of a Projective Inductive Function §31. Projective Inductive Functions of Markoff Sequences 31.1. Statement of the Problem 31.2. A Reformulation 31.3. A Sufficient Condition 31.4. A Further Sufficient Condition §32. Adjoint Processes 32.1. Conditional Distributions 32.2. The Fundamental Lemma 32.3. Restricted Solution to the Problem 32.4. The Remaining Alternative 32.5. Some Remarks §33- Application to Statistical Predictability 33.1. Resumé 33.2. Application of Theorem 32.1 33.3. Uniqueness BIBLIOGRAPHY A classic treatment of stationary processes and prediction theory from the acclaimed Annals of Mathematics Studies series Princeton University Press is proud to have published the Annals of Mathematics Studies since 1940. One of the oldest and most respected series in science publishing, it has included many of the most important and influential mathematical works of the twentieth century. The series continues this tradition as Princeton University Press publishes the major works of the twenty-first century. To mark the continued success of the series, all books are available in paperback and as ebooks.