This book is concerned with a set of related problems in probability theory that are considered in the context of Markov processes. Some of these are natural to consider, especially for Markov processes. Other problems have a broader range of validity but are convenient to pose for Markov processes. The book can be used as the basis for an interesting course on Markov processes or stationary processes. For the most part these questions are considered for discrete parameter processes, although they are also of obvious interest for continuous time parameter processes. This allows one to avoid the delicate measure theoretic questions that might arise in the continuous parameter case. There is an attempt to motivate the material in terms of applications. Many of the topics concern general questions of structure and representation of processes that have not previously been presented in book form. A set of notes comment on the many problems that are still left open and related material in the literature. It is also hoped that the book will be useful as a reference to the reader who would like an introduction to these topics as well as to the reader interested in extending and completing results of this type.
Table of ContentsI Basic Notions and Illustrations.- 0. Summary.- 1. Markov Processes and Transition Probability Functions.- 2. Markov Chains.- 3. Independent Random Variables.- 4. Some Continuous Parameter Markov Processes.- 5. Random Walks on Countable Commutative Groups.- Notes.- II Remarks on Some Applications.- 0. Summary.- 1. A Model in Statistical Mechanics.- 2. Some Models in Learning Theory.- 3. A Resource Flow Model.- Notes.- III Functions of Markov Processes.- 0. Summary.- 1. Collapsing of States and the Chapman-Kolmogorov Equation.- 2. Markovian Functions of Markov Processes.- 3. Functions of Finite State Markov Chains.- Notes.- IV Ergodic and Prediction Problems.- 0. Summary.- 1. A Markov Process Restricted to a Set A.- 2. An L1 Ergodic Theorem.- 3. Transition Operators and Invariant Measures on a Topological Space.- 4. Asymptotic Behavior of Powers of a Transition Probability Operator.- Notes.- V Random Walks and Convolution on Groups and Semigroups.- 0. Summary.- 1. A Problem of P. Lévy.- 2. Limit Theorems and the Convolution Operation.- 3. Idempotent Measures as Limiting Distributions.- 4. The Structure of Compact Semigroups.- 5. Convergent Convolution Sequences.- Notes.- VI Nonlinear Representations in Terms of Independent Random Variables.- 0. Summary.- 1. The Linear Prediction Problem for Stationary Sequences.- 2. A Nonlinear Prediction Problem.- 3. Questions for Markov Processes.- 4. Finite State Markov Chains.- 5. Real-Valued Markov Processes.- Notes.- VII Mixing and the Central Limit Theorem.- 0. Summary.- 1. Independence.- 2. Uniform Ergodicity, Strong Mixing and the Central Limit Problem.- 3. An Operator Formulation of Strong Mixing and Uniform Ergodicity.- 4. Lp Norm Conditions and a Central Limit Theorem.- Notes.- Appendix 1. Probability Theory.- Appendix 2. Topological Spaces.- Appendix 3. The Kolmogorov Extension Theorem.- Appendix 4. Spaces and Operators.- Appendix 5. Topological Groups.- Postscript.- Author Index.- Notation.