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**Markov chains and invariant probabilities.**
*(English)*
Zbl 1036.60003

Progress in Mathematics (Boston, Mass.) 211. Basel: Birkhäuser (ISBN 3-7643-7000-9/hbk). xvi, 205 p. (2003).

This book is devoted to homogeneous discrete-time Markov chains that admit invariant probability measures. The authors of the book are well-known contributors in this area. They have published a long series of original papers during the last decade. Now the readers can enjoy a systematic presentation of stability properties of Markov chains in a book form. The main concerns of the authors are different kinds of ergodicity, ergodic decompositions, ergodic theorems, individual and mean, existence and/or uniqueness of invariant measures, occupation measures.

The Markov chains are with values in countable or more general state spaces and the operator theory language is successfully used to present, with all necessary details, a wide spectrum of results in a quite compact form. The authors pay special attention to boundary cases, which are near the “trap” in the sense that due to an essential reason a statement fails to hold. Even when treating classical properties, e.g. Feller Markov chains, the authors are successful in presenting new results, or refinements of previously known results from a new point of view. Some results by the authors appear in the book for the first time.

The material is incorporated in the following chapters: 1. Preliminaries. 2. Markov chains and ergodic theorems. 3. Countable Markov chains. 4. Harris Markov chains. 5. Markov chains in metric spaces. 6. Classification of Markov chains via occupation measures. 7. Feller Markov chains. 8. The Poisson equation. 9. Strong and uniform ergodicity. 10. Existence of invariant probability measures. 11. Existence and uniqueness of fixed points for invariant measures. At the end of the book we find a quite comprehensive up-to-date reference list, as well as an index, and list of abbreviations.

The book is carefully written, its contents and style show undoubtedly that Markov chains form a beautiful branch of modern stochastics and hence of modern mathematics. There are reasons to congratulate the authors for writing this excellent work. A wide spectrum of readers will benefit from the book, in particular, researchers and PhD students in the area of probability, analysis, optimization theory and dynamical systems.

The Markov chains are with values in countable or more general state spaces and the operator theory language is successfully used to present, with all necessary details, a wide spectrum of results in a quite compact form. The authors pay special attention to boundary cases, which are near the “trap” in the sense that due to an essential reason a statement fails to hold. Even when treating classical properties, e.g. Feller Markov chains, the authors are successful in presenting new results, or refinements of previously known results from a new point of view. Some results by the authors appear in the book for the first time.

The material is incorporated in the following chapters: 1. Preliminaries. 2. Markov chains and ergodic theorems. 3. Countable Markov chains. 4. Harris Markov chains. 5. Markov chains in metric spaces. 6. Classification of Markov chains via occupation measures. 7. Feller Markov chains. 8. The Poisson equation. 9. Strong and uniform ergodicity. 10. Existence of invariant probability measures. 11. Existence and uniqueness of fixed points for invariant measures. At the end of the book we find a quite comprehensive up-to-date reference list, as well as an index, and list of abbreviations.

The book is carefully written, its contents and style show undoubtedly that Markov chains form a beautiful branch of modern stochastics and hence of modern mathematics. There are reasons to congratulate the authors for writing this excellent work. A wide spectrum of readers will benefit from the book, in particular, researchers and PhD students in the area of probability, analysis, optimization theory and dynamical systems.

Reviewer: Jordan M. Stoyanov (Newcastle upon Tyne)

### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |