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Large deviations for Markov chains. (English) Zbl 07535603

Cambridge Tracts in Mathematics 229. Cambridge: Cambridge University Press (ISBN 978-1-316-51189-3/hbk; 978-1-00-905312-9/ebook). xii, 249 p. (2022).
This book explores large deviations of empirical measures and additive functionals of Markov chains on a general state space and provides a comprehensive understanding of the connection between large deviations and recurrence of Markov chains. The subject of large deviations plays an important role in many physical and mathematical models and the author is an expert with many important contributions to the field.
The introduction emphasizes the importance of irreducibility and avoiding the need for a Polish state space, contributing to the accessibility of the material. The rest of chapters are organized logically, with lower bounds covered in the second chapter and upper bounds detailed in subsequent chapters. The identification and discussion of rate functions, along with specific discussions on empirical measures and vector-valued additive functionals, offer a comprehensive exploration of the subject. The inclusion of explicit examples in Chapter 10 serves as an illustration of the theory, emphasizing that rate functions may depend on the initial distribution or lack convexity. The technical nature of the proofs requires familiarity with classical Markov chain theory, making the provided appendix a valuable resource for readers.
In conclusion, this treatise stands as a seminal work, summarizing 40 years of the author’s research in large deviations for empirical measures of Markov chains. Its accessibility to researchers and graduate students alike, coupled with its stimulating approach to understanding the asymptotic behavior of stochastic processes, makes it a valuable addition to the literature in this domain.

MSC:

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60F10 Large deviations
60Jxx Markov processes
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