Large Deviations for Markov Chains
Large Deviations for Markov Chains
£95.00
GBPThis book studies the large deviations for empirical measures and vector-valued additive functionals of Markov chains with general state space. Under suitable recurrence conditions, the ergodic theorem for additive functionals of a Markov chain asserts the almost sure convergence of the averages of a real or vector-valued function of the chain to the mean of the function with respect to the invariant distribution. In the case of empirical measures, the ergodic theorem states the almost sure convergence in a suitable sense to the invariant distribution. The large deviation theorems provide precise asymptotic estimates at logarithmic level of the probabilities of deviating from the preponderant behavior asserted by the ergodic theorems.
- The first book to study large deviations for Markov chains in depth in the framework of the theory of irreducible nonnegative kernels on a general state space. The relevant aspects of this theory are presented in several appendices
- An essential role is played by irreducibility, its consequences, and its derivative notions, such as the convergence parameter of an irreducible nonnegative kernel
- Many results in the book have not previously appeared in the literature – this includes new results on uniformity sets and the role of invariant distributions
Reviews & endorsements
'This is a treatise on the large deviations for the empirical measure (and additive functionals) of Markov chains. Structural assumptions are kept to a minimum, and great care is taken to separately obtain upper and lower bounds (which under appropriate conditions are shown to be equal). This book summarizes the author's work over the last 40 years and contains a plethora of new, sharpest in its class, results.' Ofer Zeitouni, Weizmann Institute, Israel
'This is a wonderful book. For four decades, Alejandro de Acosta's work in large deviations has been characterized by both its elegance and generality. Large Deviations for Markov Chains synthesizes, refines, and advances the current state of the field in a treatment that is at once rigorous and readable. It belongs on the bookshelf of any mathematician interested in the current state of the subject.' Sandy L. Zabell, Northwestern University
'This is an excellent book, by a top expert in the field, on large deviations for Markov chains in a very general setting. It is a valuable resource for graduate students learning the subject and for researchers in probability theory, statistical mechanics, statistics, engineering, and the sciences.' Firas Rassoul-Agha, University of Utah
'The theory of large deviations is an important way to understand many mathematical and physical models. This book covers the fascinating topic of large deviations for empirical measures and additive functionals of Markov chains with general state space, a subject on which the author is a leading expert who has made crucial contributions. Markov chains represent a large class of stochastic models with a wide spectrum of behaviors. It is remarkable that any universal results, like the ones given in the book, can be formulated for such a large family. It is equally remarkable that the book develops a sharp link between the large deviations and the degree of recurrence of Markov chains. The book does a superb job of clarification, comparison and identification of the rate functions that govern the large deviations.' Xia Chen, University of Tennessee
Product details
October 2022Hardback
9781316511893
230 pages
235 × 158 × 22 mm
0.55kg
Available
Table of Contents
- Preface
- 1. Introduction
- 2. Lower bounds and a property of lambda
- 3. Upper bounds I
- 4. Identification and reconciliation of rate functions
- 5. Necessary conditions – bounds on the rate function, invariant measures, irreducibility and recurrence
- 6. Upper bounds II – equivalent analytic conditions
- 7. Upper bounds III – sufficient conditions
- 8. The large deviations principle for empirical measures
- 9. The case when S is countable and P is matrix irreducible
- 10. Examples
- 11. Large deviations for vector-valued additive functionals
- Appendix A
- Appendix B
- Appendix C
- Appendix D
- Appendix E
- Appendix F
- Appendix G
- Appendix H
- Appendix I
- Appendix J
- Appendix K
- References
- Author index
- Subject index.
