Large Deviations for Markov Chains

A study of large deviations for empirical measures and vector-valued additive functionals of general state space Markov chains.

Alejandro D. de Acosta (Author)

9781316511893, Cambridge University Press

Hardback, published 27 October 2022

230 pages
23.5 x 15.8 x 2.2 cm, 0.55 kg

'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

This 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.

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.

Subject Areas: Stochastics [PBWL], Topology [PBP], Functional analysis & transforms [PBKF], Real analysis, real variables [PBKB]