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Full Description
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.
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.
