Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 1 Introduction
- 2 Lower Bounds and a Property of Λ
- 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 Deviation 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 The Ergodic Theorem for Empirical Measures and Vector-Valued Functionals of a Markov Chain
- Appendix B Irreducible Kernels, Small Sets, and Petite Sets
- Appendix C The Convergence Parameter
- Appendix D Approximation of P by Pt
- Appendix E On Varadhan’s Theorem
- Appendix F The Duality Theorem for Convex Functions
- Appendix G Daniell’s Theorem
- Appendix H Relative Compactness in the V Topology
- Appendix I A Monotone Class Theorem
- Appendix J On the Axioms V.1–V.3 and V.1’–V.4
- Appendix K On Gâteaux Differentiability
- References
- Author Index
- Subject Index
5 - Necessary Conditions: Bounds on the Rate Function, Invariant Measures, Irreducibility, and Recurrence
Published online by Cambridge University Press: 03 August 2022
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 1 Introduction
- 2 Lower Bounds and a Property of Λ
- 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 Deviation 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 The Ergodic Theorem for Empirical Measures and Vector-Valued Functionals of a Markov Chain
- Appendix B Irreducible Kernels, Small Sets, and Petite Sets
- Appendix C The Convergence Parameter
- Appendix D Approximation of P by Pt
- Appendix E On Varadhan’s Theorem
- Appendix F The Duality Theorem for Convex Functions
- Appendix G Daniell’s Theorem
- Appendix H Relative Compactness in the V Topology
- Appendix I A Monotone Class Theorem
- Appendix J On the Axioms V.1–V.3 and V.1’–V.4
- Appendix K On Gâteaux Differentiability
- References
- Author Index
- Subject Index
Summary
We obtain bounds for an a priori unknown rate function. We prove the existence and uniqueness of invariant probability measures and the necessity of irreducibility.
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- Large Deviations for Markov Chains , pp. 68 - 78Publisher: Cambridge University PressPrint publication year: 2022