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1 - Introduction
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 1-13
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Summary

We present the outlook and some of the main contents of the book, including some basic definitions.

7 - Upper Bounds III: Sufficient Conditions
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 94-116
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Summary

We present several sufficient conditions for the upper bound.

Appendix B - Irreducible Kernels, Small Sets, and Petite Sets
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 200-211
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8 - The Large Deviation Principle for Empirical Measures
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 117-133
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Summary

We present several formulations of the large deviation principle for empirical measures in the V topology, depending on the initial distribution. The case V = B(S) is further studied.

Appendix E - On Varadhan’s Theorem
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 223-226
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Appendix A - The Ergodic Theorem for Empirical Measures and Vector-Valued Functionals of a Markov Chain
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 197-199
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Appendix F - The Duality Theorem for Convex Functions
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 227-229
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References
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 244-246
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Contents
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp vii-x
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4 - Identification and Reconciliation of Rate Functions
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 47-67
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Summary

We relate the rate functions introduced in Chapters 2 and 3 to the functions I and Isubindex ψ. We establish conditions for the equality I= Isubindex ψ. We introduce the conditions V.1′–V.4.

Author Index
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 247-247
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Appendix K - On Gâteaux Differentiability
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 240-243
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9 - The Case When S Is Countable and P Is Matrix Irreducible
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 134-142
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Summary

We apply the results obtained in previous chapters, with some simplifications, to the case when S is countable and P is matrix irreducible.

3 - Upper Bounds I
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 36-46
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Summary

We obtain upper bounds for random probability measures and empirical measures. We introduce the conditions V.1–V.3 for a vector subspace V of B(S).

Appendix I - A Monotone Class Theorem
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 233-235
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2 - Lower Bounds and a Property of Λ
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 14-35
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Summary

We obtain lower bounds for bounded vector-valued additive functionals and use them to obtain lower bounds for empirical measures. We prove a lower semicontinuity property of Λ.

Appendix C - The Convergence Parameter
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 212-218
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11 - Large Deviations for Vector-Valued Additive Functionals
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 154-196
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Summary

We study large deviations for general vector-valued additive functionals. The relationship between large deviations for empirical measures and large deviations for additive functionals is discussed.

Appendix H - Relative Compactness in the V Topology
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 231-232
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Appendix G - Daniell’s Theorem
Alejandro D. de Acosta, Case Western Reserve University, Ohio
Book: Large Deviations for Markov Chains
Published online: 03 August 2022
Print publication: 27 October 2022, pp 230-230
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1 - Introduction

Summary

7 - Upper Bounds III: Sufficient Conditions

Summary

Appendix B - Irreducible Kernels, Small Sets, and Petite Sets

8 - The Large Deviation Principle for Empirical Measures

Summary

Appendix E - On Varadhan’s Theorem

Appendix A - The Ergodic Theorem for Empirical Measures and Vector-Valued Functionals of a Markov Chain

Appendix F - The Duality Theorem for Convex Functions

References

Contents

4 - Identification and Reconciliation of Rate Functions

Summary

Author Index

Appendix K - On Gâteaux Differentiability

9 - The Case When S Is Countable and P Is Matrix Irreducible

Summary

3 - Upper Bounds I

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Appendix I - A Monotone Class Theorem

2 - Lower Bounds and a Property of Λ

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Appendix C - The Convergence Parameter

11 - Large Deviations for Vector-Valued Additive Functionals

Summary

Appendix H - Relative Compactness in the V Topology

Appendix G - Daniell’s Theorem

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