Large Deviations for Markov Chains

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.

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• 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

本書研究具有一般狀態空間的馬可夫鏈的經驗測度和向量值可加性泛函的大偏差。在適當的循環條件下，馬可夫鏈的可加性泛函的遞歸定理斷言了鏈的實數或向量值函數的平均數幾乎必然收斂到不變分佈的平均數。對於經驗測度的情況，遞歸定理以適當的意義斷言了幾乎必然收斂到不變分佈。大偏差定理在對數級別上提供了從遞歸定理所斷言的主導行為偏離的概率的精確漸近估計。

• 這是第一本在一般狀態空間上深入研究馬可夫鏈大偏差的書籍，並以不可約非負核理論的框架進行。該理論的相關方面在幾個附錄中介紹。


• 不可約性及其結果以及其衍生概念，如不可約非負核的收斂參數，起著重要作用。


• 本書中的許多結果在文獻中尚未出現，包括關於均勻集和不變分佈的新結果。

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.

前言
1. 引言
2. 下界和 lambda 的性質
3. 上界 I
4. 率函數的識別和調和
5. 必要條件 - 率函數的上界、不變測度、不可約性和循環性
6. 上界 II - 等效的分析條件
7. 上界 III - 充分條件
8. 對於實驗測度的大偏差原理
9. 當 S 是可數的且 P 是矩陣不可約時的情況
10. 例子
11. 向量值加法函數的大偏差
附錄 A
附錄 B
附錄 C
附錄 D
附錄 E
附錄 F
附錄 G
附錄 H
附錄 I
附錄 J
附錄 K
參考文獻
作者索引
主題索引。