eBook ISBN:  9781470414825 
Product Code:  MEMO/228/1070.E 
List Price:  $76.00 
MAA Member Price:  $68.40 
AMS Member Price:  $45.60 
eBook ISBN:  9781470414825 
Product Code:  MEMO/228/1070.E 
List Price:  $76.00 
MAA Member Price:  $68.40 
AMS Member Price:  $45.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 228; 2014; 108 ppMSC: Primary 60;
For a Markov chain \(\{X_j\}\) with general state space \(S\) and \({f:S\rightarrow\mathbf{R}^d}\), the large deviation principle for \({\{n^{1}\sum_{j=1}^nf(X_j)\}}\) is proved under a condition on the chain which is weaker than uniform recurrence but stronger than geometric recurrence and an integrability condition on \(f\), for a broad class of initial distributions. This result is extended to the case when \(f\) takes values in a separable Banach space. Assuming only geometric ergodicity and under a nondegeneracy condition, a local large deviation result is proved for bounded \(f\). A central analytical tool is the transform kernel, whose required properties, including new results, are established. The rate function in the large deviation results is expressed in terms of the convergence parameter of the transform kernel.

Table of Contents

Chapters

1. Introduction

2. The transform kernels $K_{g}$ and their convergence parameters

3. Comparison of $\Lambda (g)$ and $\phi _\mu (g)$

4. Proof of Theorem 1

5. A characteristic equation and the analyticity of $\Lambda _f$: the case when $P$ has an atom $C\in \mathcal {S}^+$ satisfying $\lambda ^*(C)>0$

6. Characteristic equations and the analyticity of $\Lambda _f$: the general case when $P$ is geometrically ergodic

7. Differentiation formulas for $u_g$ and $\Lambda _f$ in the general case and their consequences

8. Proof of Theorem 2

9. Proof of Theorem 3

10. Examples

11. Applications to an autoregressive process and to reflected random walk

Appendix

Background comments


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For a Markov chain \(\{X_j\}\) with general state space \(S\) and \({f:S\rightarrow\mathbf{R}^d}\), the large deviation principle for \({\{n^{1}\sum_{j=1}^nf(X_j)\}}\) is proved under a condition on the chain which is weaker than uniform recurrence but stronger than geometric recurrence and an integrability condition on \(f\), for a broad class of initial distributions. This result is extended to the case when \(f\) takes values in a separable Banach space. Assuming only geometric ergodicity and under a nondegeneracy condition, a local large deviation result is proved for bounded \(f\). A central analytical tool is the transform kernel, whose required properties, including new results, are established. The rate function in the large deviation results is expressed in terms of the convergence parameter of the transform kernel.

Chapters

1. Introduction

2. The transform kernels $K_{g}$ and their convergence parameters

3. Comparison of $\Lambda (g)$ and $\phi _\mu (g)$

4. Proof of Theorem 1

5. A characteristic equation and the analyticity of $\Lambda _f$: the case when $P$ has an atom $C\in \mathcal {S}^+$ satisfying $\lambda ^*(C)>0$

6. Characteristic equations and the analyticity of $\Lambda _f$: the general case when $P$ is geometrically ergodic

7. Differentiation formulas for $u_g$ and $\Lambda _f$ in the general case and their consequences

8. Proof of Theorem 2

9. Proof of Theorem 3

10. Examples

11. Applications to an autoregressive process and to reflected random walk

Appendix

Background comments