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Large deviations for additive functionals of Markov chains


About this Title

Alejandro D. de Acosta and Peter Ney

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 228, Number 1070
ISBNs: 978-0-8218-9089-9 (print); 978-1-4704-1482-5 (online)
DOI: http://dx.doi.org/10.1090/memo/1070
Published electronically: July 15, 2013
Keywords:Large deviations, Markov chains, additive functionals, transform kernels, convergence parameter

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. The transform kernels $K_{g}$ and their convergence parameters
  • Chapter 3. Comparison of $\Lambda (g)$ and $\phi _\mu (g)$
  • Chapter 4. Proof of Theorem 1
  • Chapter 5. A characteristic equation and the analyticity of $\Lambda _f$: the case when $P$ has an atom $C\in \mathcal {S}^+$ satisfying $\lambda ^*(C)>0$
  • Chapter 6. Characteristic equations and the analyticity of $\Lambda _f$: the general case when $P$ is geometrically ergodic
  • Chapter 7. Differentiation formulas for $u_g$ and $\Lambda _f$ in the general case and their consequences
  • Chapter 8. Proof of Theorem 2
  • Chapter 9. Proof of Theorem 3
  • Chapter 10. Examples
  • Chapter 11. Applications to an autoregressive process and to reflected random walk
  • Appendix
  • Background comments

Abstract


For a Markov chain with general state space and , the large deviation principle for is proved under a condition on the chain which is weaker than uniform recurrence but stronger than geometric recurrence and an integrability condition on , for a broad class of initial distributions. This result is extended to the case when takes values in a separable Banach space. Assuming only geometric ergodicity and under a non-degeneracy condition, a local large deviation result is proved for bounded . 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.

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