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Large Deviations for Additive Functionals of Markov Chains
Alejandro D. de Acosta and Peter Ney
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Memoirs of the American Mathematical Society
2014; 108 pp; softcover
Volume: 228
ISBN-10: 0-8218-9089-1
ISBN-13: 978-0-8218-9089-9
List Price: US$76
Individual Members: US$45.60
Institutional Members: US$60.80
Order Code: MEMO/228/1070
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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 non-degeneracy 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

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