# Large Deviations for Additive Functionals of Markov Chains

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*Alejandro D. de Acosta; Peter Ney*

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

# Table of Contents

## Large Deviations for Additive Functionals of Markov Chains

- Chapter 1. Introduction 18 free
- Chapter 2. The transform kernels 𝐾_{𝑔} and their convergence parameters 916 free
- Chapter 3. Comparison of Λ(𝑔) and 𝜑_{𝜇}(𝑔) 2532
- Chapter 4. Proof of Theorem 1 3138
- Chapter 5. A characteristic equation and the analyticity of Λ_{𝑓}: the case when 𝑃 has an atom 𝐶∈𝒮⁺ satisfying 𝜆*(𝒞)>0 3340
- Chapter 6. Characteristic equations and the analyticity of Λ_{𝑓}: the general case when 𝑃 is geometrically ergodic 4148
- Chapter 7. Differentiation formulas for 𝑢_{𝑔} and Λ_{𝑓} in the general case and their consequences 5158
- Chapter 8. Proof of Theorem 2 6370
- Chapter 9. Proof of Theorem 3 6774
- Chapter 10. Examples 7178
- Chapter 11. Applications to an autoregressive process and to reflected random walk 7784
- Appendix 93100
- Background comments 105112
- References 107114