Memoirs of the American Mathematical Society 2014; 108 pp; softcover Volume: 228 ISBN10: 0821890891 ISBN13: 9780821890899 List Price: US$76 Individual Members: US$45.60 Institutional Members: US$60.80 Order Code: MEMO/228/1070
 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  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
