Large Deviations for Additive Functionals of Markov Chains

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.