Spectral Theory and Limit Theorems for Geometrically Ergodic Markov Processes

Consider the partial sums {S_t} of a real-valued functional F(Phi(t)) of a Markov chain {Phi(t)} with values in a general state space. Assuming only that the Markov chain is geometrically ergodic and that the functional F is bounded, the following conclusions are obtained:

1. Spectral theory: Well-behaved solutions can be constructed for the ``multiplicative Poisson equation’'.
2. A ``multiplicative’’ mean ergodic theorem: For all complex \alpha in a neighborhood of the origin, the normalized mean of \exp(\alpha S_t) converges exponentially fast to a solution of the multiplicative Poisson equation.
3. Edgeworth Expansions: Rates are obtained for the convergence of the distribution function of the normalized partial sums S_t to the standard Gaussian distribution.
4. Large Deviations: The partial sums are shown to satisfy a large deviations principle in a neighborhood of the mean. This result, proved under geometric ergodicity alone, cannot in general be extended to the whole real line.
5. Exact Large Deviations Asymptotics: Rates of convergence are obtained for the large deviations estimates above. Extensions of these results to continuous-time Markov processes are also given.