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:
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Spectral theory. Well-behaved solutions $\cf$ can be constructed for the "multiplicative Poisson equation" $(e^{\alpha F}P)\cf=\lambda\cf$, where P is the transition kernel of the Markov chain and $\alpha\in\Co$ is a constant. The function $\cf$ is an eigenfunction, with corresponding eigenvalue $\lambda$, for the kernel $(e^{\alpha F}P)=e^{\alpha F(x)}P(x,dy)$.
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A "multiplicative" mean ergodic theorem. For all complex $\alpha$ in a neighborhood of the origin, the normalized mean of $\exp(\alpha S_t)$ (and not the logarithm of the mean) converges to $\cf$ exponentially, where $\cf$ is a solution of the multiplicative Poisson equation.
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Edgeworth expansions. Rates are obtained for the convergence of the distribution function of the normalized partial sums $S_t$ to the standard Gaussian distribution. The first term in this expansion is of order $(1/\sqrt{t})$ and it depends on the initial condition of the Markov chain through the solution $\haF$ of the associated Poisson equation (and not the solution $\cf$ of the multiplicative Poisson equation).
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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.
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Exact large deviations asymptotics. Rates of convergence are obtained for the large deviations estimates above. The polynomial preexponent is of order $(1/\sqrt{t})$ and its coefficient depends on the initial condition of the Markov chain through the solution $\cf$ of the multiplicative Poisson equation.
¶ Extensions of these results to continuous-time Markov processes are also given.