Spectral theory and limit theorems for geometrically ergodic Markov processes

Abstract

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:

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 $(e^{\alpha F}P)=e^{\alpha F(x)}P(x,dy)$.

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.

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).

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.

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.