Spectral Theory and Limit Theorems for Geometrically Ergodic Markov Processes

TL;DR

This paper develops spectral theory and limit theorems for geometrically ergodic Markov processes, providing new tools for analyzing their long-term behavior and deviations.

Contribution

It introduces solutions to the multiplicative Poisson equation and establishes exponential convergence, Edgeworth expansions, and large deviations principles under geometric ergodicity.

Findings

Solutions to the multiplicative Poisson equation constructed.

Exponential convergence of normalized exponential means shown.

Rates for distribution convergence and large deviations established.

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: 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.