Spectral Theory and Limit Theorems for Geometrically Ergodic Markov Processes

2002/09/16 by Ioannis Kontoyiannis, Sean Meyn, Kontoyiannis, Ioannis +1 · 3 citations
Mathematics · Physics and Astronomy · #34L40 #41A36 #60F10 #60J10 #60J25 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #math.PR #math.SP #msc:34L40 #msc:41A36 #msc:60F10 #msc:60J10 #msc:60J25

paper · pdf · doi:10.48550/arxiv.math/0209200

52 pages, 1 figure, to appear, Annals of Applied Probability

arxiv created 2002/09/16 · openalex publication_date 2002/09/16 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

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 αin a neighborhood of the origin, the normalized mean of exp(αS_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.

Cited by

• Statistical properties of Markov shifts: part II-LLT
• Statistical properties of Markov shifts (part I)
• Moment stability and large deviations for random dynamical systems on non-compact manifolds

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