Spectral Theory and Limit Theorems for Geometrically Ergodic Markov Processes

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dc.creatorKontoyiannis, Ioannis
dc.creatorMeyn, Sean
dc.date2002-09-16
dc.date.accessioned2026-07-07T04:50:55Z
dc.date.available2026-07-07T04:50:55Z
dc.descriptionConsider 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.
dc.description52 pages, 1 figure, to appear, Annals of Applied Probability
dc.identifierhttps://arxiv.org/abs/math/0209200
dc.identifierhttp://arxiv.org/abs/math/0209200
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/64964
dc.subjectProbability
dc.subjectSpectral Theory
dc.subject60J10; 60F10; 34L40; 60J25; 41A36
dc.titleSpectral Theory and Limit Theorems for Geometrically Ergodic Markov Processes
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