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Spectral Theory and Limit Theorems for Geometrically Ergodic Markov Processes | | Ioannis Kontoyiannis
; Sean Meyn
; | | Date: |
16 Sep 2002 | | Subject: | Probability; Spectral Theory MSC-class: 60J10; 60F10; 34L40; 60J25; 41A36 | math.PR math.SP | | 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. | | Source: | arXiv, math.PR/0209200 | | Services: | Forum | Review | PDF | Favorites | |
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