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Joint Applied Math and Probability Seminar
Large Deviations, Multiplicative Ergodicity, and Spectral Theory for Markov Chains
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Precise limit theorems and structural results are developed for Markov chains on general state spaces, under natural and often minimal assumptions. We consider the class of "multiplicatively regular" chains, which are characterized by a new Lyapunov drift criterion for the nonlinear generator. This criterion is intimately related to (but weaker than) the classical Donsker-Varadhan assumptions. For such Markov chains, we develop a "multiplicative" ergodic theory in close analogy to the classical "additive" theory. Using analytical tools we first show that the transition kernel and a related family of linear operators have a purely discrete spectrum in an appropriate Banach space, and we construct maximal, well-behaved solutions for the multiplicative Poisson equation. This structure is then exploited to prove probabilistic limit theorems. We establish a large deviations principle (LDP) for the empirical measures, in a topology stronger than the usual tau topology, and with rate-function given explicitly in terms of relative entropy. Moreover, this LDP can be refined to a precise expansion similar to the Bahadur-Ranga Rao expansion for independent random variables. Links with potential applications in stochastic networks, linear systems, and simulation may be mentioned. |