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# Large Deviations for a Class of Nonhomogeneous Markov Chains

Zach Dietz and Sunder Sethuraman
The Annals of Applied Probability
Vol. 15, No. 1A (Feb., 2005), pp. 421-486
Stable URL: http://www.jstor.org/stable/30038320
Page Count: 66
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## Abstract

Large deviation results are given for a class of perturbed nonhomogeneous Markov chains on finite state space which formally includes some stochastic optimization algorithms. Specifically, let ${P_{n}}$ be a sequence of transition matrices on a finite state space which converge to a limit transition matrix P. Let ${X_{n}}$ be the associated nonhomogeneous Markov chain where $P_{n}$ controls movement from time n - 1 to n. The main statements are a large deviation principle and bounds for additive functionals of the nonhomogeneous process under some regularity conditions. In particular, when P is reducible, three regimes that depend on the decay of certain "connection" $P_{n}$ probabilities are identified. Roughly, if the decay is too slow, too fast or in an intermediate range, the large deviation behavior is trivial, the same as the time-homogeneous chain run with P or nontrivial and involving the decay rates. Examples of anomalous behaviors are also given when the approach $P_{n} \rightarrow P$ is irregular. Results in the intermediate regime apply to geometrically fast running optimizations, and to some issues in glassy physics.

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