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Large deviations for a class of nonhomogeneous Markov chains | | Zach Dietz
; Sunder Sethuraman
; | | Date: |
12 Apr 2004 | | Journal: | Annals of Applied Probability 2005, Vol. 15, No. 1A, 421-486 DOI: 10.1214/105051604000000990 | | Subject: | Probability MSC-class: 60J10 (Primary) 60F10. (Secondary) | math.PR | | 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 o P is irregular. Results in the intermediate regime apply to geometrically fast running optimizations, and to some issues in glassy physics. | | Source: | arXiv, math.PR/0404230 | | Services: | Forum | Review | PDF | Favorites | |
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