### 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} → P is irregular. Results in the intermediate regime apply to geometrically fast running optimizations, and to some issues in glassy physics.

Original language | English (US) |
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Pages (from-to) | 421-486 |

Number of pages | 66 |

Journal | Annals of Applied Probability |

Volume | 15 |

Issue number | 1 A |

DOIs | |

State | Published - Feb 1 2005 |

Externally published | Yes |

### Keywords

- Geometric cooling
- Glassy models
- Large deviations
- Markov
- Nonhomogeneous
- Optimization

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Annals of Applied Probability*,

*15*(1 A), 421-486. https://doi.org/10.1214/105051604000000990