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