We consider finite state nonhomogeneous Markov chains with one-step transition probabilities roughly proportional to powers of a small parameter, converging to zero. We examine asymptotic properties of trajectories. The analysis is based on the so-called orders of recurrence. Transient states, recurrent classes and periodic subclasses can be identified in terms of the matrix of powers. This leads to a complete description of the tail sigma field. Our theorems generalize the classical results for homogeneous chains and can also be applied to chains generated by stochastic algorithms of the "simulated annealing" type.
"Tail Events of some Nonhomogeneous Markov Chains." Ann. Appl. Probab. 5 (1) 261 - 293, February, 1995. https://doi.org/10.1214/aoap/1177004840