Asymptotic properties of a singularly perturbed Markov chain with inclusion of transient states

This work is concerned with aggregations in a singularly perturbed Markov chain having a finite state space and fast and slow motions.The state space of the underlying Markov chain can be decomposed into several groups of recurrent states and a group of transient states.The asymptotic properties are studied through sequences of unscaled and scaled occupation measures.By treating the states within each recurrent class as a single state, an aggregated process is defined and shown to be convergent to a limit Markov chain.In addition, it is shown that a sequence of suitably rescaled occupation measures converges to a switching diffusion process weakly.