G. Yin, et al., Occupation Measures of Singularly Perturbed Markov Chains with Absorbing States, Acta mathematica Sinica. English series (Print) , 16(1), 2000, pp. 161-180
This paper develops asymptotic properties of singularly perturbed Markov chains with inclusion of absorbing states. It focuses on both unscaled and scaled occupation measures. Under mild conditions, a mean-square estimate is obtained. By averaging the fast components, we obtain an aggregated process. Although the aggregated process itself may be non-Markovian, its weak limit is a Markov chain with much smaller state space. Moreover, a suitably scaled sequence consisting of a component of scaled occupation measures and a component of the aggregated process is shown to converge to a pair of processes with a switching diffusion component.