Reduction of Markov chains with two-time-scale state transitions

In this paper, we consider a general class of two-time-scale Markov chains whose transition rate matrix depends on a parameter lambda > 0. We assume that some transition rates of the Markov chain will tend to infinity as lambda -> infinity. We divide the state space of the Markov chain X into a fast state space and a slow state space and define a reduced chain Y on the slow state space. Our main result is that the distribution of the original chain X will converge in total variation distance to that of the reduced chain Y uniformly in time t as lambda -> infinity.