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 $λ>0$. We assume that some transition rates of the Markov chain will tend to infinity as $λ\rightarrow\infty$. 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 $λ\rightarrow\infty$.

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