Lumpable Markov chain

In probability theory, lumpability is a method for reducing the size of the state space of some continuous-time Markov chains, first published by Kemeny and Snell.

Suppose that the complete state-space of a Markov chain is divided into disjoint subsets of states, where these subsets are denoted by t_i. This forms a partition ${\displaystyle \scriptstyle {T=\{t_{1},t_{2},\ldots \}}}$ of the states. Both the state-space and the collection of subsets may be either finite or countably infinite. A continuous-time Markov chain ${\displaystyle \{X_{i}\}}$ is lumpable with respect to the partition T if and only if, for any subsets t_i and t_j in the partition, and for any states n,n’ in subset t_i,

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