Lumpability - Definition

Definition

Suppose that the complete state-space of a Markov chain is divided into disjoint subsets of states, where these subsets are denoted by ti. This forms a partition of the states. Both the state-space and the collection of subsets may be either finite or countably infinite. A continuous-time Markov chain is lumpable with respect to the partition T if and only if, for any subsets ti and tj in the partition, and for any states n,n’ in subset ti,

where q(i,j) is the transition rate from state i to state j.

Similarly, for a stochastic matrix P, P is a lumpable matrix on a partition T if and only if and only if, for any subsets ti and tj in the partition, and for any states n,n’ in subset ti,

where p(i,j) is the probability of moving from state i to state j.

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