Markov chain, decomposable
2010 Mathematics Subject Classification: Primary: 60J10 Secondary: 60J27 [MSN][ZBL]
A Markov chain whose transition probabilities $p_{ij}(t)$ have the following property: There are states $i,j$ such that $p_{ij}(t) = 0$ for all $t \ge 0$. Decomposability of a Markov chain is equivalent to decomposability of its matrix of transition probabilities $P = \left( {p_{ij}} \right)$ for a discrete-time Markov chain, and of its matrix of transition probability densities $Q = \left( {p'_{ij}(0)} \right)$ for a continuous-time Markov chain. The state space of a decomposable Markov chain consists either of inessential states or of more than one class of communicating states (cf. Markov chain).
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References
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Markov chain, decomposable. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_chain,_decomposable&oldid=39530