View source for Markov chain, decomposable

{{MSC|60J10|60J27}}

[[Category:Markov processes]]

A [[Markov chain|Markov chain]] whose [[Transition probabilities|transition probabilities]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062380/m0623801.png" /> have the following property: There are states <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062380/m0623802.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062380/m0623803.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062380/m0623804.png" />. Decomposability of a Markov chain is equivalent to decomposability of its matrix of transition probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062380/m0623805.png" /> for discrete-time Markov chains, and of its matrix of transition probability densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062380/m0623806.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062380/m0623807.png" />, for continuous-time Markov chains. 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|Markov chain]]).



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====References====
{|
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