Difference between revisions of "Markov chain, non-decomposable"
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A [[Markov chain|Markov chain]] whose transition probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062410/m0624101.png" /> have the following property: For any states <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062410/m0624102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062410/m0624103.png" /> there is a time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062410/m0624104.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062410/m0624105.png" />. The non-decomposability of a Markov chain is equivalent to non-decomposability of its matrix of transition probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062410/m0624106.png" /> for a discrete-time Markov chain, and of its matrix of transition probability densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062410/m0624107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062410/m0624108.png" /> for a continuous-time Markov chain. The state space of a non-decomposable Markov chain consists of one class of communicating states (cf. [[Markov chain|Markov chain]]). | A [[Markov chain|Markov chain]] whose transition probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062410/m0624101.png" /> have the following property: For any states <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062410/m0624102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062410/m0624103.png" /> there is a time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062410/m0624104.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062410/m0624105.png" />. The non-decomposability of a Markov chain is equivalent to non-decomposability of its matrix of transition probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062410/m0624106.png" /> for a discrete-time Markov chain, and of its matrix of transition probability densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062410/m0624107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062410/m0624108.png" /> for a continuous-time Markov chain. The state space of a non-decomposable Markov chain consists of one class of communicating states (cf. [[Markov chain|Markov chain]]). | ||
Revision as of 20:22, 9 March 2012
2020 Mathematics Subject Classification: Primary: 60J10 Secondary: 60J27 [MSN][ZBL]
A Markov chain whose transition probabilities 
have the following property: For any states 
and 
there is a time 
such that 
. The non-decomposability of a Markov chain is equivalent to non-decomposability of its matrix of transition probabilities 
for a discrete-time Markov chain, and of its matrix of transition probability densities 
, 
for a continuous-time Markov chain. The state space of a non-decomposable Markov chain consists of one class of communicating states (cf. Markov chain).
Comments
Cf. also Markov chain and Markov chain, decomposable for references.
How to Cite This Entry:
Markov chain, non-decomposable. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_chain,_non-decomposable&oldid=21653
Markov chain, non-decomposable. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_chain,_non-decomposable&oldid=21653
This article was adapted from an original article by B.A. Sevast'yanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article