View source for Markov chain, periodic

{{MSC|60J10}}

[[Category:Markov chains]]

A non-decomposable homogeneous [[Markov chain|Markov chain]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062420/m0624201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062420/m0624202.png" /> in which each state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062420/m0624203.png" /> has period larger than 1, that is,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062420/m0624204.png" /></td> </tr></table>

In a non-decomposable Markov chain (cf. [[Markov chain, non-decomposable|Markov chain, non-decomposable]]) all states have the same period. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062420/m0624205.png" />, then the Markov chain is called aperiodic.



====Comments====
Cf. also [[Markov chain|Markov chain]] and [[Markov chain, decomposable|Markov chain, decomposable]] for references.