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Difference between revisions of "Markov chain, periodic"

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(MSC|60J10 Category:Markov chains)
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[[Category:Markov chains]]
 
[[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,
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A non-decomposable homogeneous [[Markov chain|Markov chain]] $  \xi ( n) $,
 
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$  n = 1 , 2 \dots $
<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>
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in which each state $  i $
 
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has period larger than 1, that is,
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.
 
  
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$$
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d _ {i}  =  \textrm{ g } . \textrm{ c } . \textrm{ d } .
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\{ {n } : { {\mathsf P} \{ \xi ( n) = i \mid  \xi ( 0) = i \}
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> 0 } \}  >  1 .
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$$
  
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In a non-decomposable Markov chain (cf. [[Markov chain, non-decomposable|Markov chain, non-decomposable]]) all states have the same period. If  $  d = 1 $,
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then the Markov chain is called aperiodic.
  
 
====Comments====
 
====Comments====
 
Cf. also [[Markov chain|Markov chain]] and [[Markov chain, decomposable|Markov chain, decomposable]] for references.
 
Cf. also [[Markov chain|Markov chain]] and [[Markov chain, decomposable|Markov chain, decomposable]] for references.

Revision as of 07:59, 6 June 2020


2020 Mathematics Subject Classification: Primary: 60J10 [MSN][ZBL]

A non-decomposable homogeneous Markov chain $ \xi ( n) $, $ n = 1 , 2 \dots $ in which each state $ i $ has period larger than 1, that is,

$$ d _ {i} = \textrm{ g } . \textrm{ c } . \textrm{ d } . \{ {n } : { {\mathsf P} \{ \xi ( n) = i \mid \xi ( 0) = i \} > 0 } \} > 1 . $$

In a non-decomposable Markov chain (cf. Markov chain, non-decomposable) all states have the same period. If $ d = 1 $, then the Markov chain is called aperiodic.

Comments

Cf. also Markov chain and Markov chain, decomposable for references.

How to Cite This Entry:
Markov chain, periodic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_chain,_periodic&oldid=21654
This article was adapted from an original article by V.P. Chistyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article