Difference between revisions of "Markov chain, periodic"
From Encyclopedia of Mathematics
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{{MSC|60J10}} | {{MSC|60J10}} | ||
[[Category:Markov chains]] | [[Category:Markov chains]] | ||
| − | A non-decomposable homogeneous [[Markov chain|Markov chain]] | + | A non-decomposable homogeneous [[Markov chain|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|Markov chain, non-decomposable]]) all states have the same period. If $ d = 1 $, | ||
| + | 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
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