Difference between revisions of "Markov chain, periodic"
From Encyclopedia of Mathematics
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| − | + | {{MSC|60J10}} | |
| − | + | [[Category:Markov chains]] | |
| + | 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, | ||
| + | $$ | ||
| + | d _ {i} = \textrm{gcd}\{ {n } : { {\mathsf P} \{ \xi ( n) = i \mid \xi ( 0) = i \} | ||
| + | > 0 } \} > 1 . | ||
| + | $$ | ||
| + | |||
| + | 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. | ||
Latest revision as of 13:47, 20 January 2021
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,
$$ d _ {i} = \textrm{gcd}\{ {n } : { {\mathsf P} \{ \xi ( n) = i \mid \xi ( 0) = i \} > 0 } \} > 1 . $$
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=11752
Markov chain, periodic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_chain,_periodic&oldid=11752
This article was adapted from an original article by V.P. Chistyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article