# Markov chain, periodic

From Encyclopedia of Mathematics

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=51452

This article was adapted from an original article by V.P. Chistyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article