Markov chain, periodic

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[edit]

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