# Markov chain, periodic

2010 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.