5.2 Markov Chain Periodicity

Periodicity of MCs

A positive recurrent Markov chain state $x\in \mathcal{X}$ is periodic if it has a period equal to the greatest common divisor of all of the time steps $n>0$ at which the Markov chain, starting from state $x$, has a positive probability of returning to $x$. Formally,
$d(x)=\gcd \{n\in {\mathbb{Z}}_{+}:P^n(x,x)>0\}.$
If $d(x)=1$, then $x$ is aperiodic.

If all of the states in a Markov chain share a common period $d=d(x),\forall x\in \mathcal{X}$, then the Markov chain is periodic with period $d$. If $d=1$, then the Markov chain is aperiodic.