Communicating class

A communicating class of a Markov chain is open when a state in the class has positive probability of transitioning to a state outside it. Once a path leaves, it cannot return to the class: a return path would make the outside state communicate with the class. Every open class consists of transient states. Conversely, a finite transient communicating class is open; an infinite closed communicating class can be transient.

Two states of a Markov chain communicate when each can be reached from the other with positive probability in finitely many steps.

A communicating class is closed when no transition from it can reach a state outside it.

In a finite chain, a communicating class that is not closed is transient: after it is left, it need not be visited again and is eventually left almost surely.

The period of a state $i$ is the greatest common divisor of its positive-probability return times:A state is aperiodic when its period is one.

If communicating states $i,j$ can be joined in $r$ and $s$ steps, then $r+s$ is a return time to $i$, and adjoining any return loop of length $n$ at $j$ makes $r+n+s$ another. Their difference shows $d(i)\mid n$, hence $d(i)\mid d(j)$; symmetry gives equality.