Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 3 / 9H (Markov Chains)

States $i,j\in S$ communicate when each is accessible from the other: there exist nonnegative integers $m,n$ such that $P^m(i,j)>0$ and $P^n(j,i)>0$. A communicating class is an equivalence class for this relation.

The period of a state in a Markov chain $a$ isA state is aperiodic when this greatest common divisor is one.

Suppose $a$ and $b$ communicate. Choose $r,s$ with $P^r(a,b)>0$ and $P^s(b,a)>0$. Then $r+s$ is a possible return time to $a$, so $d(a)$ divides $r+s$. If $n$ is any possible return time to $b$, concatenating the paths gives a return to $a$ of length $r+n+s$. Thus $d(a)$ also divides $r+n+s$, and subtraction shows that $d(a)$ divides every such $n$. Therefore $d(a)\mid d(b)$. Interchanging $a$ and $b$ gives the reverse divisibility, soHence period is constant on a communicating class.