Given this stochastic matrix:
$$ \begin{matrix} 0 & 1/3 & 2/3 \\ 2/3 & 0 & 1/3 \\ 1/2 & 1/2 & 0 \\ \end{matrix} $$
I'm trying to determine it's period. My line of thinking so far has been to draw the transition diagram, and see that every state interacts with every other state, so it's an irreducible MC.
And then in terms it's period I'm guessing it's aperiodic(??)
Because you can go from state 1 back to state 1 in {2,3,4,...} steps, and because the chain is irreducible, the period will be the same for all states. So d(1)=d(2)=d(3)=1.
But I have an unconfirmed solution that says it's got a period of 2, which I'm not able to see in this case. Can someone help clear this up? Thanks!
You are correct, this is aperiodic. As soon as you find two cycles in a class whose periods are coprime (e.g. the $2$-cycle $(1,2)$ and the $3$-cycle $(1,2,3)$), you know that class is aperiodic.