Markov chains: understanding irreducible and period definitions on example

Why is this Markov chain not a counter example to irreducible implying all states have the same period?

\begin{align} (p_{ij}) &= \begin{pmatrix} 0 & \frac{1}{2}& \frac{1}{2}\\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3}\\ \frac{1}{4} & \frac{1}{4} &\frac{1}{2} \end{pmatrix} \end{align}

This Markov chain jumps around $\{1,2,3 \}$, and can move through states 2 and 3 into any other state, so is irreducible. However, $p^{(1)}_{11} = 0$ while $p^{(1)}_{22}$ and $p^{(1)}_{33}$ are both greater than 0 so it seems state 1 has a different period to states 2 and 3?

1 Answer

All states have period $1$ in your example. Check the definition of period. It's true that $p^{(1)}_{11} = 0$, but $p^{(n)}_{11} > 0$ for all $n \ge 2$, so the period is $1$.