A Markov chain on states $\{0,1,2,3,4,5,6,7\}$ has the transition probability matrix
$$P= \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0.5 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix}. $$ How can I determine the period of each states of the Markov chain ?
I noticed that it is an irreducible Markov chain. So every state will have the same period. Is there any easy technique to find the period of the given Markov chain with the mentioned transition probability matrix $P$?
For this very special case, note that starting at 0 it must go 0,1,2,3,4 and then from 4 it might go back to 0, making the least $n>0$ with $P_{00}^{(n)}>0$ to be $n=5.$ But again resuming from 4 it might go on to 5, then it must go to 6,7, then 0. That means that also $P_{00}^{(8)}>0.$ The gcd of 5 and 8 is 1, so if I'm following the definition right it seems here the period is 1.