Period of a state i in a Markov chain

Question

I'm using the book titled, Essential of Stochastic Process by Durret which is known to be rampant with errors.

Here is a part of the text which is confusing me:

and I'd like to know if this is an error on the author's part or due to my lack of understanding.

Definition:

The period of a state i in a Markov chain is some largest number which divides the number of step n such that $p^{n}\left (i,i \right ) \forall n\geq 1$.

In the example given below, I fail to observe why $p^{n}\left (0,0 \right )>0$... From the table, $p\left (0,0 \right )=0$ and any exponent on this is 0....

Any clarification is appreciated.

Answer 1

What you want to notice is that If you go to the left, you will be back at $0$ in 3 steps. If you go to the right, you will be back at $0$ in 4 steps. Since these happen with equal probability, we must have $p^3(0,0) = 1/2 = p^4(0,0)$.

If you compute the matrix powers properly, you will see this is indeed the case.