# Can a state have zero periodicity? [closed]

I am getting my concepts cleared in Stochastic process. I understand the concept of periodicity. Just to make it clear, suppose there is a finite Markov chain with states $$1,2,3$$. Let their transition probability matrix be $$\begin{bmatrix} 0 & 1 & 0 \\ 1& 0 & 0 \\ 0& 1& 0 \end{bmatrix}$$

In this case, what is the periodicity of state 3?

• Please add the [self-study] tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. – gung - Reinstate Monica Jan 3 '19 at 14:18

A state $$i$$ has period $$k$$ if any return to state $$i$$ must occur in multiples of $$k$$ time steps. Formally, the period of a state $$i$$ is defined as
$${\displaystyle k=\gcd\{n>0:P(X_{n}=i\mid X_{0}=i)>0\}}$$
provided that this set is not empty. Otherwise the period of a state $$i$$ is not defined.
Then, using your transition matrix, compute $$P(X_n = 3 | X_0 = 3)$$ for any $$n$$, and see where the definition takes you.