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?

  • $\begingroup$ 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. $\endgroup$ – gung - Reinstate Monica Jan 3 '19 at 14:18

I assume it is a self-study type of question, so here is a hint-answer.

Check the definition of periodicity carefully (from wikipedia):

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.


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