Markov chain periodicity

Question

Can a Markov chain have 5 states, one open and one closed class and all the states be periodic (e.g. period 2)?

I tried the following: https://www.dropbox.com/s/v818oqlizaci23m/Untitled.png?dl=0

but I have doubts for the transient state. If it will move from the left part to the right side, then it will never come back, meaning that the class on the left is not periodic. Is this a possible MC at all?

Answer 1

The chain you drew is a correct example of what you were trying to do. In the second paragraph of the question, you seem to be working with a wrong idea of periodicity. Periodicity in the context of Markov chains does not refer to periodicity of the occupation probabilities of the states as a function of time (which is, as you rightly point out, not given due to the "leakage" from the open to the closed class). It requires not that the process always return to the same state after the period, only that if it does return, this can only occur after the period or a multiple thereof, which is the case in your example.

Answer 2

Assume we have the following image.

Firstly, yes, it can represent a graph of a Markov Chain.

We have the 2 communicating classes $A = \{1,2,3\}$ and $B=\{4,5\}$. All states in $A$ are transitive, while all states in $B$ are positive recurrent.

Also, it is true that all the states that belong to the same class have the same period. Clearly, the period of the class $A$ is $d_A = 2$ and the period of class $B$ is $d_B=2$. It is totally accidental that the two classes have the same period.