Markov Chain with two components

Question

I am trying to understand a question with the following Markov Chain:

As can be seen, the chain consists of two components. If I start at state 1, I understand that the steady-state probability of being in state 3 for example is zero, because all states 1,2,3,4 are transient. But what I do not understand is that is it possible to consider the second component as a separate Markov chain? And would it be correct to say that the limiting probabilities of the second chain considered separately exist? For example, if I start at state 5, then can we say that the steady-state probabilities of any of the states in the right Markov chain exist and are positive?

Answer 1

Yes, you can. Actually this Markov chain is reducible, with two communicating classes (as you have correctly observed),

1. $C_1=\{1,2,3,4\}$, which is not closed and therefore any stationary distribution assigns zero probability to it and
2. $C_2=\{5,6,7\}$ which is closed.

As stated for example in this answer,

Every stationary distribution of a Markov chain is concentrated on the closed communicating classes.

In general the following holds

Theorem: Every Markov Chain with a finite state space has a unique stationary distribution unless the chain has two or more closed communicating classes.

Note: If there are two or more communicating classes but only one closed then the stationary distribution is unique and concentrated only on the closed class.

So, here you can treat the second class as a separate chain but you do not need to. No matter where you start you can calculate the steady-state probabilities and they will be concentrated on the class $C_2$.