In an excercise I'm given the following matrix of a markov chain $\begin{pmatrix} 0&1&0 \\ 1/2 &1/2 & 0 \\ 0 & 1/2 &1/2 \end{pmatrix}$. It has the stationary distribution $\pi=(1/3,2/3,0)$ (which I think is unique).
In the answer to the problem it says that the limiting distribution aproaches the stationary independently of the initial distribution.
The only theorem that I know that talks about this is the "ergodicity theorem"(if the chain is ergodic then $p(n)\rightarrow \pi$ indpendently of the initial distribution). However it doesn't seem to me that this chain is ergodic; since no state communicates with the last state it isn't irreducible and therefore not ergodic (I think?).
What is the argument for the limiting distribution approaching the stationary independently of the initial distribution?
I think, there are two things that you should check. Firstly, you should be looking if a Markov chain is periodic or not. Secondly, check if the markov chain has only one minimal subset. If it is not periodic and it has only one minimal subset, which is the case for this problem, you can have a unique steady state, regardless of the initial distribution.