# Limiting distribution of non-irreducible markov chain

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?

• Did you happen to find the argument for this? I have the same problem.. – Biggiez Oct 27 '18 at 12:51