Im having difficulty understanding when exactly a stationary distribution can occur for a Markov chain's transition matrix.
Can a stationary distribution occur for any states in the Markov chain as long as not all of them are transient?
I have a 4x4 matrix:
0.75 0.25 0 0
0.5 0.5 0 0
0.3 0 0.3 0.3
0.16 0.16 0.3 0.3
I apologise for the lack of latex, I'm typing from my phone unfortunately. 0.3 is a third, 0.16 is a sixth, and the others are self explanatory.
So here I assume that classes one and two are recurrent and the other two are transient. Hence we have one class C={1,2} which is recurrent. So as far as I can tell, we can have a unique stationary distribution.
So for this I got my stationary distribution as (0.3) * (2,1,0,0).
This seems to make sense since states three and four are transient. Does this mean that the n step transition matrix will now tend to the matrix consisting of rows of this stationary distribution?