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This is the picture of one of the problems of my coursework.

With regards to the question in the above picture and the markov chain drawn in the question, my query is whether is it possible to conclude from Ergodic theorem that this Markov chain has an invariant stationary distribution ?

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You have a finite chain, in which case there is always a stationary distribution. In this case it looks like your chain is irreducible after removing vertex "4", so there's going to be a unique stationary distribution.

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  • $\begingroup$ But why should I remove vertex "4", sir ? Does this mean that there is a problem with the question itself ? Do I have to contact my professor to clarify about this ? $\endgroup$ – Dwaipayan Gupta Aug 2 '16 at 5:54
  • $\begingroup$ You're not removing vertex 4, it's just that vertex 4 only has outgoing paths hence it does not form a connected component with the rest of the vertices. The point is that in this case the stationary distribution will have a 0 component on vertex 4. $\endgroup$ – Alex R. Aug 2 '16 at 15:23
  • $\begingroup$ there is a path from 5 to 1 but not vice-versa, Then {1,2,3,5} is not a communicating class. Then how can i apply Ergodic theorem ? $\endgroup$ – Dwaipayan Gupta Aug 2 '16 at 16:23
  • $\begingroup$ Correct so in this case you can also remove 5. $\endgroup$ – Alex R. Aug 2 '16 at 16:27

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