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I am trying to figure out if this Markov chain is irreducible and if it is aperiodic and why or why not.

enter image description here

For me it is not irreducible Markov chain because you cannot go from state 2, 5 or 6 to any of the states 1, 3, 4. I don't know if my argument is good.

Pepto
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You totally got it... a drawing helps:

Markov chain

It’s not irreducible, {1, 4}, {3} and {2, 5, 6} are the communication classes, and {2, 5, 6} is an absorbing class...

Elvis
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    There is a mistake in the drawing I guess. In the drawing from state 3 it goes to state 1 with probability 1 and in the matrix it leads from state 3 to state 2 with probability 1. – Pepto Nov 27 '13 at 14:55
  • woups... I corrected this. Your argument is still correct... – Elvis Nov 27 '13 at 16:36
  • Thank you for help :) If you could also help me with the following problem I would be glad. I have to determine periodicity of the states of the chain and I think that the chain is aperiodic but I don't know how to argument it. Mabye there is some calculation that I can do to prove it? I also think that state one can have a period 2 because it can be visited only at the multiple of two moves. Is that correct reasoning? And the last thing. To determine stationary distribution should I first reduce the chain to states 2, 5, 6? Without such a reduction I cannot solve balanced equations. – Pepto Nov 27 '13 at 17:01
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    Pepto, your question in the comment there goes far beyond clarification, to a substantive additional question. It's a bit much to ask Elvis for one-on-one responses; you should instead address such questions to the whole site - that is, you should either edit your original question to reflect what you meant to post to begin with, or post a new question (with, if necessary, a link back to this one). – Glen_b Nov 27 '13 at 17:42
  • sorry for that and thank you for reminding me. I have posted it now as a separate question. – Pepto Nov 27 '13 at 17:56