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**Summary:**Determine the absorbing states & communication classes of the given matrix.

Hello everyone,

If we have a state space of S = {1,2,3,4} and the following matrix:

\begin{bmatrix}

0 & 1 & 0 & 0\\

0 & 0 & 1/3 & 2/3\\

1 & 0 & 0 & 0\\

0 & 1/2 & 1/2 & 0\\

\end{bmatrix}

Now, given the above, I don't think there are any absorbing states or sets of states, is that correct?

And since the above is a finite and irreducible closed set of states, then all of the states are recurrent and there are no transient states, right?

Also, since all the states communicate with each other, then the communication class is simply all of the state space, right?

Finally, not directly related to the above matrix, but just in general, if we don't have a limit law, then does that imply that we don't have a stationary distribution?

Please correct me if I am wrong, thanks.

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