Question on the definition of Markov Chain matrices: Is it possible to have an absorbing state (i.e. a state where the probability of returning to itself is 1) within an irreducible set?
I.e., if we know this is an irreducible set:
M = \begin{bmatrix} \frac{5}{6} & \frac{1}{6} & 0 \\[0.3em] \frac{5}{6} & 0 & \frac{1}{6} \\[0.3em] 0 & \frac{5}{6} & \frac{1}{6} \end{bmatrix}
Is this one too?
Z = \begin{bmatrix} \frac{5}{10} & \frac{1}{10} & 0 & \frac{4}{10} \\[0.3em] \frac{5}{10} & 0 & \frac{1}{10} & \frac{4}{10}\\[0.3em] 0 & \frac{5}{10} & \frac{1}{10} & \frac{4}{10}\\ 0&0&0&1\end{bmatrix}
Upon reflection, Matrix Z cannot be irreducible -- the upper left 3x3 matrix is an irreducible set in itself.
Furthermore, if we have an irreducible set, all states within the set must be recurrent (i.e. the probability of revisiting that state is 1). This cannot happen if we have an absorbing state in the matrix.