91Ó°ÊÓ

Consider the Markov chain with general \(2 \times 2\) transition matrix $$ \mathbf{P}=\left(\begin{array}{cc} 1-a & a \\ b & 1-b \end{array}\right) $$ (a) Under what conditions is \(\mathbf{P}\) absorbing? (b) Under what conditions is \(\mathbf{P}\) ergodic but not regular? (c) Under what conditions is \(\mathbf{P}\) regular?

Show that any ergodic Markov chain with a symmetric transition matrix (i.e., \(\left.p_{i j}=p_{j i}\right)\) is reversible.