Let (left{X_{0}, X_{1}, ldots ight}) be an irreducible Markov chain with state space (mathbf{Z}={1,2, ldots, n}), (n

Let \(\left\{X_{0}, X_{1}, \ldots\right\}\) be an irreducible Markov chain with state space \(\mathbf{Z}=\{1,2, \ldots, n\}\), \(n<\infty\), and with the doubly stochastic transition matrix \(\mathbf{P}=\left(\left(p_{i j}\right)\right)\), i.e.,

\[\sum_{j \in \mathbf{Z}} p_{i j}=1 \text { for all } i \in \mathbf{Z} \text { and } \sum_{i \in \mathbf{Z}} p_{i j}=1 \text { for all } j \in \mathbf{Z}\]

(1) Prove that the stationary distribution of \(\left\{X_{0}, X_{1}, \ldots\right\}\) is \(\left\{\pi_{j}=1 / n, j \in \mathbf{Z}\right\}\).

(2) Can \(\left\{X_{0}, X_{1}, \ldots\right\}\) be a transient Markov chain?