A markov chain has an initial distribution $u^{(0)}$ = {1/6 1/2 1/3} and the following transition matrix
P= $\left(\begin{matrix}0&&0.5&&0.5\\0.5&&0&&0.5\\0.5&&0.5&&0\end{matrix}\right)$
Find its stationary distribution. Is it unique? Verify that the limiting distribution of the chain is stationary.
Since the matrix is doubly stochastic, the stationary distribution straight away works out to
($\pi_1$ $\pi_2$ $\pi_3$)= (1/3 1/3 1/3).... and also the distribution is unique.
But who do we verify that the limiting distribution is stationary?