Let P be the transition matrix $$ P = \begin{bmatrix} 0 & 0.2 & 0.2 & 0.2& 0.2 & 0.2 \\ 0.2 & 0 & 0.2 & 0.2 & 0.2 &0.2 \\ 0.2 & 0.2 & 0 & 0.2 & 0.2 & 0.2 \\ 0.2 & 0.2 & 0.2 & 0 & 0.2 & 0.2 \\ 0.2 & 0.2 & 0.2 & 0.2 & 0 & 0.2 \\ 0.2 & 0.2 & 0.2 & 0.2 & 0.2 & 0 \\ \end{bmatrix}. $$
I notice that $P_{i,i}=0$ for $i\in\{1,2,3,4,5,6\}$ Does this imply that a limiting distribution does not exist for transition matrix P?
This is a doubly stochastic matrix - notice that both the rows AND columns to 1. It is easy to verify that $$\pi = \left(\frac16, \frac16, \frac16, \frac16, \frac16, \frac16\right) $$ is a stationary distribution (i.e. satisfies $\pi=\pi P$).