Consider the one-step transition matrix

$\displaystyle P = \begin{pmatrix} q & p & 0 \\0 & q & p \\ p & 0 & q \\ \end{pmatrix} $

where $\displaystyle q = 1 - p$ and $\displaystyle 0 < p < 1$.

Find all stationary distributions of a chain with state space $\displaystyle E = \{0,1,2\}$ and one-step transition matrix P.

Are any stationary distributions also limiting distributions?

Thanks