I have the following stochastic matrix with $p_{ij} > 0$ and $\sum_j p_{ij} = 1$
$$ P = \begin{bmatrix} p_{11} & p_{12} & 0 & 0 & 0 & 0 \\ p_{21} & p_{22} & p_{23} & 0 & 0 & 0 \\ 0 & p_{32} & p_{33} & p_{34} & 0 & 0 \\ 0 & 0 & p_{43} & p_{44} & p_{45} & 0 \\ 0 & 0 & 0 & p_{54} & p_{55} & p_{56} \\ 0 & 0 & 0 & 0 & p_{65} & p_{66} \end{bmatrix} $$
Now this stochastic matrix has a limiting distribution because it is irreducible, it is possible to get to any state from any state, and it is aperiodic, $(P^1)_{ii} > 0$.
Now I have the following stochastic distribution.
$$ P = \begin{bmatrix} p_{11} & p_{12} & 0 & 0 & 0 & 0 \\ p_{21} & p_{22} & p_{23} & 0 & 0 & 0 \\ 0 & p_{32} & p_{33} & p_{34} & 0 & 0 \\ 0 & 0 & p_{43} & p_{44} & p_{45} & 0 \\ 0 & 0 & 0 & p_{54} & p_{55} & p_{56} \\ 1 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} $$
While $P$ is irreducible it is not aperiodic, its period is not $1$ but $6$ because $(P^6)_{ii} > 0$.
I want to prove that the stochastic matrix also has a limiting distribution. How do I do this?