# Stationary distribution of a DTMC that has recurrent and transient states

I need to construct a stationary distribution for $$\{X_n, n\in \mathcal{N}\}$$ with state space $$E=\{0,1,2,3,4,5\}$$ and transition matrix: $$\begin{bmatrix} 0.8 & 0.2 & 0&0&0&0\\ 0&0.1&0.4&0.5&0&0\\ 0&0.3&0.3&0.4&0&0\\ 0&0.6&0.3&0.1&0&0\\ 0&0&0&0&0.7&0.3\\ 0&0&0&0&0.5&0.5 \end{bmatrix}$$

In the previous steps of this problem I am asked to find the stationary distribution of all positive recurrent classes. For the positive recurrent class $$\{1,2,3\}$$ I was able to find the stationary distribution to be $$1/3$$ for each state, and for $$\{4,5\}$$ to be $$5/8$$ and $$3/8$$, respectively. However, I need to put these distributions together to find the stationary distribution for $$X_n$$ and I am not sure how to do so. This is just a practice problem as I study for my exam. Thank you for any help

States $$\{0,1,2,3\}$$ are isolated from $$\{4,5\}$$, so the stationary distribution isn't unique: any weighted combination of the stationary distributions for the subsystems will be stationary. The probabilities will evolve to a combination of the two, with weights equal to the total probability each subsystem started with. For instance, if the initial probabilities are $$(1/3,1/3,0,0,0,1/3),$$ the limiting probabilities are $$(0,2/9,2/9,2/9,5/24,3/24).$$
• Thank you and you were right about the limiting probabilities of $\{4,5\}$. I corrected my mistake. Dec 24, 2020 at 0:12