Again a probability exercise:
Let $X=U \cup V$ be the finite state space of a Markov chain, where $U$ and $V$ are disjoint subsets of $X$ and $p_{ij}=0$ if both $i,j \in U$ or both $i,j \in V$. Otherwise, they are positive. Find a stationary distribution?
I'm assuming that it has multiple distributions but I wrote down the system that I'm trying to solve and either I'm really tired or I should be thinking of smth else: if $\pi$ is the stationary distribution then I'm trying to find we have $\pi_{j} = \sum_{i \in V} {\pi_{i} p_{ij}}$ for all $j \in U$ and also $\pi_{j} = \sum_{i \in U} {\pi_{i} p_{ij}}$ for all $j \in V$... Can someone please help again?
Thanks!