Do $r_{ij}(n)$ converge to some $\pi_{j}$ (independent of i) ? where $\pi_{j}$ denotes the steady state probability of occupancy of state $j$, or $P(X_{n} = j)$ for large $n$.
Yes if,
Assuming yes, \begin{align} r_{ij}(n) &= \sum_{k} r_{ik}(n-1)p_{kj}\newline \lim_{n \to \infty} r_{ij}(n) &= \lim_{n \to \infty} \sum_{k} r_{ik}(n-1)p_{kj}\newline \pi_{j} &= \sum_{k} \pi_{k} p_{kj} \quad\text{balance equations} \newline \mbox{and,} \sum_{i} \pi_{i} &= 1 \newline \text{frequency of transitions $k \rightarrow j$} &= \pi_{k} p_{kj} \quad\text{in one step}\newline \text{frequency of transitions into $j$} &= \sum_{k} \pi_{k} p_{kj} \quad\text{influx from all connected states} \end{align}
The $pi_{j}$ sum up to 1 and form a probability distribution called the stationary distribution of the chain (because if the initial distribution $P(X_{0} = j) = \pi_{j}$, the occupancy distribution of the states is constant for all steps and can be verified using total probability theorem on any of the nodes).
In the steady state,