Showing $QP=Q$ where $P$ is stochastic and $Q$ is the limiting matrix.

Question

Let $P$ be an $N\times N$ stochastic matrix, $Q$ is defined by $q_{i,j}= \lim_{n\to \infty} 1/n \sum_{k=0}^{n-1} p_{i,j}^{(k)}$ where $p_{i,j}^{(k)}$ is the $i,j$ entry of $P^k$. Under certain conditions of ergodicity we have that the limit exists for each $i$ and $j$. Then I am asked to show $QP=Q$. I have tried but am just stuck on one step which I am unable to justify, namely does $\sum_{r=1}^{N} p_{r,j} \lim_{n\to \infty} 1/n \sum_{k=0}^{n-1} p_{i,r}^{(k)}$ = $ \lim_{n\to \infty} 1/n \sum_{k=0}^{n-1} \sum_{r=1}^{N}p_{i,r}^{(k)} p_{r,j}$? Because if I can interchange the sums in this case then I think I can prove it. Thanks in advance!

Answer 1

$$QP=\lim_n\left({1\over n}\sum_{j=0}^{n-1}P^j\right)P =\lim_n\left({1\over n}\sum_{j=0}^{n-1}P^{j+1}\right) =\lim_n\left[\left({1\over n}\sum_{k=0}^{n-1}P^k\right)-\underbrace{{1\over n}(I-P^n)}_{\to\, 0}\right] =Q.$$