Any ideas on how to prove the following statement?
Let $P$ be an irreducible transition matrix. $\nu$ an initial distribution and $\displaystyle\pi=\lim_{n\rightarrow\infty} \nu P^n$ and denote $A_n:=\frac{1}{n}\sum_{m=0}^{n-1} P^m$.
Assume that $j$ is positive recurrent, and set $C:=\{i:i\leftrightarrow j\}$. Given a probability vector $\mu$ with the property that $ \sum_{i\notin C}(\mu)_i=0$, show that, in general, $(\mu A_n)_i\rightarrow\pi_{ii}$ and, when $j$ is aperiodic, $(\mu P^n)_i \rightarrow \pi_{ii}$ for each $i\in C$.
You can find the question in: "Introduction to Markov Processes" by Daniel W. Stroock. Second Edition, section 4.2 exercise 4.2.5 or in other editions in Chapter 3, exercise section.