Consider a homogeneous Markov chain $\{X_n\}_{n\in\mathbb N}$ with a countable (but not necessarily finite) state space $S$. Suppose that there exists a limit distribution $\pi$, namely: $$\pi(j)=\lim_{n\to\infty} P(X_n=j)\qquad\forall j\in S$$ I want to prove that $\pi$ is an invariant distrubution, that is the following equation must be satisfied:
$$\pi(j)=\sum_{i\in S}\pi(i)P_{ij}\quad\textrm{where}\quad P_{ij}:=P(X_{n+1}=j|X_n=i)$$
The following proof of the above statement has a mysterious passage for me:
$$\pi(j)=\lim_{n\to\infty} P(X_n=j)=\lim_{n\to\infty}\sum_{i\in S}P(X_n=i)P_{ij}=$$ $$=\sum_{i\in S}\left(\lim_{n\to\infty} P(X_n=i)\right)P_{ij}=\sum_{i\in S}\pi(i)P_{ij}$$
I don't understand why it is possible to exchange the limit and the summation! I some books I read "thanks to dominated convergence theorem" but I can't realize how to apply this result.
Thanks in advance.
You know that $\lim\limits_{n \to \infty} P(X_n=i) = \pi(i)$ and that $\sup_n P(X_n = i)P_{ij} = \sup_n |P(X_n = i) P_{ij}| \leq P_{ij}$. This means that you use the dominating function $i \mapsto P_{ij}$ noting that $\sum_i P_{ij} = 1$ (hence $i \mapsto P_{ij}$ is in $L^1$) and apply the DCT. Does this clarify the use of the DCT?