ergodic theorem proof questions

Question

I have questions concerning the proof of the ergodic theorem by Norris. http://www.statslab.cam.ac.uk/~james/Markov/s110.pdf

Let $V_i(n)=\sum_{k=0}^{n-1}1_{\{X_k=i\}}$,

$T_i^{(r+1)}:=\inf\{n\ge T_i^{(r)}+1: X_n=i\}$ with $T_i^{(0)}=0$ and

$S_i^{(r)}:=\begin{cases} T_i^{(r)}-T_i^{(r-1)} &\mbox{if } T_i^{(r-1)} < \infty \\ 0 & \mbox{otherwise} \end{cases}$.

Then $T_i^{(V_i(n)-1)}=\sum_{k=1}^{(V_i(n)-1)}S_i^{(k)}\le n-1$, since $T_i^{(V_i(n)-1)}$ is the second last time the chain is in $i$ before time $n$?? But why is $T_i^{(V_i(n))}=\sum_{k=1}^{V_i(n)}S_i^{(k)}\ge n$????

2nd question concerning the second part of the proof. We have to choose this $J$ as a finite set since then $\lim_{n\to\infty}\sum_{i\in J}\left|\dfrac{V_i(n)}{n}-\pi_i\right|=\sum_{i\in J}\underbrace{\lim_{n\to\infty}\left|\dfrac{V_i(n)}{n}-\pi_i\right|}_{=0}=0.$ If we would have an infinite $J$, then we cannot change $\lim$ and the sum. (by dominated convergence or any other theorem?) Am I right?

I appreciate all answers.

Zitrone

Answer 1

1. You are miscounting. Since the process is assumed ($\lambda=\delta_i$) to begin in state $i$, the first visit to $i$ is at time $0$. Therefore $T_i^{V_i(n)-1}$ is the last visit to state $i$ before time $n$, while $T^{V_i(n)}_i$ is the first visit to $i$ after time $n-1$. Norris explains this quite clearly.

2. You are correct. This argument is not valid for infinite $J$.