By strong markov property $$S_{i}^{(2)},S_{i}^{(3)},\ldots$$ are iid. With common distribution is distribution of $S_{i}^{(2)}$ conditional on $X_{0}=i$

Assume $X_{1}=i$ and $P$ recurrent, then: 1. $S_{i}^{(2)},S_{i}^{(3)},\ldots$ are iid 2. $P\left(\lim_{r\to\infty} \frac{1}{r}T_{i}^{(r)}=m_{i}\right)=1$ where $m_{i}=E_{i}[T_{i}^{(1)}]$