Lemma

Let $i\in S$ and $r\ge0$. Then the passage time $T_{i}^{(r)}$ is a stopping time.

Lemma

Let $i\in S$ and $r\ge1$. Then

1. For $r\ge1$, $$P(T_{i}^{(r+1)}<\infty|T_{i}^{(r)}<\infty)=P_{i}(T_{i}^{(1)}<\infty)=h_{i}$$or the probability that we pass state $i$ again given we did it $r$ times is equivalent to probability that starting from $i$ we return to $i$ which is equivalent to the Hitting probability.
2. Consequently, $r\ge0$, $$P_{i}(T_{i}^{(r)}<\infty)=(P_{i}(T_{i}^{(1)}<\infty))^{r}=(h_{i})^{r}$$