I have a rather naïve question about the reflection principle and I wanted to be sure, if my understanding is correct.
[Feller page 70]. The number of paths from $A(a,\alpha)$ to $B(b,\beta)$ which touch or cross the $x$-axis equal the number of paths from $A'$ to $B$.
Proof. Let $t$ be the abscissa of the first such vertex, where $s_a>0,\ldots,s_{t-1}>0,s_{t}=0.$ The particle first hits the $x$-axis at epoch time $T(t,0)$. The sections $AT$ and $A'T$ being reflections of each other, there exists a one-to-one correspondence between all paths from $A'$ to $B$ and such paths from $A$ to $B$ as have a vertex on the $x$-axis. The lemma is proved.
The example figure 2. in Feller's book shows a path, that touches the $x$-axis, $k=1$ times. I mocked up another blue path, that crosses the $x$-axis several times; below. Sections $AT$ and $A'T$ are reflections about the $x$-axis.
Question. The geometric argument [Feller] still holds true, because, in principle, the same logic can be applied to each section isolated, between the crossings from the negative to the positive region and vice versa. Every blue path maps to a unique green path. Is my understanding broadly correct?
