Can someone explain to me, why the probability of returning to the origin of a Dyck path with length $2n$ is:
$\mathbb{P}(\tau=2n) = 2C_{n-1}4^{-n}$?
$C_n$ stands for the Catalan number and $\tau :=$inf$\{n\geq1:S_n=0\}$ is the distribution of the first entry time/ stopping time.
This is an example of a Dyck path:
If I understand correctly, you need to count the Dyck paths whose first return to $0$ occurs after $2n$ steps. The first step must be up, from $\langle 0,0\rangle$ to $\langle 1,1\rangle$, the last step must be down, from $\langle 2n-1,1\rangle$ to $\langle 2n,0\rangle$, and between $\langle 1,1\rangle$ and $\langle 2n-1,1\rangle$ the path must not drop below the line $y=1$. But this means that the path from $\langle 1,1\rangle$ to $\langle 2n-1,1\rangle$ is just a Dyck path from $\langle 0,0\rangle$ to $\langle 2n-2,0\rangle$ that has been translated one unit up and to the right, and there are $C_{n-1}$ such paths. There are altogether $2^{2n}=4^n$ possible paths of length $2n$, all equally probable, so the probability of traversing a Dyck path that begins with an up-step and first returns to height $0$ after $2n$ steps is $C_{n-1}4^{-n}$.
The only way that I see to justify the factor of $2$ in $2C_{n-1}4^{-n}$ is to assume that we also want to include the paths that begin with a down-step and first return to the axis after $2n$ steps. This makes sense if we’re actually talking about a linear random walk along the $y$-axis that is permitted to start in either direction, and we’re using the $x$-axis as a time axis to represent the path of the walk in two dimensions and turn those that we want to count into Dyck paths or the reflections in the $x$-axis of Dyck paths.