Let $S_n$ be the simple nearest-neighbor random walk on the integers started at $S_0=1$. Define $T$ to be the time of the first visit to the origin, that is, the smallest $n\geqslant 1$ such that $S_n=0$. Define $Z_0=1$ and $$Z_k=\sum_{n=0}^{T-1}\mathsf 1_{\{X_n=k\text{ and } X_{n+1}=k+1}\}. $$ In words, $Z_k$ is the number of times that the random walk $X_n$ crosses from $k$ to $k+1$ before first visiting $0$.
(A) Prove that the sequence $\{Z_k\}_{k\geqslant0}$ is a Galton-Watson process, and identify the offspring distribution as a geometric distribution.
I have no idea how to give a solution to this problem. Since the Galton-Watson process is about offspring and extinction. I don't know why it can be used to solve simple random walk problems. And also I'm not sure why there is offspring distribution and why it is geometric? Thanks a lot! I'm totally torched by the stochastic process.
If you are talking about a random walk reflected at the x-axis or stopped when hitting $0$, or just about the first excursion of a random walk:
So, what you see (in red) is (a possible realization of) a binary Galton-Watson tree ($0$ or 2 children with probability $1/2$) with $exp(1)$-distributed lifetime. Given a tree like that, you can draw the random walk (which stops when hitting zero): A branching point is a local minimum, a leaf is a local maximum. With the right scaling you get a reflected brownian motion.
Of course, the construction is possible for all distributions and the random walk will just be something else!
The key message is, what Ted Harris said 1952: "Walks and trees are abstractly identical objects ... "
And regarding your exercise: You have fixed lifetimes of one period. So instead of $\exp(1)$ we look at $\delta_1$. The expected number of periods till the next downstep are two, so the offspring distribution should be $geo(1/2)$.