Coupon collector's problem as Random Walk

Question

I see in a book the following as coupon collector's problem. We have $N$ coupons labelled $1,2,\dots,N$ from which we pick with replacement. I could not understand what is the random walk here.

Answer 1

The state space is $\{0,1,\ldots,N\}$ and the steps of the random walk are $0$ and $+1$. When at some state $k$, the transition $k\to k+1$ has probability $1-\frac{k}N$ and the transition $k\to k$ has probability $\frac{k}N$. Thus, the transitions $0\to1$ and $N\to N$ both have probability $1$ and the state $N$ is absorbing. (Note that this process is not usually described as a (inhomogenous) random walk but rather as a Markov chain.)