Problem : Let $Y_{1}, Y_{2}, \cdots$ be independent identically distributed random variables with $\mathbb{P}\left(Y_{1}=1\right)=\mathbb{P}\left(Y_{1}=-1\right)=\frac{1}{2}$ and set $X_{0}=1$ and $X_{n}=X_{0}+Y_{1}+\cdots+Y_{n}$ for $n \geq 1 .$ Define the stopping time $$ H_{0}=\inf \left\{n \geq 0 | X_{n}=0\right\} $$ Find the probability generating function $\phi(s)=\mathbb{E}\left[s^{H_{0}}\right]$
Solution : [...] Returning to our original process $X_{n}=1+Y_{1}+\cdots Y_{n},$ and conditioning on $X_{1}=2,$ we have that $H_{0}=1+\bar{H}_{0}$ where $\bar{H}_{0}$ is the time taken, starting from 2 to reach $0,$ and so is distributed as $H_{0}$ under $\mathbb{P}_{2} .$ So we get $$ \begin{aligned} \phi(s) &=\mathbb{E}_{1}\left[s^{H_{0}}\right]=\frac{1}{2} \mathbb{E}_{1}\left[s^{H_{0}} | X_{1}=2\right]+\frac{1}{2} \mathbb{E}_{1}\left[s^{H_{0}} | X_{1}=0\right] \\ &=\frac{1}{2} \mathbb{E}_{1}\left[s^{1+\bar{H}_{0}} | X_{1}=2\right]+\frac{1}{2} \mathbb{E}_{1}\left[s^{H_{0}} | X_{1}=0\right] \\ &=\frac{1}{2} s \mathbb{E}_{2}\left[s^{H_{0}}\right]+\frac{1}{2} s \\ &=\frac{1}{2} s \phi(s)^{2}+\frac{1}{2} s \end{aligned} $$ Solving this, we get $$ \phi(s)=\frac{1 \pm \sqrt{1-s^{2}}}{s} $$
Hello, I am still starting the course on stochastic processes and I am having a hard time understanding the first sentence of the solution. If I understood correctly, to reach state $0$, starting from state $1$, and under condition that after $1$ we went to $2$, is it the same as going from $2$ to $0+ 1$ step?