I am reading a proof of the assertion that counting processes with stationary and independent increments are necessarily Poisson processes. There is some argument in it that I do not quite follow. I hope someone can explain why it is true.
Let $(X_t)_{t\geq 0}$ be a counting process with stationary and independent increments. Define the random times $T_k = \inf\{t \geq 0: X_t \geq k\}$ for $k \geq 0$ and $Y_n = T_n - T_{n-1}$ for $n \geq 1$.
The author first shows that $Y_1$ has exponential distribution and that the process $(X_{Y_1+t} - X_{Y_1})_{t\geq 0}$ is independent of $Y_1$ and that it has the same distribution as $(X_t)_{t\geq 0}$.
Then the author claims that from the same reasoning that proved these assertions it also follows that $Y_2$ is independent of $Y_1$ and it also has exponential distribution. Furthermore, $Y_3$ is independent of $Y_1$ and $Y_2$ and has exponential distribution. I don't know how he arrives at these conclusions but my guess is as follows.
\begin{align} Y_2 &= \inf\{t\geq 0 : X_{Y_1+t} - X_{Y_1} \geq 1\} \\ &\stackrel{d}{=} \inf\{t\geq 0 : X_t \geq 1\} \end{align} By the first equality it is clear that $Y_2$ is independent of $Y_1$. By the second equality, it is clear that it has the same distribution as $Y_1$.
This argument does not work when I consider $Y_3$. \begin{align} Y_3 &= \inf\{t\geq 0 : X_{Y_1+Y_2+t} - X_{Y_1+Y_2} \geq 1\} \\ &\stackrel{d}{=} \inf\{t\geq 0 : X_t \geq 1\} \end{align} Now I can only say $Y_3$ is independent of $Y_1+Y_2$. I think the problem here is that the author only proved the independence of $(X_{Y_1+t} - X_{Y_1})_{t\geq 0}$ and $Y_1$. I think if he proved the independence of $(X_{Y_1+t} - X_{Y_1})_{t\geq 0}$ and $\mathcal{F}_{Y_1}$, which I am almost certain is true, my argument would work.
I would appreciate it if someone commented on my (as well as the author's) reasoning.