I encountered a question and I am having difficulty understanding why the starting point of the process will determine if the process will be Poisson or not.
So here is the process, let's say we start with $N_0$, and the next N will be determined by:
$$ N_{n+1} = Binomial(N_n, p) + P(\lambda) $$ Where $P(\lambda)$ is a Poisson random variable.
So my attempt is this, let $x=Binomial(N_n, p), y = N_(n+1)$:
$$ \begin{align} P(N_{n+1} = y) &= \sum_{x=0}^{N_n}P_{binom}(x;p)*P_{poisson}(y-x;\lambda)\\ &=\sum_{x=0}^{N_n}\frac{N_n!}{(N_n-x)!x!}p^x(1-p)^{N_n-x}*\frac{\lambda^{(y-x)}e^{-\lambda}}{(y-x)!}\\ &=(N_n)!e^{-\lambda}\sum_{x=0}^{N_n}\frac{p^x}{x!}\frac{(1-p)^{N_n-x}}{(N_n-x)!}\frac{\lambda^{(y-x)}}{(y-x)!} \end{align} $$
This is as far as I get. But I was told by the fact that I am supposed to prove, that if we starts from $N_0 = 0$, then going forward $N_n$ shall all be Poisson. Here is what I don't get it: Is there any way I can further reduce the above formula? Am I missing something?
I kind of understand the intuition behind it. $N_1$ must be Poisson, consider this as number of certain event happened, then $N_2$ is the sum of two things: 1) another Poisson, 2) a "Poisson" with rate of $\lambda p$ (events happened with a smaller probability), but I can't prove it.