The problem is the following:
Suppose we have a branching process so that the initial generation $Z_0\sim \operatorname{Poisson}(\lambda)$, and the number of offsprings of each individual is also Poisson with parameter $\lambda$. Find a function $f$ such that if $H_n(x)$ is the (probability) generating function of the total number of individuals $Z_0 + Z_1 + \dots + Z_n$, then
$$H_n(s) = f\left (sH_{n-1}(s)\right )
$$
My attempt so far:
Let's look at $H_1$ first and then try to discover a generalised solution. Let $G_X$ denote the probability generating function of an r.v. $X$. Let, also, $C_1, C_2, \dots C_{Z_0}$ be the offsprings of $Z_0$ s.t.
$$
Z_1 = C_1 + \dots + C_{Z_0}
$$
$$
H_1(s) = G_{Z_0+Z_1}(s) = G_{Z_0 +C_1 + \dots + C_{Z_0} }(s)
$$
Here I make the following claim even though I am not sure I can, so please do let me know if it is valid or not : $Z_0, C_1, \dots , C_{Z_0}$ are independent.
Hence:
$$
H_1(s) = G_{Z_0}(s)\cdot G_{C_1 + \dots + C_{Z_0}} (s) = G_{Z_0}(s)\cdot G_{Z_0}\left (G(s)\right ),
$$
where $G(s)$ is the probability generating function of the (identical) offsprings. Given that $Z_0$ and the offsprings have the same distribution, the above equation can be written as:
$$
H_1(s) = G(s)\cdot G(G(s))
$$
However I get stuck here. I would appreciate some hints as to how to move forward.
The result they give is that $f(s) = G(s)$, however it is not helpful…
Best Answer
I mean, your claim is strange when put in that form since $Z_0$ determines how many of the $C_i$s you have. With that said, you can consider $C_i$ to be $X_i 1_{i \leq Z_0}$, where $X_i$ is independent from $Z_0$. Now, I will do the base case:
$H_1(s) = \mathbb{E} s^{Z_0 + Z_1} = \mathbb{E}[ \mathbb{E}[ s^{Z_0 + Z_1} \mid Z_0]] = \mathbb{E}[ s^{Z_0} \mathbb{E}[ s^{Z_1} \mid Z_0]] = \mathbb{E}[ s^{Z_0} \mathbb{E}[ s^{\sum_{i=1}^{\infty} X_i 1_{i \leq Z_0}} \mid Z_0]] = \mathbb{E}[ s^{Z_0} G(s)^{Z_0}] = G(sG(s))$.
From this, you should be able to generalize (though it seems like for the proof go through, at time $i$, number of people having children is $Z_1 + Z_2 + ... + Z_{i-1}$, not just $Z_{i-1}$ which is not a convention I am used to in terms of branching processes; I am more used to the parent "disappearing" after splitting.)