Proving a recursive relation of Probability Generating Functions

Question

I have been working on the following problem for a while now, and I'm starting to think that I'm overcomplicating it. The problem is the following: consider a branching process (discrete time steps), and let $P_t$ be the total population up to and including time $t$, $Z_t$ the population of generation $t$, and $P_0=1$. $Y$ is the random variable describing the offspring for a given individual. I want to prove the following recursive relation $$f_{P_t}(s)=sf_Y(f_{P_{t-1}}(s))$$ with $f_{P_0}(s)=s$. Here, $f$ is the probability generating function.
I have shown that this holds for $t=1$. I tried to prove the general case through induction. However, in assuming that the statements holds for some $t$, that means I can assume $\mathbb{E}(s^{P_t})=s\sum_{k=0}^\infty\mathbb{E}(s^{P_{t-1}})^k\mathbb{P}(Y=k)$. Due to the $k$-th power of the expectation, I did not consider this fact useful, so I abandoned my attempt at using induction.
Instead, I tried to prove it directly: $\mathbb{E}(s^{P_t})=\mathbb{E}(s^{P_{t-1}+Z_t})=\mathbb{E}(\mathbb{E}(s^{P_{t-1}+Z_t}|P_{t-1}))=\sum_{k=0}^\infty\mathbb{P}(P_{t-1}=k)\mathbb{E}(s^{P_{t-1}+Z_t}|P_{t-1}=k)$. I am not sure what to do with this expectation, as $Z_t$ is not independent from $P_{t-1}$.
I have been trying to figure out how to convert this into the desired expression, but I can't find anything to get me on the right track. I would appreciate any advice on how to proceed.

Answer 1

Rather than condition on the first $t - 1$ generations, try going in the other direction. The population at time $t$ consists of:

• the root

• and $Z_1$ many i.i.d. copies of $P_{t-1}$.

Can you translate this into the recursive relation you seek?