So I'm reading about a topic called "Branching process" and I don't completely understand some stuff. Here's what's written in the textbook:
Let $(X_n)_{n \geq 0}$ be a stochastic process where :
$X_n$ is the number of individuals at time $n$ $\Rightarrow$ $X_n \in \Bbb N \bigcup \{0\}$
And at each timestep the $i$ -th individual in the population gives birth to a $Z$ distributed number of offspring (independent of others).
$Z_i^n \sim Z$ , where $Z \geq 0$ , $\mu = \Bbb E(Z) \geq 0$
$Z^n_i$ = In the $n$-th generation, the $i$ -th person gives birth to something that's got distribution $Z4
Hence, given $X_n=k$ $$X_{n+1}= \sum^k_{i=1} Z_i^n$$
The Probability matrix :
$p_{ij}= \Bbb P(X_{n+1} = j | X_n = i) = \Bbb P( \sum^i_{k=1}Z^n_k =j)$
Denoted the probability generating function as
$G(s) = \Bbb E[s^z] = \sum^{\infty}_{k=0}s^k \Bbb P(z=k)$
Define the PGF for $X_{n+1}$ as $(n\geq 0)$:
$$F_{n+1}(s) = \Bbb [S^{X_{n+1}} | X_0 =1]=$$
$$\sum^{\infty}_{k=0} \Bbb E[S^{X_{n+1}} | X_n=k] \Bbb P_1(X_n=k)=$$
(here's my first question. Why does $\Bbb P$ have a subscript $1$ here? In my textbook, $\Bbb P_1(X_n=k)$ would mean "probability of getting from $1$-st to $k$-th state in $n$ timesteps)
$$\sum^{\infty}_{k=0} \Bbb[s^{\sum^k_{j=1}Z^n_j}] \Bbb P_1(X_n=k)=$$
and by independence of the $Z_j$'s
$$\sum^{\infty}_{k=0}\underbrace{ \prod^k_{j=1} \Bbb E[s^{Z^n_j}]}_{G(s)} \Bbb P_1(X_n=k)=$$
$$\sum G(s)^k \Bbb P_1(X_n=k)=$$
$$F_n(G(s))\tag{**}$$
So, $F_0(s)=s$ (How is this derived??)
$\Rightarrow$ by $($**$)$ $F_1(s)=F_0(G(s))=G(s)$. So, im not completely sure how this is derived since my confusion of why $F_0(s)=s$
$F_2(s)=F_1(G(s))=G(G(s))$ - So this I understand algebraically- it's just a deduction from the line above.
$\Rightarrow$
$$F_n(s)= F_{n-1}(G(s))=F_{n-2}(G \circ G(s))$$ $$.$$
$$.$$
$$.$$
$$= \underbrace{G \circ G \circ ... \circ G(s)}_{n \text{ times}}=$$ (the equality follows because $F_1 = G$ )
Could someone please explain how this works? I am very confused why this equals to the composition of those $G$'s... Algebraic or analytic explanations are welcome
$\Rightarrow$ $$=G(F_{n-1}(s))$$
The subscript in $P_1$ indicates that the process has size $1$ at time $0$.
Since $X_0=1$ by assumption we get $F_0(s)=Es^{1}=s$.
We have $F_1(s)=G(s)$ and $F_{n+1}(s)=F_n(G(s))$. Use induction to prove that $F_n(s)=G(G(...(s))...)$ (composition if $n$ functions). [For example $F_2(s)=F_1(G(s))=G(G(s))$].