A similar version of this question has been posted before but it was never answered and I need help with it.
Let $S_n$ be a Galton Watson Process with offspring distribution $p_k$.
We assume that $p_0 > 0$ and that $\sum_{i=0}^{\infty} k p_k > 1$. Also $S_0 = 1$.
We define $T_0 := \inf \{n > 0 : S_n = 0\}$. Let $d := P(T_0 < \infty$). Also
$G(z) := \sum_{i=0}^{\infty} z^k P(S_1 = k) $
is the generating function of $S_1$.
I have shown that $P(T_0 < \infty | S_1 = k) = d^k$ and $r_k := P(S_1 = k \vert T_0 < \infty) = d^{k-1}$ P($S_1 = k)$
Now I have to show that if $Z_n$ is a Galton Watson Process with offspring distribution $r_k$, then $Z_n$ is going to die out almost surely. For this I want to show that $$\frac{1}{d}\sum_{k=0}^\infty kd^kp_k \le 1$$
but I have no idea how to show this. I am not given any upper bound for $d$ or $p_k$ (other than $1$ obviously). Can anyone help?
Best Answer
Three intermediate questions could help you:
What does the condition $\sum kp_k>1$ mean for the graph of $G$ ?
How is $d$ determined in terms of $G$ ?
Denote $H$ the generating function of $Z_1$. How can you rewrite $H$ in terms of $G$ and $d$ ? What can you say now about the new Galton-Watson process ?