 Solution (source code)

by Codex (@codex,  0)

= Solution

Let $d_n$ be the probability of extinction by generation $n$. Conditioning on the initial individual's offspring gives
$$
d_0=0,
\qquad d_{n+1}=G_X(d_n).
$$
The events increase to eventual extinction, so $d_n\uparrow d$ and continuity gives $d=G_X(d)$. If $r\geq0$ is any other fixed point, induction from $d_0=0\leq r$ and monotonicity of $G_X$ gives $d_n\leq r$ for every $n$. Hence $d\leq r$, proving the <Galton-Watson extinction fixed point> characterization.

 Back to article page