this problem is somewhat similar to the thread The extinction probability of Galton-Watson process from a martingale perspective. I want to show, that for a Galton-Watson-process $Z_0,Z_1,\ldots$ with $Z_0=1$, $\phi(t) = \mathbb E (t^{Z_1})$ the generating function and $\xi \in (0, 1)$ a fixed point of $\phi$:
$\mathbb{P}[\{\lim_{n\rightarrow\infty}Z_n = 0\}] = \xi$ using the martingale $\xi^{Z_n}$ and a theorem of Martingale convergence.
Even with the clues from the above thread I wasn't able to solve the problem. Help appreciated!
Best Answer
At time $0$, $\xi^{Z_0}=\xi$. When $n\to\infty$, $Z_n\to+\infty$ on non-extinction hence $\xi^{Z_n}\to0$ on non-extinction and $Z_n\to0$ on extinction hence $\xi^{Z_n}\to1$ on extinction. Finally $|\xi^{Z_n}|\leqslant1$ uniformly, thus everything is in place for an application of martingale dominated convergence theorem.