My question is about a homework question that I found interesting. It gives another proof (without using martingales) for that the critical Galton Watson tree dies out eventually. But it has given a recurrent relation which turns out to be tricky in the problem 5 (Part (a)) of this problem set. Is there any idea for that? It gives a relation between two different order terms which makes it easy to use induction.
Let $(Z_n)$ be a branching process with $Z_0 = 1$ and offspring distribution $ξ$ with $P(ξ = 0) = P(ξ = 2) = 1/2.$ Show that there exists $ C > 0 $ such that for any $\epsilon > 0$ $P(Z_{3n} > 0) \le C\sqrt{\epsilon n} + \epsilon\, nP(Z_n > 0)^2$ .