TITLE: Branching process survival probability
QUESTION [5 upvotes]: I have a time-inhomogeneous Galton-Watson binary branching process over a finite number of generations $n$. In each generation $i$, there is a probability $p_i$ of a child surviving; so each node has 2 children with probability $p_i^2$, 1 child with probability $2 p_i (1-p_i)$, and zero children with probability $(1-p_i)^2$.  Furthermore $p_i$ is a decreasing function of $i$. So the process starts out as a super-critical branching and ends as a sub-critical branching.
I want to show that the process survives to time $n$ with probability say $\Omega(1)$ or $\Omega(1/\text{poly}(n))$. What is the easiest criterion to show this? 
The process is inhomogeneous. The number of expected survivors at level $i$ is $\mu_i = 2^i p_1 \dots p_i$, which is a unimodular function of $i$. It seems that a sufficient criterion should be $\mu_n \geq 1$ or maybe $\mu_n = \Omega(\text{poly}(n))$.
Thanks!

REPLY [2 votes]: A new answer for the new version of the question.
Under the constraint that $p_i$ are decreasing and $\mu_n\ge 1$, the minimal survival probability is obtained when all $p_i=1/2$. You can see this by showing that for any level $j$, if you fix all the $p_i$ except for $p_j$ and $p_{j+1}$ then the minimum is obtained when $p_j=p_{j+1}$.