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!



Branching process survival probability
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 $p\_j=p\_{j+1}$.


