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!