I have the following problem about a sort of "reversed" time-inhomogeneous branching process that I'm trying to solve.
Let $Z_0,Z_1,\ldots$ be a time-inhomogeneous branching process with $Z_0=1$.
Every individual in my process is complete graph, whose number of vertices depends on the current generation. In particular, at generation 0, the single individual $Z_0 = 1$ is a complete graph on $n$ vertices, and $\binom{n}{2}$ edges with values sampled in $U(0,1)$, denoted by $K_n$.
At generation $Z_1$, the individuals are complete graph on $n-1$ vertices, where a vertex has been removed and its connected edges.
Similarly, at generation $Z_2$, the individuals are complete graph on $n-2$ vertices where I removed another vertex (and its edges) and so on.
I call it reversed because of this dependency on $n$ which decrease with the generations.
Fix the initial individual $K_n$. In this graph, every vertex independently can give rise to a certain number of offsprings. Denoting the vertex $i$ with the random variable $D_i$, we have the following probability of having 0 or 1 offspring.
\begin{align*} p_0 = \frac{2e-1}{e^2} \\ p_1 = 1-\frac{2e-1}{e^2} \end{align*}
Thus, $D_i \sim Bernoulli(p_1)$.
If $D_i = 1$, the offspring on generation $Z_1$ will be the complete graph $K_{n-1}$ where I removed vertex $i$ and all its adjacent edges.
Now, since every vertex can give rise to an offspring independently, I have
$D = \sum_{i=1}^n D_i \sim Binomial(n, p_1)$
Thus, the offspring distribution at a given generation $t$ is $Binomial(n-t,p_1)$, since the size of the graph decreases with the increasing generation.
I can calculate the expected number of offsprings at every generation
\begin{align*} \mathbb{E}\left[ Z_1 \right] = \sum_{i=1}^n \mathbb{E}[D_i] = p_1 \cdot n \\ \mathbb{E}\left[ Z_2 \right] = Z_1 \cdot \sum_{i=1}^{n-1} \mathbb{E}[D_i] = Z_1 p_1 \cdot (n-1) \\ \cdots \\ \end{align*}
I would like to prove that at generation $n-3$
$P(Z_{n-3} \geq 1) = 1$ almost as surely.
I found this article supercritical bpre where Theorem 2 seems very applicable in my case.
I tried to build a (sub)Martingale from it but I'm not able to conclude my claim. Do you have any idea how to proceed? Thanks!
