Let $Z\_n$ denote the population of the $n$-th generation of a Galton Watson Process, with


* $Z\_0=N>1$
* $p\_0 \in (0,1)$
* supercritical, i.e. the mean of descendeants is above $1$
* $(Z\_n$) is conditioned on survival


Now let $K\in \mathbb{N}$. Uniformly choose an individual of the $K$-th generation, kill it and everyone sharing a blood line with it. Let $prop(K)$ denote the proportion of the size of the $K$-th generation killed in this way. 


How large is $prop(K)$? Will it converge as $K\rightarrow \infty$? It seems plausible to assume $prop(K)>\frac1n$, but can we say more? My guess would be that $prop(K)$ simply converges to one over the number of lines that survive eventually.


To clarify: The $K$-th generation is constituted by all individuals of distance $K$ from any of the $N$ root vertices. I'm interested in the proportion of killed individuals in the $K$-th generation relative to the size of the $K$-th generation. 


Inspired by a webcomic about role playing games.



Extinguish a blood line
If I understand correctly, your problem is equivalent to the following one. Take a random number $M$ (in the original problem $M$ is the size of the $K$-th generation), consider $M$ independent G-W processes $Z\_n^{(j)}$, $j = 1, 2, \ldots, M$, and condition on event that at least one of them never dies out. What is the limit of $Q\_n = Z\_n^{(1)} / (Z\_n^{(1)} + \ldots + Z\_n^{(M)})$?


It is known that $Z\_n^{(j)} / \mu^j$ converges to a random variable $W^{(j)}$ with probability one. In addition, if the offspring distribution has finite variance, then $W^{(j)} = 0$ if and only if $Z\_n^{(j)}$ dies out at some point. In this case the limiting distribution of $Q\_n$ is equal to the distribution of $W^{(1)} / (W^{(1)} + \ldots + W^{(M)})$, conditioned on the event that the denominator is non-zero. Here $W^{(1)}, \ldots, W^{(M)}$ have the same distribution, and $M, W^{(1)}, \ldots, W^{(M)}$ are independent.


