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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.

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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.

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  • $\begingroup$ I believe all individuals at a given generation develop their genealogical trees independently (except of course for conditioning on joint survival), so I do not see a reason for any bias. However, I may fail to understand the problem correctly: if a fixed, randomly chosen individual from $K$-th generation is chosen, and then what exactly $\operatorname{prop}(K)$ is? Is this a "proportion of the size of the $n$-th generation killed in this way" as $n \to \infty$? I find "$K$-th generation" in the statement of the problem a bit confusing. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Sep 15, 2017 at 11:39
  • $\begingroup$ Is this right? I don't see why it's correct to have the numerator as a bloodline chosen in advance. More prolific bloodlines are more likely to be chosen? $\endgroup$
    – Anthony Quas
    Commented Sep 16, 2017 at 16:25
  • $\begingroup$ @AnthonyQuas: I am sorry, I do not understand your point: $Z_n^{(j)}$, $j = 1, 2, \ldots, M$, are perfectly interchangeable. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Sep 16, 2017 at 20:17
  • $\begingroup$ @AnthonyQuas: I realise that I might have completely misunderstood what "sharing a blood line" means. Now I think the original question corresponds to $M = N$ and the distribution of a random variable that is equal to $Q_K^{(j)} = Z_K^{(j)}/(Z_K^{(1)}+\ldots+Z_K^{(M)})$ with probability $Q_K^{(j)}$ for $j = 1, 2, \ldots, M$. By a similar argument, this random variable still converges a.s. to a non-deterministic limit, but obviously the answer is different. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Sep 16, 2017 at 21:12
  • $\begingroup$ I agree with this. $\endgroup$
    – Anthony Quas
    Commented Sep 16, 2017 at 22:52

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