# Asymptotic Genealogy of a Critical Branching Process

Report Number
628
Authors
Lea Popovic
Citation
PDf
Abstract

Let $\T_{t,n}$ be a continuous-time critical branching process conditioned to have population $n$ at time $t$. Consider $T_{t,n}$ as a random rooted tree with edge-lengths. We define the genealogy $\G(T_t)$ of the population at time $t$ to be the smallest subtree of $\T_{t,n}$ containing all the edges at a distance $t$ from the root. We also consider a Bernoulli($p$) sampling process on the leaves of $\T_{t,n}$, and define the $p$-sampled history $\H_p(\T_{t,n})$ to be the smallest subtree of $\T_{t,n}$ containing all the sampled leaves at a distance less than $t$ from the root. We first give a representation of $\G(\T_{t,n})$ and $\H_p(\T_{t,n})$ in terms of point-processes, and then prove their convergence as $n\rightarrow\infty$, $\frac{t}{n}\rightarrow t_0$, and $np\rightarrow p_0$. The resulting asymptotic processes are related to a Brownian excursion conditioned to have local time at $0$ equal to $1$, sampled at times of a Poisson($\frac{p_0}{2}$) process.

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