Asymptotic genealogy of a critical branching process

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Abstract
Let Tt;n be a continuous-time critical branching process conditioned to have population n at time t .C onsiderTt;n as a random rooted tree with edge-lengths. We dene the genealogy G(Tt;n )o f the population at time t to be the smallest subtree of Tt;n contain- ing all the edges at a distance t from the root. We also consider a Bernoulli(p) sampling process on the leaves of Tt;n, and dene the p-sampled history Hp(Tt;n) to be the smallest subtree of Tt;n containing all the sampled leaves at a distance less than t from the root. We rst give a representation of G(Tt;n )a ndHp(Tt;n )i n terms of point-processes, and then provide their asymptotic behavior as n !1 , t n ! t0 ,a ndnp ! p0. The resulting asymptotic processes are related to a Brownian excursion conditioned to have local time at t0 equal to 1, sampled at times of a Poisson( p 0 2 ) process.
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