Abstract
Take a continuous-time Galton-Watson tree and pick $k$ distinct particles uniformly from those alive at a time $T$. What does their genealogical tree look like? The case $k=2$ has been studied by several authors, and the near-critical asymptotics for general $k$ appear in Harris, Johnston and Roberts (2018) [9]. Here we give the full picture.
Highlights
Let L be a random variable taking values in {0, 1, 2, . . .}
Consider a continuous-time GaltonWatson tree starting with one initial particle, branching at rate 1, and with offspring distributed like L
On the overwhelmingly rare event that a subcritical tree manages to survive until a large time T, the law of the number of particles alive conditioned on survival converges to a quasi-stationary limit [4, Section III.7]
Summary
Let L be a random variable taking values in {0, 1, 2, . . .}. Consider a continuous-time GaltonWatson tree starting with one initial particle, branching at rate 1, and with offspring distributed like L. Different qualitative behaviours arise depending on whether the underlying Galton-Watson tree is supercritical, critical, or subcritical These cases correspond to m > 1, m = 1, and m < 1 respectively (where m = f ′(1)). Like in the supercritical case, it is possible to use limit theory for critical trees in conjunction with (2.1) to obtain the law of τCrit. On the overwhelmingly rare event that a subcritical tree manages to survive until a large time T , the law of the number of particles alive conditioned on survival converges to a quasi-stationary limit [4, Section III.7]. Lambert showed in [14] (and Athreya in [2]) that conditioned on {NT ≥ 2}, the difference υL,T := T − τ L,T converges in distribution to a [0, ∞)-valued random variable υL as T → ∞. It is straightforward to sketch a proof of (2.8)
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