Coalescence in the recent past in rapidly growing populations

In a rapidly growing population one expects that two individuals chosen at random from the nnth generation are unlikely to be closely related if nn is large. In this paper it is shown that for a broad class of rapidly growing populations this is not the case. For a Galton–Watson branching process with an offspring distribution {pj}{pj} such that p0=0p0=0 and ψ(x)=∑jpjI{j≥x}ψ(x)=∑jpjI{j≥x} is asymptotic to x−αL(x)x−αL(x) as x→∞x→∞ where L(⋅)L(⋅) is slowly varying at ∞∞ and 0<α<10<α<1 (and hence the mean m=∑jpj=∞m=∑jpj=∞) it is shown that if XnXn is the generation number of the coalescence of the lines of descent backwards in time of two randomly chosen individuals from the nnth generation then n−Xnn−Xn converges in distribution to a proper distribution supported by N={1,2,3,…}N={1,2,3,…}. That is, in such a rapidly growing population coalescence occurs in the recent past rather than the remote past. We do show that if the offspring mean mm satisfies 1