Coalescence in the recent past in rapidly growing populations

In a rapidly growing population one expects that two individuals chosen at random from the n th generation are unlikely to be closely related if n 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 { p j } such that p 0 = 0 and ψ ( x ) = ∑ j p j I { j ≥ x } is asymptotic to x − α L ( x ) as x → ∞ where L ( ⋅ ) is slowly varying at ∞ and 0 < α < 1 (and hence the mean m = ∑ j p j = ∞ ) it is shown that if X n is the generation number of the coalescence of the lines of descent backwards in time of two randomly chosen individuals from the n th generation then n − X n converges in distribution to a proper distribution supported by 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 m satisfies 1 < m ≡ ∑ j p j < ∞ and p 0 = 0 then coalescence time X n does converge to a proper distribution as n → ∞ , i.e., coalescence does take place in the remote past.