Coalescence in the recent past in rapidly growing populations

K.B. Athreya

Stochastic Processes and their Applications, 2012, vol. 122, issue 11, 3757-3766

Abstract: In a rapidly growing population one expects that two individuals chosen at random from the nth 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 {pj} such that p0=0 and ψ(x)=∑jpjI{j≥x} is asymptotic to x−αL(x) as x→∞ where L(⋅) is slowly varying at ∞ and 0<α<1 (and hence the mean m=∑jpj=∞) it is shown that if Xn is the generation number of the coalescence of the lines of descent backwards in time of two randomly chosen individuals from the nth generation then n−Xn 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 1Keywords: Coalescence; Recent past; Rapidly growing populations; Branching processes; Regular variation (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1016/j.spa.2012.06.015

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