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Coalescence in the recent past in rapidly growing populations

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  • Athreya, K.B.

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 1

Suggested Citation

  • Athreya, K.B., 2012. "Coalescence in the recent past in rapidly growing populations," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3757-3766.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:11:p:3757-3766
    DOI: 10.1016/j.spa.2012.06.015
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    1. Grey, D. R., 1979. "On regular branching processes with infinite mean," Stochastic Processes and their Applications, Elsevier, vol. 8(3), pages 257-267, May.
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