Athreya, K.B.
(2012)
*Coalescence in the recent past in rapidly growing populations*
Stochastic Processes and their Applications, 122
(11).
pp. 3757-3766.
ISSN 03044149

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192kB |

Official URL: http://doi.org/10.1016/j.spa.2012.06.015

Related URL: http://dx.doi.org/10.1016/j.spa.2012.06.015

## Abstract

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=∑jp j =∞) it is shown that if X n is the generation number of the coalescence of the lines of descents 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 show that if the offspring mean m satisfies 1<m≡∑jp j <∞ and p 0 =0 then the coalescence time X n does converge to a proper distribution as n→∞, i.e., coalescence does take place in the remote past.

Item Type: | Article |
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Source: | Copyright of this article belongs to Elsevier B.V. |

Keywords: | Coalescence, Recent past, Rapidly growing populations, Branching processes, Regular variation |

ID Code: | 131590 |

Deposited On: | 07 Dec 2022 08:29 |

Last Modified: | 07 Dec 2022 08:29 |

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