Coalescence on Supercritical Bellman-Harris Branching Processes

We consider a continuous-time single-type age-dependent Bellman-Harris branching process {Z(t) : t ≥ 0} with offspring distribution {pj}j≥0 and lifetime distribution G. Let k ≥ 2 be a positive integer. If Z(t) ≥ k, we pick k individuals from those who are alive at time t by simple random sampling wi...

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Bibliographic Details
Published in:Taiwanese journal of mathematics 2018-02, Vol.22 (1), p.245-261
Main Authors: Athreya, Krishna B., Hong, Jyy-I
Format: Article
Language:English
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Summary:We consider a continuous-time single-type age-dependent Bellman-Harris branching process {Z(t) : t ≥ 0} with offspring distribution {pj}j≥0 and lifetime distribution G. Let k ≥ 2 be a positive integer. If Z(t) ≥ k, we pick k individuals from those who are alive at time t by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let Dk(t) be the coalescence time (the death time of the most recent common ancestor)and let Xk(t) be the generation number of the most recent common ancestor of thesek random chosen individuals. In this paper, we study the distributions of Dk(t) andXk(t) and their limit distributions as t → ∞.
ISSN:1027-5487
2224-6851
DOI:10.11650/tjm/8123