The coalescent of a sample from a binary branching process

Amaury Lambert

Theoretical Population Biology, 2018, vol. 122, issue C, 30-35

Abstract: At time 0, start a time-continuous binary branching process, where particles give birth to a single particle independently (at a possibly time-dependent rate) and die independently (at a possibly time-dependent and age-dependent rate). A particular case is the classical birth–death process. Stop this process at time T>0. It is known that the tree spanned by the N tips alive at time T of the tree thus obtained (called a reduced tree or coalescent tree) is a coalescent point process (CPP), which basically means that the depths of interior nodes are independent and identically distributed (iid). Now select each of the N tips independently with probability y (Bernoulli sample). It is known that the tree generated by the selected tips, which we will call the Bernoulli sampled CPP, is again a CPP. Now instead, select exactly k tips uniformly at random among the N tips (a k-sample). We show that the tree generated by the selected tips is a mixture of Bernoulli sampled CPPs with the same parent CPP, over some explicit distribution of the sampling probability y. An immediate consequence is that the genealogy of a k-sample can be obtained by the realization of k random variables, first the random sampling probability Y and then the k−1 node depths which are iid conditional on Y=y.

Keywords: Splitting tree; Random tree; Birth–death process; Incomplete sampling; Coalescent point process; Finite exchangeable sequence. (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:thpobi:v:122:y:2018:i:c:p:30-35

DOI: 10.1016/j.tpb.2018.04.005

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