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2016 Moments of the frequency spectrum of a splitting tree with neutral Poissonian mutations
Nicolas Champagnat, Benoît Henry
Electron. J. Probab. 21: 1-34 (2016). DOI: 10.1214/16-EJP4577


We consider a branching population where individuals live and reproduce independently. Their lifetimes are i.i.d. and they give birth at a constant rate $b$. The genealogical tree spanned by this process is called a splitting tree, and the population counting process is a homogeneous, binary Crump-Mode-Jagers process. We suppose that mutations affect individuals independently at a constant rate $\theta $ during their lifetimes, under the infinite-alleles assumption: each new mutation gives a new type, called allele, to his carrier. We study the allele frequency spectrum which is the numbers $A(k,t)$ of types represented by $k$ alive individuals in the population at time $t$. Thanks to a new construction of the coalescent point process describing the genealogy of individuals in the splitting tree, we are able to compute recursively all joint factorial moments of $\left (A(k,t)\right )_{k\geq 1}$. These moments allow us to give an elementary proof of the almost sure convergence of the frequency spectrum in a supercritical splitting tree.


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Nicolas Champagnat. Benoît Henry. "Moments of the frequency spectrum of a splitting tree with neutral Poissonian mutations." Electron. J. Probab. 21 1 - 34, 2016.


Received: 30 September 2015; Accepted: 24 July 2016; Published: 2016
First available in Project Euclid: 2 September 2016

zbMATH: 1348.60124
MathSciNet: MR3546390
Digital Object Identifier: 10.1214/16-EJP4577

Primary: 60J80
Secondary: 60F15 , 60G51 , 60G57 , 60J85 , 92D10

Keywords: allelic partition , branching process , Campbell’s formula , Coalescent point process , Crump–Mode–Jagers process , frequency spectrum , infinite alleles model , Lévy process , linear birth–death process , Palm measure , random measure , scale function , Splitting tree


Vol.21 • 2016
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