Electronic Journal of Probability

Moments of the frequency spectrum of a splitting tree with neutral Poissonian mutations

Nicolas Champagnat and Benoît Henry

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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.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 53, 34 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 92D10: Genetics {For genetic algebras, see 17D92} 60J85: Applications of branching processes [See also 92Dxx] 60G51: Processes with independent increments; Lévy processes 60G57: Random measures 60F15: Strong theorems

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

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Champagnat, Nicolas; Henry, Benoît. Moments of the frequency spectrum of a splitting tree with neutral Poissonian mutations. Electron. J. Probab. 21 (2016), paper no. 53, 34 pp. doi:10.1214/16-EJP4577. https://projecteuclid.org/euclid.ejp/1472830615

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