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

Nicolas Champagnat (1, 2) , Benoît Henry (1, 2)

1 TOSCA - TO Simulate and CAlibrate stochastic models
2 IECL - Institut Élie Cartan de Lorraine

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 θ 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 (A(k, t)) k≥1. These moments allow us to give an elementary proof of the almost sure convergence of the frequency spectrum in a supercritical splitting tree.

Mots clés: allelic partition -- frequency spectrum; linear birth--death process; infinite alleles model; coalescent point process; Crump--Mode--Jagers process; splitting tree; branching process; Lévy process; scale function; random measure; Palm measure; Campbell's formula

Domaines: Probabilités [math.PR]

Nicolas Champagnat, Benoît Henry. Moments of the frequency spectrum of a splitting tree with neutral Poissonian mutations. Electronic Journal of Probability, 2016, 21 (53), pp.1-34. ⟨10.1214/16-EJP4577⟩. ⟨hal-01202732v2⟩