Splitting trees with neutral Poissonian mutations II: Largest and oldest families

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:123

Article

Champagnat, Nicolas^1,2  Lambert, Amaury^3,4 

[1] Univ Lorraine, IECN, F-54506 Vandoeuvre Les Nancy, France

[2] Inria, F-54600 Villers Les Nancy, France

[3] CNRS, UMR 7599, Lab Probabil & Modeles Aleatoires, F-75252 Paris 05, France

[4] Univ Paris 06, F-75252 Paris 05, France

关键词: Branching process;    Coalescent point process;    Splitting tree;    Crump-Mode-Jagers process;    Linear birth death process;    Allelic partition;    Infinite alleles model;    Extreme values;    Mixed Poisson point process;    Cox process;    Levy process;    Scale function;   

DOI  :  10.1016/j.spa.2012.11.013

来源: Elsevier

【 摘 要 】

We consider a supercritical branching population, where individuals have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate. We assume that individuals independently experience neutral mutations, at constant rate theta during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called allele or haplotype, to its carrier. The type carried by a mother at the time when she gives birth is transmitted to the newborn. We are interested in the sizes and ages at time t of the clonal families carrying the most abundant alleles or the oldest ones, as t -> infinity, on the survival event. Intuitively, the results must depend on how the mutation rate theta and the Malthusian parameter alpha > 0 compare. Hereafter, N N-t is the population size at time t, constants a, c are scaling constants, whereas k, k' are explicit positive constants which depend on the parameters of the model. When alpha > 0, the most abundant families are also the oldest ones, they have size cN(1-theta/alpha) and age t - a. When alpha < theta, the oldest families have age (alpha/theta)t + a and tight sizes; the most abundant families have sizes k log(N) - k' log log(N) + c and all have age (theta - alpha)(-1) log(t). When alpha = theta, the oldest families have age kt - k' log(t) + a and tight sizes; the most abundant families have sizes (k log(N) - k' log log(N) + c)(2) and all have age t/2. Those informal results can be stated rigorously in expectation. Relying heavily on the theory of coalescent point processes (Popovic, 2004, Lambert, 2010), we are also able, when alpha < theta, to show convergence in distribution of the joint, properly scaled ages and sizes of the most abundant/oldest families and to specify the limits as some explicit Cox processes. This is in deep contrast with the largest/oldest families in the standard coalescent with Poissonian mutations, which converge to some point processes after being rescaled by N (Donnelly and Tavare 1986, Ewens, 2005, Durrett, 2008). (C) 2012 Elsevier B.V. All rights reserved.

【 授权许可】

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