Approximation of the allelic frequency spectrum in general supercritical branching populations
type: Item
instance of: scholarly article
title: Approximation of the allelic frequency spectrum in general supercritical branching populations
zbMATH Open document ID: 1475.60167
author: Benoit Henry
published in: Stochastic Processes and their Applications
publication date: 2021-02-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/1701.07281
review text: The author considers a branching population model with the following dynamics: starting with a single individual (called the ancestor) whose lifetime is distributed according to an arbitrary probability distribution on \((0,\infty]\), this ancestor, during its lifetime, gives birth to new individuals according to a Poisson process with a constant rate. Each birth event gives a single new individual. Then each child of the ancestor lives and gives birth according to the same mechanism independently of the other individuals in the population. In addition, neutral mutations can occur on individuals and each new mutation confers to its holder a brand new type (i.e., never seen in the population): this is the infinitely many alleles assumption. It is also supposed that every individual inherits the type of its parent. This model leads to a partition of the population by types. The frequency spectrum of the population alive at time t is defined as the sequence \(A(k,t)\), \(k\in\{1,2,\ldots\}\), where, for each \(k\), \(A(k,t)\) is the number of families (i.e., sets of individuals carrying the same type) of size \(k\) in the population. In general the law of the frequency spectrum \(A(k,t)\), \(k\in\{1,2,\ldots\}\), is hardly tractable. Motivated by this, the author introduced two approximations of the frequency spectrum and investigated their errors. The first approximation is \[ (A(k,t))_{k\in\{1,2,\ldots\}} \approx (c_k)_{k\in\{1,2,\ldots\}}{\mathrm{e}}^{\alpha t} \mathcal{E}, \] where \(c_k\), \(k\in\{1,2,\ldots\}\), are some explicitly given deterministic constants, \(\alpha>0\) (the largest root of the Laplace exponent of the corresponding splitting tree), and \(\mathcal{E}\) is a conditionally exponentially distributed random variable given non-extinction. Here \(\alpha>0\) corresponds to the supercritical case, when \(\alpha\) is also called the Malthusian parameter. The second approximation is \[ (A(k,t))_{k\in\{1,2,\ldots\}} \approx (c_k)_{k\in\{1,2,\ldots\}} N_…
reviewed by: Mátyás Barczy
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60J85
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 60K15
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 92D10
zbMATH DE Number: 7312679
zbMATH Keywords: allelic partition
zbMATH Keywords: allelic frequency spectrum
zbMATH Keywords: branching processes
zbMATH Keywords: neutral mutations
zbMATH Keywords: splitting tree
zbMATH Keywords: central limit theorems
zbMATH Keywords: Laplace distribution
MaRDI profile type: Publication
OpenAlex ID: W3103796807
arXiv ID: 1701.07281
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DOI: 10.1016/J.SPA.2020.10.008
description: scientific article