Asymptotic line-of-descent distributions
type: Item
instance of: scholarly article
title: Asymptotic line-of-descent distributions
zbMATH Open document ID: 0577.92011
DOI: 10.1007/BF00275223
published in: Journal of Mathematical Biology
publication date: 1984-01-01T00:00:00Z
review text: In a diffusion approximation, assume that the process is extended in the past to -\(\infty\) from a time origin 0. Let \(T_{\ell}\) be the time until there are \(\ell\) lines of descent left, beginning from n at time 0. Then the quantity \(L_ n(t)\) given by \(P(L_ n(t)\leq \ell)=P(T_{\ell}\leq t)\) represents the number of lines of descent at t in the past from the time origin. Introducing a mutation parameter \(\theta >0\) in the death-process, the author observes the random variable \(T_{\ell}\) with moment generating function \[ E(e^{\phi T_{\ell}})=\prod^{n}_{j=\ell +1}(1-2\phi /j(j+\theta -1))^{-\quad 1} \] and derives asymptotic distributions for the number of non-mutant ancestors of a sample of n from a neutral infinite alleles model: Either \(L_ n(t)\) has a normal distribution or \(n-L_ n(t)\) has a Poisson distribution as \(n\to \infty\), \(t\to 0\).
Mathematics Subject Classification ID: 92D10
Mathematics Subject Classification ID: 60J70
zbMATH DE Number: 3924576
zbMATH Keywords: coalescent
zbMATH Keywords: population genetics
zbMATH Keywords: diffusion approximation
zbMATH Keywords: lines of descent
zbMATH Keywords: mutation parameter
zbMATH Keywords: death-process
zbMATH Keywords: moment generating function
zbMATH Keywords: asymptotic distributions for the number of non-mutant ancestors
zbMATH Keywords: neutral infinite alleles model
zbMATH Keywords: normal distribution
zbMATH Keywords: Poisson distribution
author: Robert Griffiths
reviewed by: Yûsaku Komatu
Wikidata QID: Q113909325
MaRDI profile type: Publication
cites work: Q4197800
cites work: The transient behaviour of the Moran model in population genetics
cites work: Lines of descent in the diffusion approximation of neutral Wright-Fisher models
cites work: The coalescent
cites work: Line-of-descent and genealogical processes, and their applications in population genetics models
cites work: Asymptotic Analysis of Diffusion Equations in Population Genetics
cites work: Asymptotic analysis of stochastic models in population genetics
cites work: Lines of descent and the coalescent
description: scientific article; zbMATH DE number 3924576