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Recent common ancestors of all present-day individuals

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Joseph T. Chang
Joseph T. Chang*
Affiliation:
Yale University
*
∗ Postal address: Statistics Department, Yale University, Box 208290, New Haven, CT 06520, USA. Email address: joseph.chang@yale.edu
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Abstract

Previous study of the time to a common ancestor of all present-day individuals has focused on models in which each individual has just one parent in the previous generation. For example, ‘mitochondrial Eve’ is the most recent common ancestor (MRCA) when ancestry is defined only through maternal lines. In the standard Wright-Fisher model with population size n, the expected number of generations to the MRCA is about 2n, and the standard deviation of this time is also of order n. Here we study a two-parent analog of the Wright-Fisher model that defines ancestry using both parents. In this model, if the population size n is large, the number of generations, 𝒯n, back to a MRCA has a distribution that is concentrated around lgn (where lg denotes base-2 logarithm), in the sense that the ratio 𝒯n(lgn) converges in probability to 1 as n→∞. Also, continuing to trace back further into the past, at about 1.77 lgn generations before the present, all partial ancestry of the current population ends, in the following sense: with high probability for large n, in each generation at least 1.77lgn generations before the present, all individuals who have any descendants among the present-day individuals are actually ancestors of all present-day individuals.

Keywords

CoalescentWright-Fisher modelGalton-Watson processgenealogical modelspopulation genetics

MSC classification

Primary: 92D25: Population dynamics (general)
Secondary: 60J85: Applications of branching processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 4 , December 1999 , pp. 1002 - 1026
Copyright
Copyright © Applied Probability Trust 1999 

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