RECENT COMMON ANCESTORS OF ALL PRESENT-DAY INDIVIDUALS
Joseph T. Chang
Department of Statistics, Yale University
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 lg n (where lg denotes base-2 logarithm), in the sense that the ratio Tn/(lg n) converges in probability to 1 as n → ∞. Also, continuing to trace back further into the past, at about 1.77 lg n 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 1.77 lg n generations before the present, all individuals who have any descendants among the present-day individuals are actually ancestors of all present-day individuals.
COALESCENT, WRIGHT-FISHER MODEL, GALTON-WATSON PROCESS, GENEALOGICAL MODELS, POPULATION GENETICS
AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 92D25
SECONDARY 60J85
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