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Coalescent lineage distributions

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Robert C. Griffiths
Robert C. Griffiths*
Affiliation:
University of Oxford
*
∗ Postal address: Department of Statistics, University of Oxford, 1 South Parks Rd, Oxford OX1 3TG, UK. Email address: griff@stats.ox.ac.uk
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Abstract

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We study identities for the distribution of the number of edges at time t back (i.e. measured backwards) in a coalescent tree whose subtrees have no mutations. This distribution is important in the infinitely-many-alleles model of mutation, where every mutation is unique. The model includes, as a special case, the number of edges in a coalescent tree at time t back when mutation is ignored. The identities take the form of expected values of functions of Zt=eiXt, where Xt is distributed as standard Brownian motion. Associated identities are also found for the distributions of the time to the most recent common ancestor, the time until loss of ancestral lines by coalescence or mutation, and the age of a mutation. Hypergeometric functions play an important role in the identities. The identities are of mathematical interest, as well as potentially being formulae to use for numerical integration or simulation to compute distributions that are usually expressed as alternating-sign series expansions, which are difficult to compute.

Keywords

Age of a mutationcoalescent lineage distributioncoalescent processhypergeometric functioninfinitely-many-alleles modelJacobi polynomialline of descent

MSC classification

Primary: 92D15: Problems related to evolution
Secondary: 60G99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 2 , June 2006 , pp. 405 - 429
Copyright
Copyright © Applied Probability Trust 2006 

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