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Dirichlet approximation of equilibrium distributions in Cannings models with mutation

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  08 September 2017

Han L. Gan ,
Adrian Röllin  and
Nathan Ross
Han L. Gan*
Affiliation:
Washington University in St. Louis
Adrian Röllin*
Affiliation:
National University of Singapore
Nathan Ross*
Affiliation:
University of Melbourne
*
* Current address: Mathematics Department, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA. Email address: ganhl@math.northwestern.edu
** Postal address: Department of Statistics and Applied Probability, National University of Singapore, 6 Science Drive 2, 117546, Singapore. Email address: adrian.roellin@nus.edu.sg
*** Postal address: School of Mathematics and Statistics, University of Melbourne, Peter Hall Building, Melbourne, VIC 3010, Australia. Email address: nathan.ross@unimelb.edu.au
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Abstract

Consider a haploid population of fixed finite size with a finite number of allele types and having Cannings exchangeable genealogy with neutral mutation. The stationary distribution of the Markov chain of allele counts in each generation is an important quantity in population genetics but has no tractable description in general. We provide upper bounds on the distributional distance between the Dirichlet distribution and this finite-population stationary distribution for the Wright–Fisher genealogy with general mutation structure and the Cannings exchangeable genealogy with parent independent mutation structure. In the first case, the bound is small if the population is large and the mutations do not depend too much on parent type; 'too much' is naturally quantified by our bound. In the second case, the bound is small if the population is large and the chance of three-mergers in the Cannings genealogy is small relative to the chance of two-mergers; this is the same condition to ensure convergence of the genealogy to Kingman's coalescent. These results follow from a new development of Stein's method for the Dirichlet distribution based on Barbour's generator approach and a probabilistic description of the semigroup of the Wright–Fisher diffusion due to Griffiths and Li (1983) and Tavaré (1984).

Keywords

Wright–Fisher processCannings modelequilibrium distributionDirichlet distributiondistributional approximationStein's method

MSC classification

Primary: 92D25: Population dynamics (general) 60F05: Central limit and other weak theorems
Secondary: 60K99: None of the above, but in this section
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 3 , September 2017 , pp. 927 - 959
Copyright
Copyright © Applied Probability Trust 2017 

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