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Looking Forwards and Backwards in the Multi-Allelic Neutral Cannings Population Model

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

M. Möhle
M. Möhle*
Affiliation:
Eberhard Karls Universität Tübingen
*
∗Postal address: Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany. Email address: martin.moehle@uni-tuebingen.de
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Abstract

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We look forwards and backwards in the multi-allelic neutral exchangeable Cannings model with fixed population size and nonoverlapping generations. The Markov chain X is studied which describes the allelic composition of the population forward in time. A duality relation (inversion formula) between the transition matrix of X and an appropriate backward matrix is discussed. The probabilities of the backward matrix are explicitly expressed in terms of the offspring distribution, complementing the work of Gladstien (1978). The results are applied to fundamental multi-allelic Cannings models, among them the Moran model, the Wright-Fisher model, the Kimura model, and the Karlin and McGregor model. As a side effect, number theoretical sieve formulae occur in these examples.

Keywords

Cannings modeldualityexchangeabilitygeneralized Stirling numberinversion formulamulti-allele modelneutralityprinciple of inclusion and exclusionSilvester's sieve formula

MSC classification

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 92D10: Genetics 60K35: Interacting random processes; statistical mechanics type models; percolation theory 92D25: Population dynamics (general)
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 3 , September 2010 , pp. 713 - 731
Copyright
Copyright © Applied Probability Trust 2010 

References

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