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

  • Part of
  • M. Möhle
  • Journal:
    Journal of Applied Probability / Volume 47 / Issue 3 / September 2010
    Published online by Cambridge University Press:
    14 July 2016, pp. 713-731
    Print publication:
    September 2010
    • Article
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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.