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Asymptotic behavior of a Feller evolution family involved in the Fisher-Wright model

Part of: Miscellaneous applications of operator theory

Published online by Cambridge University Press:  01 July 2016

Adam Bobrowski
Adam Bobrowski*
Affiliation:
Polish Academy of Sciences and Lublin University of Technology
*
∗ Postal address: Nadbystrzycka 38A, 20-618 Lublin, Poland. Email address: a.bobrowski@pollub.pl
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Abstract

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We study the evolution in time of the joint distribution of a pair of Feller processes, related by the fact that some random time ago they were identical, evolving as a single Feller process; from that time on, they began to evolve independently, conditional on a state at the time of split, according to the same Feller transition probabilities. Such processes are involved in the Fisher-Wright model: the distribution of the time counted backwards from the present to the time of split in the past is a function of deterministic but time-varying effective size 2N of the population from which the two processes are sampled. In terms of a corresponding family of Feller operators, assuming asymptotic stability or ergodicity of the process of mutation, we find the limit form of the distribution of such pairs of processes sampled from decaying, asymptotically constant, and growing populations. In the case where mutation is not asymptotically stable or ergodic, limit distributions are found for the distribution of relative differences.

Keywords

Fisher-Wright-Moran modelFeller processevolution familymutationgenetic driftcoalescence

MSC classification

Primary: 92D25: Population dynamics (general)
Secondary: 47N60: Applications in chemistry and life sciences
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 3 , September 2008 , pp. 734 - 758
Copyright
Copyright © Applied Probability Trust 2008 

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