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A Central Limit Theorem Associated with the Transformed Two-Parameter Poisson–Dirichlet Distribution

Part of: Limit theorems

Published online by Cambridge University Press:  14 July 2016

Fang Xu
Fang Xu*
Affiliation:
Beijing Normal University and McMaster University
*
∗ Postal address: Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4K1. Email address: xuf23@math.mcmaster.ca
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Abstract

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In this paper we introduce the transformed two-parameter Poisson–Dirichlet distribution on the ordered infinite simplex. Furthermore, we prove the central limit theorem related to this distribution when both the mutation rate θ and the selection rate σ become large in a specified manner. As a consequence, we find that the properly scaled homozygosities have asymptotical normal behavior. In particular, there is a certain phase transition with the limit depending on the relative strength of σ and θ.

Keywords

Poisson–Dirichlet distributiontwo-parameter Poisson–Dirichlet distributionhomozygosityGEM representationselection

MSC classification

Secondary: 60F05: Central limit and other weak theorems 92D10: Genetics

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 2 , June 2009 , pp. 392 - 401
Copyright
Copyright © Applied Probability Trust 2009 

References

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