The Annals of Applied Probability

Large deviations associated with Poisson–Dirichlet distribution and Ewens sampling formula

Shui Feng

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Abstract

Several results of large deviations are obtained for distributions that are associated with the Poisson–Dirichlet distribution and the Ewens sampling formula when the parameter θ approaches infinity. The motivation for these results comes from a desire of understanding the exact meaning of θ going to infinity. In terms of the law of large numbers and the central limit theorem, the limiting procedure of θ going to infinity in a Poisson–Dirichlet distribution corresponds to a finite allele model where the mutation rate per individual is fixed and the number of alleles going to infinity. We call this the finite allele approximation. The first main result of this article is concerned with the relation between this finite allele approximation and the Poisson–Dirichlet distribution in terms of large deviations. Large θ can also be viewed as a limiting procedure of the effective population size going to infinity. In the second result a comparison is done between the sample size and the effective population size based on the Ewens sampling formula.

Article information

Source
Ann. Appl. Probab., Volume 17, Number 5-6 (2007), 1570-1595.

Dates
First available in Project Euclid: 3 October 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1191419176

Digital Object Identifier
doi:10.1214/105051607000000230

Mathematical Reviews number (MathSciNet)
MR2358634

Zentralblatt MATH identifier
1145.92025

Subjects
Primary: 60F10: Large deviations
Secondary: 92D10: Genetics {For genetic algebras, see 17D92}

Keywords
Ewens sampling formula Dirichlet distribution Poisson–Dirichlet distribution GEM representation large deviation

Citation

Feng, Shui. Large deviations associated with Poisson–Dirichlet distribution and Ewens sampling formula. Ann. Appl. Probab. 17 (2007), no. 5-6, 1570--1595. doi:10.1214/105051607000000230. https://projecteuclid.org/euclid.aoap/1191419176


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