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A functional central limit theorem for the Ewens sampling formula

Published online by Cambridge University Press:  14 July 2016

Jennie C. Hansen
Jennie C. Hansen*
Affiliation:
Tufts University
*
∗Present address: Mathematics Department, Northeastern University, Boston, MA 02115, USA.
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Abstract

For each n > 0, the Ewens sampling formula from population genetics is a measure on the set of all partitions of the integer n. To determine the limiting distributions for the part sizes of a partition with respect to the measures given by this formula, we associate to each partition a step function on [0, 1]. Each jump in the function equals the number of parts in the partition of a certain size. We normalize these functions and show that the induced measures on D[0, 1] converge to Wiener measure. This result complements Kingman's frequency limit theorem [10] for the Ewens partition structure.

Keywords

PARTITION STRUCTURESWIENER MEASURERANDOM PARTITIONSPÓLYA-LIKE URN
Type
Research Papers
Information
Journal of Applied Probability , Volume 27 , Issue 1 , March 1990 , pp. 28 - 43
Copyright
Copyright © Applied Probability Trust 1990 

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References

[1] Billingsley, P. (1976) Convergence of Probability Measures. Wiley, New York.Google Scholar
[2] Billingsley, P. (1979) Probability and Measure. Wiley, New York.Google Scholar
[3] Delaurentis, J. M. and Pittel, B. (1985) Random permutations and Brownian motion. Pacific J. Math. 119, 287–301.CrossRefGoogle Scholar
[4] Donnelly, P. (1986) Partition structures, Pólya urns, the Ewens sampling formula, and the age of alleles. Theoret. Popn Biol. 30, 271–288.Google Scholar
[5] Ewens, W. J. (1972) The sampling theory of selectively neutral alleles. Theoret. Popn Biol. 3, 87–112.Google Scholar
[6] Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. Wiley, New York.Google Scholar
[7] Hansen, J. C. (1989) A functional central limit theorem for random mappings. Ann. Prob. 17, 317–332.Google Scholar
[8] Hoppe, F. M. (1984) Pólya-like urns and Ewens' sampling formula. J. Math. Biol. 20, 91–94.Google Scholar
[9] Kingman, J. F. C. (1975) Random discrete distributions. J. R. Statist. Soc. B 37, 1–22.Google Scholar
[10] Kingman, J. F. C. (1978) Random partitions in population genetics. Proc. Roy. Soc. London A 361, 1–20.Google Scholar
[11] Kingman, J. F. C. (1980) Mathematics of Genetic Diversity. SIAM, Philadelphia.CrossRefGoogle Scholar
[12] Watterson, G. A. (1974) The sampling theory of selectively neutral alleles. Adv. Appl. Prob. 6, 463–488.CrossRefGoogle Scholar
[13] Watterson, G. A. (1976) The stationary distribution of the infinitely-many neutral alleles diffusion model. J. Appl. Prob. 13, 639–651.Google Scholar