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Article

Weak limit for the stationary distribution of the infinite-alleles model

  • Haim Avni (a1)
Abstract

The limit behavior of the stationary distribution of the infinite-alleles model is reduced to a single Laplace transform formula. Some known results, such as Ewens' sampling formula, the distribution of the order-statistics and the frequency spectrum density are shown to follow from this relation. All the results are obtained within the framework of the configuration process, without recourse to finite alleles models.

In view of some recent results by Kingman (1977), the results apply wherever the Ewens' sampling formula is valid.

Copyright
COPYRIGHT: © Applied Probability Trust 1979 
Corresponding author
∗ Currently at the Israel Institute for Biological Research, P.O.B. 19, Nes-Ziona, Israel.
References
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Ewens, W. J. (1972) The sampling theory of selectively neutral alleles. Theoret. Popn Biol. 3, 87–112.
Karlin, S. and Mcgregor, J. L. (1965) The number of mutant forms maintained in a population. Proc. 5th Berkeley Symp. Math. Statist. Prob. 4, 415–438.
Karlin, S. and Mcgregor, J. L. (1972) An addendum to a paper by W. J. Ewens. Theoret. Popn Biol. 3, 113–117.
Kingman, J. F. C. (1975) Random discrete distributions. J. R. Statist. Soc. B 37, 1–15.
Kingman, J. F. C. (1977) The population structure associated with the Ewens sampling formula. Theoret. Popn Biol. 11, 274–283.
Trajstman, A. C. (1974) On a conjecture of G. A. Watterson. Adv. Appl. Prob. 6, 489–493.
Watterson, G. A. (1974) The sampling theory of selectively neutral alleles. Adv. Appl. Prob. 6, 463–488.
Watterson, G. A. (1976) The stationary distribution of the infinitely many neutral alleles diffusion model. J. Appl. Prob. 13, 639–651.
Watterson, G. A. and Guess, H. A. (1977) Is the most frequent allele the oldest? Theoret. Popn Biol. 11, 141–160.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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