Convergence time to the Ewens sampling formula in the infinite alleles Moran model

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

In this paper, we establish an upper bound for time to convergence to stationarity for the discrete time infinite alleles Moran model. If M is the population size and μ is the mutation rate, this bound gives a cutoff time of log(M μ)/μ generations. The stationary distribution for this process in the case of sampling without replacement is the Ewens sampling formula. We show that the bound for the total variation distance from the generation t distribution to the Ewens sampling formula is well approximated by one of the extreme value distributions, namely, a standard Gumbel distribution. Beginning with the card shuffling examples of Aldous and Diaconis and extending the ideas of Donnelly and Rodrigues for the two allele model, this model adds to the list of Markov chains that show evidence for the cutoff phenomenon. Because of the broad use of infinite alleles models, this cutoff sets the time scale of applicability for statistical tests based on the Ewens sampling formula and other tests of neutrality in a number of population genetic studies.

Original languageEnglish (US)
Pages (from-to)189-206
Number of pages18
JournalJournal of Mathematical Biology
Volume60
Issue number2
DOIs
StatePublished - Sep 2010

Fingerprint

Ewens Sampling Formula
Convergence Time
Alleles
Sampling
alleles
Sampling without Replacement
Gumbel Distribution
Total Variation Distance
sampling
Extreme Value Distribution
Markov Chains
Neutrality
Population Genetics
t-distribution
Statistical tests
Stationarity
Mutation Rate
Statistical test
Stationary Distribution
Population Density

Keywords

  • Cutoff phenomena
  • Ewens sampling formula
  • Extreme value distribution
  • Hoppe's urn
  • Infinite alleles Moran model
  • Lines of descent
  • Markov chains

ASJC Scopus subject areas

  • Agricultural and Biological Sciences (miscellaneous)
  • Applied Mathematics
  • Modeling and Simulation

Cite this

Convergence time to the Ewens sampling formula in the infinite alleles Moran model. / Watkins, Joseph C.

In: Journal of Mathematical Biology, Vol. 60, No. 2, 09.2010, p. 189-206.

Research output: Contribution to journalArticle

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