### 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 language | English (US) |
---|---|

Pages (from-to) | 189-206 |

Number of pages | 18 |

Journal | Journal of Mathematical Biology |

Volume | 60 |

Issue number | 2 |

DOIs | |

State | Published - Sep 2010 |

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### 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.

Research output: Contribution to journal › Article

*Journal of Mathematical Biology*, vol. 60, no. 2, pp. 189-206. https://doi.org/10.1007/s00285-009-0255-x

}

TY - JOUR

T1 - Convergence time to the Ewens sampling formula in the infinite alleles Moran model

AU - Watkins, Joseph C

PY - 2010/9

Y1 - 2010/9

N2 - 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.

AB - 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.

KW - Cutoff phenomena

KW - Ewens sampling formula

KW - Extreme value distribution

KW - Hoppe's urn

KW - Infinite alleles Moran model

KW - Lines of descent

KW - Markov chains

UR - http://www.scopus.com/inward/record.url?scp=74049099360&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=74049099360&partnerID=8YFLogxK

U2 - 10.1007/s00285-009-0255-x

DO - 10.1007/s00285-009-0255-x

M3 - Article

C2 - 19288263

AN - SCOPUS:74049099360

VL - 60

SP - 189

EP - 206

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 2

ER -