Bernoulli

On sampling distributions for coalescent processes with simultaneous multiple collisions

M. Möhle

Full-text: Open access

Abstract

Recursions for a class of sampling distributions of allele configurations are derived for the situation where the genealogy of the underlying population is modelled by a coalescent process with simultaneous multiple collisions of ancestral lineages. These recursions describe a new family of partition structures in terms of the composition probability function, parametrized by the infinitesimal rates of the coalescent process. For the Kingman coalescent process with only binary mergers of ancestral lines, the recursion reduces to that known for the classical Ewens sampling distribution. We solve the recursion for the star-shaped coalescent. The asymptotic behaviour of the number Kn of alleles (types) for large sample size n is studied, in particular for the star-shaped coalescent and the Bolthausen-Sznitman coalescent.

Article information

Source
Bernoulli, Volume 12, Number 1 (2006), 35-53.

Dates
First available in Project Euclid: 28 February 2006

Permanent link to this document
https://projecteuclid.org/euclid.bj/1141136647

Mathematical Reviews number (MathSciNet)
MR2202319

Zentralblatt MATH identifier
1099.92052

Keywords
Bolthausen-Sznitman coalescent composition probability function Ewens sampling formula mutation rate neutral infinite alleles model partition structure sampling distribution simultaneous multiple collisions star-shaped tree

Citation

Möhle, M. On sampling distributions for coalescent processes with simultaneous multiple collisions. Bernoulli 12 (2006), no. 1, 35--53. https://projecteuclid.org/euclid.bj/1141136647


Export citation

References

  • [1] Bertoin, J. and Le Gall, J.F. (2000) The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes. Probab. Theory Related Fields, 117, 249-266.
  • [2] Bolthausen, E. and Sznitman, A.-S. (1998) On Ruellés probability cascades and an abstract cavity method. Comm. Math. Phys., 197, 247-276.
  • [3] Cannings C. (1974) The latent roots of certain Markov chains arising in genetics: a new approach, I. Haploid models. Adv. Appl. Probab., 6, 260-290.
  • [4] Gnedin, A. and Pitman, J. (2005) Regenerative composition structures. Ann. Probab., 33, 445-479.
  • [5] Griffiths, R.C. and Lessard, S. (2005) Ewens´ sampling formula and related formulae: combinatorial proofs, extensions to variable population size and applications to ages of alleles. Theoret. Popul. Biol. To appear.
  • [6] Griffiths, R.C. and Tavaré, S. (1996) Monte Carlo inference methods in population genetics. Math. Comput. Modelling, 23, 141-158.
  • [7] Johnson, N.L., Kotz, S. and Kemp A.W. (1992) Univariate Discrete Distributions, 2nd edn. New York: Wiley.
  • [8] Joyce, P., Krone, S.M. and Kurtz, T.G. (2002) Gaussian limits associated with the Poisson-Dirichlet distribution and the Ewens sampling formula. Ann. Appl. Probab., 12, 101-124.
  • [9] Kingman, J.F.C. (1977) The population structure associated with the Ewens sampling formula. Theoret. Popul. Biol., 11, 274-283.
  • [10] Kingman, J.F.C. (1982a) On the genealogy of large populations. J. Appl. Probab., 19A, 27-43.
  • [11] Kingman, J.F.C. (1982b) Exchangeability and the evolution of large populations. In G. Koch and F. Spizzichino (eds), Exchangeability in Probability and Statistics, pp. 97-112. Amsterdam: North-Holland.
  • [12] Kingman, J.F.C. (1982c) The coalescent. Stochastic Process. Appl., 13, 235-248.
  • [13] Kingman, J.F.C. (2000) Origins of the coalescent: 1974-1982. Genetics, 156, 1461-1463.
  • [14] Möhle, M. and Sagitov, S. (2001) A classification of coalescent processes for haploid exchangeable population models. Ann. Probab., 29, 1547-1562.
  • [15] Pitman, J. (1995) Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields, 102, 145-158.
  • [16] Pitman, J. (1999) Coalescents with multiple collisions. Ann. Probab., 27, 1870-1902.
  • [17] Pitman, J. (2002) Combinatorial stochastic processes. Technical Report 621, Department of Statistics, University of California, Berkeley.
  • [18] Rosenblatt, M. (1959) Functions of a Markov process that are Markovian. J. Math. Mech., 8, 585-596.
  • [19] Sagitov, S. (1999) The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab., 36, 1116-1125.
  • [20] Schweinsberg, J. (2000a) A necessary and sufficient condition for the Λ-coalescent to come down from infinity. Electron. Comm. Probab., 5, 1-11.
  • [21] Schweinsberg, J. (2000b) Coalescents with simultaneous multiple collisions. Electron. J. Probab., 5, 1-50.