## Electronic Communications in Probability

### A population model for $\Lambda$-coalescents with neutral mutations

Andreas Lagerås

#### Abstract

Bertoin and Le Gall (2003) introduced a certain probability measure valued Markov process that describes the evolution of a population, such that a sample from this population would exhibit a genealogy given by the so-called $\Lambda$-coalescent, or coalescent with multiple collisions, introduced independently by Pitman (1999) and Sagitov (1999). We show how this process can be extended to the case where lineages can experience mutations. Regenerative compositions enter naturally into this model, which is somewhat surprising, considering a negative result by Möhle (2007).

#### Article information

Source
Electron. Commun. Probab., Volume 12 (2007), paper no. 2, 9-20.

Dates
Accepted: 4 February 2007
First available in Project Euclid: 6 June 2016

https://projecteuclid.org/euclid.ecp/1465224945

Digital Object Identifier
doi:10.1214/ECP.v12-1245

Mathematical Reviews number (MathSciNet)
MR2284043

Zentralblatt MATH identifier
1130.60057

Subjects
Primary: 60G09: Exchangeability
Secondary: 60G57: Random measures 92D25: Population dynamics (general)

Rights

#### Citation

Lagerås, Andreas. A population model for $\Lambda$-coalescents with neutral mutations. Electron. Commun. Probab. 12 (2007), paper no. 2, 9--20. doi:10.1214/ECP.v12-1245. https://projecteuclid.org/euclid.ecp/1465224945

#### References

• Bertoin, Jean; Le Gall, Jean-François. Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 (2003), no. 2, 261–288.
• Dong, Rui; Gnedin, Alexander; Pitman, Jim. Exchangeable partitions derived from Markovian coalescents. Ann. Appl. Probab. 17 (2007), no. 4, 1172–1201. http://front.math.ucdavis.edu/math.PR/0603745
• Gnedin, Alexander; Pitman, Jim. Regenerative composition structures. Ann. Probab. 33 (2005), no. 2, 445–479.
• Kingman, J. F. C. The coalescent. Stochastic Process. Appl. 13 (1982), no. 3, 235–248.
• Kingman, J. F. C. Poisson processes. Oxford Studies in Probability, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. viii+104 pp. ISBN: 0-19-853693-3
• Möhle, M. On sampling distributions for coalescent processes with simultaneous multiple collisions. Bernoulli 12 (2006), no. 1, 35–53.
• Möhle, Martin. On a class of non-regenerative sampling distributions. Combin. Probab. Comput. 16 (2007), no. 3, 435–444. http://dx.doi.org/10.1017/S0963548306008212
• Möhle, Martin; Sagitov, Serik. A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29 (2001), no. 4, 1547–1562.
• Pitman, Jim. Coalescents with multiple collisions. Ann. Probab. 27 (1999), no. 4, 1870–1902.
• Sagitov, Serik. The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 (1999), no. 4, 1116–1125.
• Schweinsberg, Jason. Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5 (2000), Paper no. 12, 50 pp. (electronic). http://www.math.washington.edu/~ejpecp/