## The Annals of Probability

### A Classification of Coalescent Processes for Haploid Exchangeable Population Models

#### Abstract

We consider a class of haploid population models with nonoverlapping generations and fixed population size $N$ assuming that the family sizes within a generation are exchangeable random variables. A weak convergence criterion is established for a properly scaled ancestral process as $N \to \infty$. It results in a full classification of the coalescent generators in the case of exchangeable reproduction. In general the coalescent process allows for simultaneous multiple mergers of ancestral lines.

#### Article information

Source
Ann. Probab., Volume 29, Number 4 (2001), 1547-1562.

Dates
First available in Project Euclid: 5 March 2002

https://projecteuclid.org/euclid.aop/1015345761

Digital Object Identifier
doi:10.1214/aop/1015345761

Mathematical Reviews number (MathSciNet)
MR1880231

Zentralblatt MATH identifier
1013.92029

#### Citation

Möhle, Martin; Sagitov, Serik. A Classification of Coalescent Processes for Haploid Exchangeable Population Models. Ann. Probab. 29 (2001), no. 4, 1547--1562. doi:10.1214/aop/1015345761. https://projecteuclid.org/euclid.aop/1015345761

#### References

• Aldous, D. J. (1997). Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25 812-854.
• Aldous, D. J. and Limic, V. (1998). The entrance boundary of the multiplicative coalescent. Electron. J. Probab. 3.
• Aldous, D. J. and Pitman, J. (1998). The standard additive coalescent. Ann. Probab. 26 1703-1726.
• 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.
• Bertoin, J. and Pitman, J. (2000). Two coalescents derived from the ranges of stable subordinators. Electron J. Probab. 5.
• Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
• Bolthausen, E. and Sznitman, A.-S. (1998). On Ruelle's probability cascades and an abstract cavity method. Comm. Math. Phys. 197 247-276.
• Cannings, C. (1974). The latent roots of certain Markovchains arising in genetics: a new approach I. Haploid models. Adv. in Appl. Probab. 6 260-290.
• Cannings, C. (1975). The latent roots of certain Markovchains arising in genetics: a new approach II. Further haploid models. Adv. in Appl. Probab. 7 264-282.
• Donnelly, P. and Tavar´e, S. (1995). Coalescents and genealogical structure under neutrality. Ann. Rev. Genet. 29 401-421.
• Evans, S. N. and Pitman, J. (1998). Construction of Markovian coalescents. Ann. Inst. H. Poincar´e Probab. Statist. 34 339-383.
• Feller, W. (1971). An Introduction to Probability Theory and Its Applications, 1, 2nd ed. Wiley, New York.
• Hudson, R. R. (1991). Gene genealogies and the coalescent process. Oxford Survey Evol. Biol. 7 1-44. Kingman, J. F. C. (1982a). On the genealogy of large populations. J. Appl. Probab. 19A 27-43. Kingman, J. F. C. (1982b). Exchangeability and the evolution of large populations. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, eds.) 97-112. North- Holland, Amsterdam. Kingman, J. F. C. (1982c). The coalescent. Stochastic Process. Appl. 13 235-248.
• Li, W. H. and Fu, Y. X. (1999). Coalescent theory and its applications in population genetics. In Statistics in Genetics (M. E. Halloran and S. Geisser, eds.) Springer, Berlin.
• Möhle, M. (1998). Robustness results for the coalescent. J. Appl. Probab. 35 438-447.
• Möhle, M. (1999). Weak convergence to the coalescent in neutral population models. J. Appl. Probab. 36 446-460.
• Möhle, M. (2000). Ancestral processes in population genetics: the coalescent. J. Theoret. Biol. 204 629-638.
• Möhle, M. and Sagitov, S. (1998). A characterisation of ancestral limit processes arising in haploid population genetics models. Preprint. Johannes Gutenberg-Univ., Mainz.
• Nordborg, M. (2001). Coalescent theory. In Handbook of Statistical Genetics (D. J. Balding, C. Cannings and M. Bishop, eds.) Wiley, Chichester. To appear.
• Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870-1902.
• Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 1116-1125.
• Schweinsberg, J. (2000a). A necessary and sufficient condition for the 2-coalescent to come down from infinity. Electron. Comm. Probab. 5 1-11.
• Schweinsberg, J. (2000b). Coalescents with simultaneous multiple collision. Technical Report Electron. J. Probab. 5.