The Annals of Applied Probability

Exchangeable partitions derived from Markovian coalescents

Rui Dong, Alexander Gnedin, and Jim Pitman

Full-text: Open access

Abstract

Kingman derived the Ewens sampling formula for random partitions describing the genetic variation in a neutral mutation model defined by a Poisson process of mutations along lines of descent governed by a simple coalescent process and observed that similar methods could be applied to more complex models. Möhle described the recursion which determines the generalization of the Ewens sampling formula in the situation where the lines of descent are governed by a Λ-coalescent, which allows multiple mergers. Here, we show that the basic integral representation of transition rates for the Λ-coalescent is forced by sampling consistency under more general assumptions on the coalescent process. Exploiting an analogy with the theory of regenerative partition structures, we provide various characterizations of the associated partition structures in terms of discrete-time Markov chains.

Article information

Source
Ann. Appl. Probab. Volume 17, Number 4 (2007), 1172-1201.

Dates
First available in Project Euclid: 10 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1186755236

Digital Object Identifier
doi:10.1214/105051607000000069

Mathematical Reviews number (MathSciNet)
MR2344303

Zentralblatt MATH identifier
1147.60022

Subjects
Primary: 60G09: Exchangeability
Secondary: 60C05: Combinatorial probability

Keywords
Exchangeable partitions Λ-coalescent with freeze consistency decrement matrix

Citation

Dong, Rui; Gnedin, Alexander; Pitman, Jim. Exchangeable partitions derived from Markovian coalescents. Ann. Appl. Probab. 17 (2007), no. 4, 1172--1201. doi:10.1214/105051607000000069. https://projecteuclid.org/euclid.aoap/1186755236


Export citation

References

  • Aldous, D. J. (1985). Exchangeability and related topics. École d'Été de Probabilités de Saint-Flour XIII---1983. Lecture Notes in Math. 1117 1--198. Springer, Berlin.
  • Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Univ. Press.
  • Bertoin, J. and Goldschmidt, C. (2004). Dual random fragmentation and coagulation and an application to the genealogy of Yule processes. In Math. and Comp. Sci. III. Trends Math. 295--308. Birkhäuser, Basel.
  • 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 Markov chains arising in genetics: A new approach. I. Haploid models. Adv. in Appl. Probab. 6 260--290.
  • Dong, R., Goldschmidt, C. and Martin, J. B. (2005). Coagulation-fragmentation duality, Poisson--Dirichlet distributions and random recursive trees. Available at http://front.math.ucdavis.edu/math.PR/0507591.
  • Donnelly, P. and Joyce, P. (1991). Consistent ordered sampling distributions: characterization and convergence. Adv. in Appl. Probab. 23 229--258.
  • Donnelly, P. and Tavaré, S. (1986). The ages of alleles and a coalescent. Adv. in Appl. Probab. 18 1--19.
  • Evans, S. N. and Pitman, J. (1998). Construction of Markovian coalescents. Ann. Inst. H. Poincaré Probab. Statist. 34 339--383.
  • Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Population Biology 3 87--112. (Erratum, ibid. 3 240; erratum, ibid. 3 376.)
  • Ford, D. J. (2005). Probabilities on cladograms: introduction to the alpha model. Available at http://front.math.ucdavis.edu/math.PR/0511246.
  • Gnedin, A. (1997). The representation of composition structures. Ann. Probab. 25 1437--1450.
  • Gnedin, A. and Pitman, J. (2004). Regenerative partition structures. Electron. J. Combin. 11 Research Paper 12 21.
  • Gnedin, A. and Pitman, J. (2005). Markov and self-similar composition structures. Zapiski Nauchnych Seminarov POMI 326 59--84. Available at http://www.pdmi.ras.ru/znsl/2005/v326.html.
  • Gnedin, A. and Pitman, J. (2005). Regenerative composition structures. Ann. Probab. 33 445--479.
  • Gnedin, A. and Pitman, J. (2006). Moments of convex distribution functions and completely alternating sequences. Available at http://front.math.ucdavis.edu/math.PR/0602091.
  • Gnedin, A. and Yakubovich, Y. (2006). Recursive partition structures. Ann. Probab. 34 2203--2218.
  • Haas, B., Miermont, G., Pitman, J. and Winkel, M. (2006). Asymptotics of discrete fragmentation trees and applications to phylogenetic models. Available at http://front.math.ucdavis.edu/math.PR/0604350.
  • Kingman, J. F. C. (1978). The representation of partition structures. J. London Math. Soc. (2) 18 374--380.
  • Kingman, J. F. C. (1982). The coalescent. Stochastic Process. Appl. 13 235--248.
  • Kingman, J. F. C. (1982). Exchangeability and the evolution of large populations. In Exchangeability in Probability and Statistics (Rome, 1981) 97--112. North-Holland, Amsterdam.
  • Kingman, J. F. C. (1982). On the genealogy of large populations. J. Appl. Probab. 19A 27--43.
  • Möhle, M. (2006). On sampling distributions for coalescent processes with simultaneous multiple collisions. Bernoulli 12 35--53.
  • Möhle, M. (2006). On a class of non-regenerative sampling distributions. Combin. Probab. Comput. 16 435--444.
  • Möhle, M. and Sagitov, S. (2001). A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29 1547--1562.
  • Moran, P. A. P. (1958). Random processes in genetics. Proc. Camb. Phil. Soc. 54 60--71.
  • Nordborg, M. (2001). Coalescent theory. In Handbook of Statistical Genetics (D. J. Balding et al., eds.) 179--208. Wiley, New York.
  • Pitman, J. (2006). Combinatorial stochastic processes. École d'Été de Probabilités de Saint-Flour XXXII---2002. Lecture Notes Math. 1875.
  • 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.
  • Sagitov, S. (2003). Convergence to the coalescent with simultaneous multiple mergers. J. Appl. Probab. 40 839--854.
  • Schweinsberg, J. (2000). A necessary and sufficient condition for the $\Lambda$-coalescent to come down from infinity. Electron. Comm. Probab. 5 1--11.
  • Schweinsberg, J. (2000). Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5 paper 12 50.
  • Tavaré, S. (1984). Line-of-descent and genealogical processes, and their applications in population genetics models. Theoret. Population Biol. 26 119--164.
  • Watterson, G. A. (1976). Reversibility and the age of an allele. Theoret. Population Biol. 10 239--253.
  • Watterson, G. A. (1984). Lines of descent and the coalescent. Theoret. Population Biol. 26 77--92.
  • Young, J. E. (1995). Partition-valued stochastic processes with applications. Ph.D. thesis, Univ. California, Berkeley.