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Excheangable partitions derived from Markovian coalescents. (English) Zbl 1147.60022
This paper investigates the problems of random coalescent process. The idea comes from biological studies for genealogy of haploid model: given a large population with many generations. As you track further, the family lines coalesce with each other, eventually all terminating at a common ancestor of current generation. The authors show that the basic integral representation of transition rates for the so-called \(\Lambda\)-coalescent is forced by sampling consistency under more general assumptions on the coalescent process. Exploiting an analogy with the theory of regenerative partition structures, they provide various characterizations of associated partition structures in terms of discrete-time Markov chains.

MSC:
60G09 Exchangeability for stochastic processes
60C05 Combinatorial probability
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[1] 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. · Zbl 0562.60042
[2] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Univ. Press. · Zbl 1107.60002
[3] 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. · Zbl 1064.60177
[4] Bolthausen, E. and Sznitman, A. S. (1998). On Ruelle’s probability cascades and an abstract cavity method. Comm. Math. Phys. 197 247–276. · Zbl 0927.60071
[5] 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. JSTOR: · Zbl 0284.60064
[6] 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. · Zbl 1123.60061
[7] Donnelly, P. and Joyce, P. (1991). Consistent ordered sampling distributions: characterization and convergence. Adv. in Appl. Probab. 23 229–258. JSTOR: · Zbl 0724.60040
[8] Donnelly, P. and Tavaré, S. (1986). The ages of alleles and a coalescent. Adv. in Appl. Probab. 18 1–19. JSTOR: · Zbl 0588.92013
[9] Evans, S. N. and Pitman, J. (1998). Construction of Markovian coalescents. Ann. Inst. H. Poincaré Probab. Statist. 34 339–383. · Zbl 0906.60058
[10] 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.) · Zbl 0245.92009
[11] Ford, D. J. (2005). Probabilities on cladograms: introduction to the alpha model. Available at http://front.math.ucdavis.edu/math.PR/0511246.
[12] Gnedin, A. (1997). The representation of composition structures. Ann. Probab. 25 1437–1450. · Zbl 0895.60037
[13] Gnedin, A. and Pitman, J. (2004). Regenerative partition structures. Electron. J. Combin. 11 Research Paper 12 21. · Zbl 1078.60009
[14] 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. · Zbl 1105.60011
[15] Gnedin, A. and Pitman, J. (2005). Regenerative composition structures. Ann. Probab. 33 445–479. · Zbl 1070.60034
[16] 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. · Zbl 1176.60027
[17] Gnedin, A. and Yakubovich, Y. (2006). Recursive partition structures. Ann. Probab. 34 2203–2218. · Zbl 1119.60025
[18] 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.
[19] Kingman, J. F. C. (1978). The representation of partition structures. J. London Math. Soc. (2) 18 374–380. · Zbl 0415.92009
[20] Kingman, J. F. C. (1982). The coalescent. Stochastic Process. Appl. 13 235–248. · Zbl 0491.60076
[21] 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. · Zbl 0494.92011
[22] Kingman, J. F. C. (1982). On the genealogy of large populations. J. Appl. Probab. 19A 27–43. · Zbl 0516.92011
[23] Möhle, M. (2006). On sampling distributions for coalescent processes with simultaneous multiple collisions. Bernoulli 12 35–53. · Zbl 1099.92052
[24] Möhle, M. (2006). On a class of non-regenerative sampling distributions. Combin. Probab. Comput. 16 435–444. · Zbl 1125.62122
[25] Möhle, M. and Sagitov, S. (2001). A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29 1547–1562. · Zbl 1013.92029
[26] Moran, P. A. P. (1958). Random processes in genetics. Proc. Camb. Phil. Soc. 54 60–71. · Zbl 0091.15701
[27] Nordborg, M. (2001). Coalescent theory. In Handbook of Statistical Genetics (D. J. Balding et al., eds.) 179–208. Wiley, New York.
[28] Pitman, J. (2006). Combinatorial stochastic processes . École d ’ Été de Probabilités de Saint-Flour XXXII—2002 . Lecture Notes Math. 1875 . · Zbl 1103.60004
[29] Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870–1902. · Zbl 0963.60079
[30] Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 1116–1125. · Zbl 0962.92026
[31] Sagitov, S. (2003). Convergence to the coalescent with simultaneous multiple mergers. J. Appl. Probab. 40 839–854. · Zbl 1052.92044
[32] Schweinsberg, J. (2000). A necessary and sufficient condition for the \(\Lambda\)-coalescent to come down from infinity. Electron. Comm. Probab. 5 1–11. · Zbl 0953.60072
[33] Schweinsberg, J. (2000). Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5 paper 12 50. · Zbl 0959.60065
[34] Tavaré, S. (1984). Line-of-descent and genealogical processes, and their applications in population genetics models. Theoret. Population Biol. 26 119–164. · Zbl 0555.92011
[35] Watterson, G. A. (1976). Reversibility and the age of an allele. Theoret. Population Biol. 10 239–253. · Zbl 0351.92015
[36] Watterson, G. A. (1984). Lines of descent and the coalescent. Theoret. Population Biol. 26 77–92. · Zbl 0542.92015
[37] Young, J. E. (1995). Partition-valued stochastic processes with applications. Ph.D. thesis, Univ. California, Berkeley.
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