Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T00:34:53.521Z Has data issue: false hasContentIssue false
  • English
  • Français

Λ-coalescents: a survey

Part of: Markov processes Stochastic processes Combinatorial probability

Published online by Cambridge University Press:  30 March 2016

Alexander Gnedin ,
Alexander Iksanov  and
Alexander Marynych
Alexander Gnedin
Affiliation:
School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK. Email address: a.gnedin@qmul.ac.uk.
Alexander Iksanov
Affiliation:
Faculty of Cybernetics, National Taras Shevchenko University of Kiev, 01033 Kiev, Ukraine. Email address: iksan@univ.kiev.ua.
Alexander Marynych
Affiliation:
Faculty of Cybernetics, National Taras Shevchenko University of Kiev, 01033 Kiev, Ukraine. Email address: marynych@unicyb.kiev.ua.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Λ-coalescents model the evolution of a coalescing system in which any number of components randomly sampled from the whole may merge into larger blocks. This survey focuses on related combinatorial constructions and the large-sample behaviour of the functionals which characterize in some way the speed of coalescence.

Keywords

Coalescent with multiple mergersexchangeable partitionslarge-sample asymptotics

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60C05: Combinatorial probability 60G09: Exchangeability 60G51: Processes with independent increments; L'evy processes
Type
Part 2. The 2014 AP lectures
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 23 - 40
Copyright
Copyright © Applied Probability Trust 2014 

References

Abraham, R., and Delmas, J.-F. (2013). β-coalescents and stable Galton-Watson trees. Preprint. Available at http://arxiv.org/abs/1303.6882v1.Google Scholar
Abraham, R., and Delmas, J.-F. (2013). A construction of a β-coalescent via the pruning of binary trees. J. Appl. Prob. 50, 772–790.CrossRefGoogle Scholar
Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5, 3–48.CrossRefGoogle Scholar
Barbour, A. D., and Gnedin, A. V. (2006). Regenerative compositions in the case of slow variation. Stoch. Process. Appl. 116, 1012–1047.CrossRefGoogle Scholar
Basdevant, A.-L., and Goldschmidt, C. (2008). Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent. Electron. J. Prob. 13, 486–512.CrossRefGoogle Scholar
Berestycki, J., Berestycki, N., and Limic, V. (2010). The Λ-coalescent speed of coming down from infinity. Ann. Prob. 38, 207–233.CrossRefGoogle Scholar
Berestycki, J., Berestycki, N., and Limic, V. (2014). Asymptotic sampling formulae for Λ-coalescents. Ann. Inst. H. Poincaré Prob. Statist. 50, 715–731.CrossRefGoogle Scholar
Berestycki, J., Berestycki, N., and Schweinsberg, J. (2007). Beta-coalescents and continuous stable random trees. Ann. Prob. 35, 1835–1887.CrossRefGoogle Scholar
Berestycki, J., Berestycki, N., and Schweinsberg, J. (2008). Small-time behavior of beta coalescents. Ann. Inst. H. Poincaré Prob. Statist. 44, 214–238.CrossRefGoogle Scholar
Berestycki, N. (2009). Recent Progress In Coalescent Theory (Math. Surveys 16). Sociedade Brasileira de Matemática, Rio de Janeiro.Google Scholar
Bertoin, J. (2010). Exchangeable coalescents. Nachdiplom Lectures, ETH Zürich.Google Scholar
Bertoin, J., and Le Gall, J.-F. (2000). The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes. Prob. Theory Relat. Fields 117, 249–266.CrossRefGoogle Scholar
Bertoin, J., and Le Gall, J.-F. (2006). Stochastic flows associated to coalescent processes. {III}. Limit theorems. Illinois J. Math. 50, 147–181.CrossRefGoogle Scholar
Birkner, M. et al.(2005). Alpha-stable branching and beta-coalescents. Electron. J. Prob. 10, 303–325.CrossRefGoogle Scholar
Bolthausen, E., and Sznitman, A.-S. (1998). On Ruelle's probability cascades and an abstract cavity method. Commum. Math. Phys. 197, 247–276.CrossRefGoogle Scholar
Delmas, J.-F., Dhersin, J.-S., and Siri-Jegousse, A. (2008). Asymptotic results on the length of coalescent trees. Ann. Appl. Prob. 18, 997–1025.CrossRefGoogle Scholar
Dong, R., Gnedin, A., and Pitman, J. (2007). Exchangeable partitions derived from Markovian coalescents. Ann. Appl. Prob. 17, 1172–1201.CrossRefGoogle Scholar
Drmota, M., Iksanov, A., Moehle, M., and Roesler, U. (2007). Asymptotic results concerning the total branch length of the Bolthausen-Sznitman coalescent. Stoch. Process. Appl. 117, 1404–1421.CrossRefGoogle Scholar
Drmota, M., Iksanov, A., Moehle, M., and Roesler, U. (2009). A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree. Random Structures Algorithms 34, 319–336.CrossRefGoogle Scholar
Freund, F. and Möhle, M. (2009). On the number of allelic types for samples taken from exchangeable coalescents with mutation. Adv. Appl. Prob. 41, 1082–1101.CrossRefGoogle Scholar
Gnedin, A., and Iksanov, A. (2012). Regenerative compositions in the case of slow variation: a renewal theory approach. Electron. J. Prob. 17, 19 pp.CrossRefGoogle Scholar
Gnedin, A., and Pitman, J. (2005). Regenerative composition structures. Ann. Prob. 33, 445–479.CrossRefGoogle Scholar
Gnedin, A., and Yakubovich, Y. (2007). Collisions and freezing events in coalescents with multiple mergers. Tech. Rep. 46/2007, Mathematisches Forschungsinstitut Oberwolfach.Google Scholar
Gnedin, A., and Yakubovich, Y. (2007). On the number of collisions in Λ-coalescents. Electron. J. Prob. 12, 1547–1567.CrossRefGoogle Scholar
Gnedin, A., Hansen, B., and Pitman, J. (2007). Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws. Prob. Surveys 4, 146–171.CrossRefGoogle Scholar
Gnedin, A., Iksanov, A., and Marynych, A. (2010). Limit theorems for the number of occupied boxes in the Bernoulli sieve. Theory Stoch. Process. 16, 44–57.Google Scholar
Gnedin, A., Iksanov, A., and Marynych, A. (2011). On Λ-coalescents with dust component. J. Appl. Prob. 48, 1133–1151.CrossRefGoogle Scholar
Gnedin, A., Iksanov, A. and Möhle, M. (2008). On asymptotics of exchangeable coalescents with multiple collisions. J. Appl. Prob. 45, 1186–1195.CrossRefGoogle Scholar
Gnedin, A., Pitman, J., and Yor, M. (2006). Asymptotic laws for compositions derived from transformed subordinators. Ann. Prob. 34, 468–492.CrossRefGoogle Scholar
Gnedin, A., Iksanov, A., Marynych, A. and Möhle, M. (2014). On asymptotics of the beta coalescents. Adv. Appl. Prob. 46, 496–515.CrossRefGoogle Scholar
Goldschmidt, C., and Martin, J. B. (2005). Random recursive trees and the Bolthausen-Sznitman coalescent. Electron. J. Prob. 10, 718–745.CrossRefGoogle Scholar
Haas, B., and Miermont, G. (2011). Self-similar scaling limits of non-increasing Markov chains. Bernoulli 17, 1217–1247.CrossRefGoogle Scholar
Hénard, O. (2013). The fixation line. Preprint. Available at http://arxiv.org/abs/1307.0784v2.Google Scholar
Huillet, T. E. (2014). Pareto genealogies arising from a Poisson branching evolution model with selection. J. Math. Biol. 68, 727–761.CrossRefGoogle ScholarPubMed
Iksanov, A. and Möhle, M. (2008). On the number of jumps of random walks with a barrier. Adv. Appl. Prob. 40, 206–228.CrossRefGoogle Scholar
Iksanov, A. and Möhle, M. (2007). A probabilistic proof of a weak limit law for the number of cuts needed to isolate the root of a random recursive tree. Electron. Commun. Prob. 12, 28–35.CrossRefGoogle Scholar
Iksanov, A., Marynych, A. and Möhle, M. (2009). On the number of collisions in β(2,b)∧-coalescents. Bernoulli 15, 829–845.CrossRefGoogle Scholar
Kersting, G. (2012). The asymptotic distribution of the length of beta-coalescent trees. Ann. Appl. Prob. 22, 2086–2107.CrossRefGoogle Scholar
Kersting, G., Schweinsberg, J., and Wakolbinger, A. (2014). The evolving beta coalescent. Preprint. Available at http://arxiv.org/abs/1402.4534v1.Google Scholar
Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.CrossRefGoogle Scholar
Kingman, J. F. C. (1982). On the genealogy of large populations. In Essays in Statistical Science (J. Appl. Prob. Spec. Vol. 19A), eds Gani, J. and Hannan, E. J., Applied Probability Trust, Sheffield, pp. 27–43.Google Scholar
Limic, V., and Talarczyk, A. (2014). Second-order asymptotics for the block counting process in a class of regularly varying λ-coalescents. Preprint. Available at http://arxiv.org/abs/1304.5183v2.Google Scholar
Meir, A., and Moon, J. W. (1974). Cutting down recursive trees. Math. Biosci. 21, 173–181.CrossRefGoogle Scholar
Möhle, M. (2006). On sampling distributions for coalescent processes with simultaneous multiple collisions. Bernoulli 12, 35–53.Google Scholar
Möhle, M. (2006). On the number of segregating sites for populations with large family sizes. Adv. Appl. Prob. 38, 750–767.CrossRefGoogle Scholar
Möhle, M., and Sagitov, S. (2001). A classification of coalescent processes for haploid exchangeable population models. Ann. Prob. 29, 1547–1562.CrossRefGoogle Scholar
Panholzer, A. (2003). Non-crossing trees revisited: cutting down and spanning subtrees. In Discrete random walks (Paris, 2003; Discrete Math. Theoret. Comput. Sci. Proc. AC), Assoc. Discrete Math. Theoret. Comput. Sci., Nancy, pp. 265–276.Google Scholar
Pfaffelhuber, P., Wakolbinger, A., and Weisshaupt, H. (2011). The tree length of an evolving coalescent. Prob. Theory Relat. Fields 151, 529–557.CrossRefGoogle Scholar
Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27, 1870–1902.CrossRefGoogle Scholar
Pitman, J. (2006). Combinatorial Stochastic Processes (Lecture Notes Math. 1875). Springer, Berlin.Google Scholar
Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36, 1116–1125.CrossRefGoogle Scholar
Schweinsberg, J. (2000). Coalescents with simultaneous multiple collisions. Electron. J. Prob. 5, 50 pp.CrossRefGoogle Scholar
Schweinsberg, J. (2000). A necessary and sufficient condition for the Λ-coalescent to come down from infinity. Electron. Commun. Prob. 5, 11 pp.CrossRefGoogle Scholar
Schweinsberg, J. (2003). Coalescent processes obtained from supercritical Galton-Watson processes. Stoch. Process. Appl. 106, 107–139.CrossRefGoogle Scholar
Tavaré, S. (2004). Ancestral inference in population genetics. In Lectures on Probability Theory and Statistics (Lecture Notes Math. 1837), Springer, Berlin, pp. 1–188.Google Scholar