Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T15:40:51.006Z Has data issue: false hasContentIssue false
  • English
  • Français

On Asymptotics of the Beta Coalescents

Part of: Combinatorial probability Stochastic processes Markov processes Limit theorems

Published online by Cambridge University Press:  22 February 2016

Alexander Gnedin ,
Alexander Iksanov ,
Alexander Marynych  and
Martin Möhle
Alexander Gnedin*
Affiliation:
University of London
Alexander Iksanov*
Affiliation:
National Taras Shevchenko University of Kyiv
Alexander Marynych*
Affiliation:
National Taras Shevchenko University of Kyiv
Martin Möhle*
Affiliation:
University of Tübingen
*
∗ Postal address: School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK. Email address: [email protected]
∗∗ Postal address: Faculty of Cybernetics, National Taras Shevchenko University of Kyiv, 01601 Kyiv, Ukraine. Email address: [email protected]
∗∗∗ Postal address: National Taras Shevchenko University of Kyiv, 01601 Kyiv, Ukraine. Email address: [email protected]
∗∗∗∗ Postal address: Mathematical Institute, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany. Email address: [email protected]
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.

We show that the total number of collisions in the exchangeable coalescent process driven by the beta (1, b) measure converges in distribution to a 1-stable law, as the initial number of particles goes to ∞. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance b = 1, which corresponds to the Bolthausen-Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta (a, b)-coalescents with 0 < a < 1 leads to a simplified derivation of the known (2 - a)-stable limit. We furthermore derive asymptotic expansions for the moments of the number of collisions and of the total branch length for the beta (1, b)-coalescent by exploiting the method of sequential approximations.

Keywords

Absorption timeasymptotic expansionbeta coalescentcouplingnumber of collisionstotal branch lengthWasserstein distance

MSC classification

Primary: 60C05: Combinatorial probability 60G09: Exchangeability
Secondary: 60F05: Central limit and other weak theorems 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 2 , June 2014 , pp. 496 - 515
Copyright
© Applied Probability Trust 

References

Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U.S. Government Printing Office, Washington, DC.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.Google Scholar
Bertoin, J. and Pitman, J. (2000). Two coalescents derived from the ranges of stable subordinators. Electron. J. Prob. 5, 17pp.Google Scholar
Berestycki, N. (2009). Recent Progress in Coalescent Theory (Ensaios Mathematicos 16), Sociedade Brasileira de Matemática, Rio de Janeiro.Google Scholar
Birkner, M. et al. (2005). Alpha-stable branching and beta-coalescents. Electron. J. Prob. 10, 303–325.Google Scholar
Bolthausen, E. and Sznitman, A.-S. (1998). On Ruelle's probability cascades and an abstract cavity method. Commun. Math. Phys. 197, 247–276.Google 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.Google 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.Google 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.Google Scholar
Feller, W. (1949). Fluctuation theory of recurrent events. Trans. Amer. Math. Soc. 67, 98–119.Google Scholar
Freund, F. and Möhle, M. (2009). On the time back to the most recent common ancestor and the external branch length of the Bolthausen–Sznitman coalescent. Markov Process. Relat. Fields 15, 387–416.Google Scholar
Givens, C. R. and Shortt, R. M. (1984). A class of Wasserstein metrics for probability distributions. Michigan Math. J. 31, 231–240.Google Scholar
Gnedin, A. and Yakubovich, Y. (2007). On the number of collisions in Λ-coalescents. Electron. J. Prob. 12, 1547–1567.Google Scholar
Gnedin, A., Iksanov, A. and Marynych, A. (2011). On Λ-coalescents with dust component. J. Appl. Prob. 48, 1133–1151.Google Scholar
Gnedin, A., Iksanov, A. and Möhle, M. (2008). On asymptotics of exchangeable coalescents with multiple collisions. J. Appl. Prob. 45, 1186–1195.Google Scholar
Gnedin, A., Iksanov, A., Marynych, A. and Moehle, M. (2012). On asymptotics of the beta-coalescents. Preprint. Available at http://uk.arxiv.org/abs/1203.3110.Google Scholar
Goldschmidt, C. and Martin, J. B. (2005). Random recursive trees and the Bolthausen–Sznitman coalescent. Electron. J. Prob. 10, 718–745.Google Scholar
Haas, B. and Miermont, G. (2011). Self-similar scaling limits of non-increasing Markov chains. Bernoulli 17, 1217–1247.Google Scholar
Huillet, T. and Möhle, M. (2013). On the extended Moran model and its relation to coalescents with multiple collisions. Theoret. Pop. Biol. 87, 5–14.Google 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.Google Scholar
Iksanov, A. and Möhle, M. (2008). On the number of Jumps of random walks with a barrier. Adv. Appl. Prob. 40, 206–228.Google Scholar
Iksanov, A., Marynych, A. and Möhle, M. (2009). On the number of collisions in beta(2,b)-coalescents. Bernoulli 15, 829–845.Google Scholar
Johnson, O. and Samworth, R. (2005). Central limit theorem and convergence to stable laws in Mallows distance. Bernoulli 11, 829–845.Google Scholar
Kersting, G. (2012). The asymptotic distribution of the length of beta-coalescent trees. Ann. Appl. Prob. 22, 2086–2107.Google Scholar
Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.Google Scholar
Möhle, M. (2006). On the number of segregating sites for populations with large family sizes. Adv. Appl. Prob. 38, 750–767.Google Scholar
Möhle, M. (2010). Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson–Dirichlet coalescent. Stoch. Process. Appl. 120, 2159–2173.Google Scholar
Panholzer, A. (2004). Destruction of recursive trees. In Mathematics and Computer Science, Vol. III, Birkhäuser, Basel, pp. 267–280.Google Scholar
Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27, 1870–1902.Google Scholar
Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36, 1116–1125.Google Scholar
Schweinsberg, J. (2000). A necessary and sufficient condition for the Λ-coalescent to come down from infinity. Electron. Commun. Prob. 5, 1–11.Google 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