The Annals of Probability

Coalescents With Multiple Collisions

Jim Pitman

Full-text: Open access


For each finite measure $\Lambda$ on [0,1] a coalescent Markov process, with state space the compact set of all partitions of the set $\mathbb{N}$ of positive integers, is constructed so the restriction of the partition to each finite subset of $\mathbb{N}$ is a Markov chain with the following transition rates: when the partition has b blocks, each $k$-tuple of blocks is merging to form a single block at rate $\int_0^1x^{k-2}(1-x)^{b-k}\Lambda(dx)$. Call this process a $\Lambda$-coalescent. Discrete measure-valued processes derived from the $\Lambda$-coalescent model a system of masses undergoing coalescent collisions. Kingman’s coalescent, which has numerous applications in population genetics, is the $\delta_0$-coalescent for $\delta_0$ a unit mass at 0. The coalescent recently derived by Bolthausen and Sznitman from Ruelle’s probability cascades, in the context of the Sherrington–Kirkpatrick spin glass model in mathematical physics, is the $U$-coalescent for $U$ uniform on [0,1]. For $\Lambda=U$, and whenever an infinite number of masses are present, each collision in $\Lambda$-coalescent involves an infinite number of masses almost surely, and the proportion of masses involved exists as a limit almost surely and is distributed proportionally to $\Lambda$. The two-parameter Poisson–Dirichlet family of random discrete distributions derived from a stable subordinator, and corresponding exchangeable random partitions of $\mathbb{N}$ governed by a generalization of the Ewens sampling formula, are applied to describe transition mechanisms for processes of coalescence and fragmentation, including the $U$-coalescent and its time reversal.

Article information

Ann. Probab., Volume 27, Number 4 (1999), 1870-1902.

First available in Project Euclid: 31 May 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J75
Secondary: 60G09: Exchangeability 60G57: Random measures 05A18: Partitions of sets

Exchangeable random partition ranked frequencies random discrete distribution two-parameter Poisson –Dirichlet stable subordinator coagulation,fragmentation time reversal Ewens sampling formula


Pitman, Jim. Coalescents With Multiple Collisions. Ann. Probab. 27 (1999), no. 4, 1870--1902. doi:10.1214/aop/1022874819.

Export citation


  • [1] Aldous, D. (1997). Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25 812-854.
  • [2] Aldous, D. J. (1985). Exchangeability and related topics. ´Ecole d' ´Et´e de Probabilit´es de Saint-Flour Lecture Notes in Math. XIII 1117 1-198. Springer, Berlin.
  • [3] Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): a reviewof the mean-field theory for probabilists. Bernoulli 5 3-48.
  • [4] Aldous, D. J. and Limic, V. (1998). The entrance boundary of the multiplicative coalescent. Electron. J. Probab. 3 1-59.
  • [5] Aldous, D. J. and Pitman, J. (1998). The standard additive coalescent. Ann. Probab. 26 1703-1726.
  • [6] Arratia, R. Barbour, A. D. and Tavar´e, S. (1997). Random combinatorial structures and prime factorizations. Notices Amer. Math. Soc. 44 903-910.
  • [7] Barlow, M. Pitman, J. and Yor, M. (1989). Une extension multidimensionnelle de la loi de l'arc sinus. S´eminaire de Probabilit´es XXIII. Lecture Notes in Math. 1372. 294-314. Springer, Berlin.
  • [8] Bertoin, J. and Le Gall, J. F. (1999). The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes. Probab. Theory Related Fields. To appear.
  • [9] Bolthausen, E. and Sznitman, A.-S. (1998). On Ruelle's probability cascades and an abstract cavity method. Comm. Math. Phys. 197 247-276.
  • [10] Derrida, B. (1987). Statistical properties of randomly broken objects and of multivalley structures in disordered systems. J. Phys. A. 20 5273-5288.
  • [11] Derrida, B. (1994). Non-self-averaging effects in sums of random variables, spin glasses, random maps and random walks. In On Three Levels: Micro-, Meso-, and MacroApproaches in Physics (M. Fannes, C. Maes and A. Verbeure, eds.) 125-137. Plenum Press, NewYork.
  • [12] Doksum, K. (1974). Tailfree and neutral random probabilities and their posterior distributions. Ann. Probab. 2 183-201.
  • [13] Dynkin, E. B. (1965). Markov Processes 1. Springer, Berlin.
  • [14] Dynkin, E. B. (1978). Sufficient statistics and extreme points. Ann. Probab. 6 705-730.
  • [15] Evans, S. N. and Pitman, J. (1988). Construction of Markovian coalescents. Ann. Inst. H. Poincar´e 34 339-383.
  • [16] Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Population Biol. 3 87-112.
  • [17] Ewens, W. J. (1990). Population genetics theory-the past and the future. In Mathematical and Statistical Developments of Evolutionary Theory (S. Lessard, ed.) 177-227. Kluwer, Dordrecht.
  • [18] Feller, W. (1966). An Introduction to Probability Theory and Its Applications 2. Wiley, New York.
  • [19] Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209-230.
  • [20] Johnson, N. L., Kotz, S. and Balakrishnan, N. (1997). Discrete Multivariate Distributions. Wiley, NewYork.
  • [21] Kerov, S. (1995). Coherent random allocations and the Ewens-Pitman formula. PDMI Preprint, Steklov Math. Institute, St. Petersburg.
  • [22] Kingman, J. F. C. (1975). Random discrete distributions. J. Roy. Statist. Soc B 37 1-22.
  • [23] Kingman, J. F. C. (1978). Random partitions in population genetics. Proc. Roy. Soc. London Ser. A 361 1-20.
  • [24] Kingman, J. F. C. (1978). The representation of partition structures. J. London Math. Soc. 18 374-380.
  • [25] Kingman, J. F. C. (1982). The coalescent. Stochastic Process. Appl. 13 235-248.
  • [26] Kingman, J. F. C. (1982). Exchangeability and the evolution of large populations. In Exchangeability in Probability and Statistics (G. Koch and F. Spizzichino, ed.) 97-112. North-Holland, Amsterdam.
  • [27] Kingman, J. F. C. (1982). On the genealogy of large populations. In Essays in Statistical Science (J. Gani and E. J. Hannan, eds.) 27-43. Applied Probability Trust, Sheffield, England.
  • [28] Lushnikov, A. A. (1978). Coagulation in finite systems. J. Colloid and Interface Science 65 276-285.
  • [29] Marcus, A. H. (1968). Stochastic coalescence. Technometrics 10 133-143.
  • [30] M ¨ohle, M. and Sagitov, S. (1999). A classification of coalescent processes for haploid exchangeable population models. Preprint. Available via serik/coal.html.
  • [31] Perman, M., Pitman, J. and Yor, M. (1992). Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92 21-39.
  • [32] Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields 102 145-158.
  • [33] Pitman, J. (1996). Random discrete distributions invariant under size-biased permutation. Adv. Appl. Probab. 28 525-539.
  • [34] Pitman, J. (1996). Some developments of the Blackwell-MacQueen urn scheme. In Statistics, Probability and Game Theory. Papers in Honor of David Blackwell (T. S. Ferguson, ed.) 245-267. IMS, Hayward, California.
  • [35] Pitman, J. (1997). Partition structures derived from Brownian motion and stable subordinators. Bernoulli 3 79-96.
  • [36] Pitman, J. and Yor, M. (1996). Random discrete distributions derived from self-similar random sets. Electron. J. Probab. 1 1-28.
  • [37] Pitman, J. and Yor, M. (1997). On the relative lengths of excursions derived from a stable subordinator. S´eminaire de Probabilit´es XXXI. Lecture Notes in Math. 1655 287-305. Springer, Berlin.
  • [38] Pitman, J. and Yor, M. (1997). The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 855-900.
  • [39] Rosenblatt, M. (1974). Random Processes. Springer, NewYork.
  • [40] Ruelle, D. (1987). A mathematical reformulation of Derrida's REM and GREM. Comm. Math. Phys. 108 225-239.
  • [41] Sagitov, S. (1998). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. To appear. Available via serik/coal.html.
  • [42] Schweinsberg, J. (1999). A necessary and sufficient condition for the -coalescent to come down from infinity. Technical Report 568, Dept. Statistics, U.C. Berkeley. Available via
  • [43] Tavar´e, S. (1984). Line-of-descent and genealogical processes and their applications in population genetics. Theoret. Population Biol. 26 119-164.
  • [44] Tsilevich, N. V. (1997). Distribution of the mean value for certain random measures. Zapiski Nauchn. Sem. POMI 240 269-280.