## The Annals of Probability

#### Abstract

Regard an element of the set $$\Delta := {(x_1, x_2, \dots): x_1 \geq x_2 \geq \dots \geq 0, \Sigma_i x_i = 1}$$ as a fragmentation of unit mass into clusters of masses $x_i$. The additive coalescent of Evans and Pitman is the $\Delta$-valued Markov process in which pairs of clusters of masses ${x_i, x_j}$ merge into a cluster of mass $x_i + x_j$ at rate $x_i + x_j$. They showed that a version $(\rm X^{\infty}(t), -\infty < t < \infty)$ of this process arises as a $n \to \infty$ weak limit of the process started at time $-1/2 \log n$ with $n$ clusters of mass $1/n$. We show this standard additive coalescent may be constructed from the continuum random tree of Aldous by Poisson splitting along the skeleton of the tree. We describe the distribution of $\rm X^{\infty}(t)$ on $\Delta$ at a fixed time $t$. We show that the size of the cluster containing a given atom, as a process in $t$, has a simple representation in terms of the stable subordinator of index 1/2. As $t \to -\infty$, we establish a Gaussian limit for (centered and normalized) cluster sizes and study the size of the largest cluster.

#### Article information

Source
Ann. Probab., Volume 26, Number 4 (1998), 1703-1726.

Dates
First available in Project Euclid: 31 May 2002

https://projecteuclid.org/euclid.aop/1022855879

Digital Object Identifier
doi:10.1214/aop/1022855879

Mathematical Reviews number (MathSciNet)
MR1675063

Zentralblatt MATH identifier
0936.60064

#### Citation

Aldous, David; Pitman, Jim. The standard additive coalescent. Ann. Probab. 26 (1998), no. 4, 1703--1726. doi:10.1214/aop/1022855879. https://projecteuclid.org/euclid.aop/1022855879

#### References

• [1] Aldous, D. (1989). Stopping times and tightness II. Ann. Probab. 17 586-595.
• [2] Aldous, D. (1991). The continuum random tree I. Ann. Probab. 19 1-28.
• [3] Aldous, D. (1991). The continuum random tree II: an overview. In Stochastic Analysis (M. Barlow and N. Bingham, eds.) 23-70. Cambridge Univ. Press.
• [4] Aldous, D. (1993). The continuum random tree III. Ann. Probab. 21 248-289.
• [5] Aldous, D. (1994). Recursive self-similarity for random trees, random triangulations and Brownian excursion. Ann. Probab. 22 527-545.
• [6] Aldous, D. (1997). Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25 812-854.
• [7] Aldous, D. (1997). Deterministic and stochastic models for coalescence: a review of the mean-field theory for probabilists. Bernoulli. To appear. Available via http://www. stat.berkeley.edu/users/aldous.
• [8] Aldous, D. and Pitman, J. (1994). Brownian bridge asymptotics for random mappings. Random Structures Algorithms 5 487-512.
• [9] Aldous, D. and Pitman, J. (1998). Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent. Technical Report 525, Dept. Statistics, Univ. California, Berkeley. Available via http://www.stat.berkeley.edu/users/pitman.
• [10] Donnelly, P. and Joyce, P. (1989). Continuity and weak convergence of ranked and sizebiased permutations on the infinite simplex. Stochastic Process. Appl. 31 89-103.
• [11] Dwass, M. and Karlin, S. (1963). Conditioned limit theorems. Ann. Math. Statist. 34 1147- 1167.
• [12] Dynkin, E. and Kuznetsov, S. (1995). Markov snakes and superprocesses. Probab. Theory Related Fields 103 433-473.
• [13] Evans, S. and Pitman, J. (1997). Stationary Markov processes related to stable Ornstein- Uhlenbeck processes and the additive coalescent. Stoch. Process Appl. To appear. Available via http://www.stat.berkeley.edu/users/pitman.
• [14] Evans, S. and Pitman, J. (1998). Construction of Markovian coalescents. Ann. Inst. H. Poincar´e 34 339-383.
• [15] Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics. Krieger, Malabar, FL.
• [16] Gall, J.-F. L. (1993). The uniform random tree in a Brownian excursion. Probab. Theory Related Fields 96 369-384.
• [17] Gall, J.-F. L. (1995). The Brownian snake and solutions of u = u2 in a domain. Probab. Theory Related Fields 102 393-432.
• [18] Gall, J. L. (1991). Brownian excursions, trees and measure-valued branching processes. Ann. Probab. 19 1399-1439.
• [19] Golovin, A. (1963). The solution of the coagulating equation for cloud droplets in a rising air current. Izv. Geophys. Ser. 5 482-487.
• [20] Hendriks, E., Spouge, J., Eibl, M. and Shreckenberg, M. (1985). Exact solutions for random coagulation processes.Phys. B 58 219-227.
• [21] Knight, F. (1966). A pair of conditional normal convergence theorems. Theory Probab. Appl. 11 191-210.
• [22] K ¨uchler, U. and Lauritzen, S. L. (1989). Exponential families, extreme point models and minimal space-time invariant functions for stochastic processes with stationary and independent increments. Scand. J. Statist. 16 237-261.
• [23] Lushnikov, A. (1978). Coagulation in finite systems. J. Colloid and Interface Science 65 276-285.
• [24] Pavlov, Y. (1979). A case of the limit distribution of the maximal volume on a tree in a random forest. Math. Notes of the Acad. Sci. USSR 25 387-392.
• [25] Perman, M. (1993). Order statistics for jumps of normalized subordinators. Stochastic Process Appl. 46 267-281.
• [26] Perman, M., Pitman, J. and Yor, M. (1992). Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92 21-39.
• [27] Pitman, J. (1996). Coalescent random forests. J. Combin. Theory Ser. A. To appear. Available via http://www.stat.berkeley.edu/users/pitman.
• [28] Pitman, J. (1996). Random discrete distributions invariant under size-biased permutation. Adv. in Appl. Probab. 28 525-539.
• [29] Pitman, J. (1997). Partition structures derived from Brownian motion and stable subordinators. Bernoulli 3 79-96.
• [30] Pitman, J. and Yor, M. (1992). Arcsine laws and interval partitions derived from a stable subordinator. Proc. London Math. Soc. 65 326-356.
• [31] Pitman, J. and Yor, M. (1997). The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 855-900.
• [32] Sheth, R. and Pitman, J. (1997). Coagulation and branching process models of gravitational clustering. Mon. Not. Roy. Astron. Soc. 289 66-80.
• [33] Tanaka, H. and Nakazawa, K. (1994). Validity of the stochastic coagulation equation and runaway growth of protoplanets. Icarus 107 404-412.
• [34] van Dongen, P. (1987). Fluctuations in coagulating systems II. J. Statist. Phys. 49 927-975.
• [35] Yao, A. (1976). On the average behavior of set merging algorithms. In Proceedings of the Eighth ACM Symposium on Theory of Computing 192-195.