The Annals of Probability

The standard additive coalescent

David Aldous and Jim Pitman

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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

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855879

Digital Object Identifier
doi:10.1214/aop/1022855879

Mathematical Reviews number (MathSciNet)
MR1675063

Zentralblatt MATH identifier
0936.60064

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60C05: Combinatorial probability 60J65: Brownian motion [See also 58J65]

Keywords
Brownian excursion continuum random tree random forest splitting stable subordinator stochastic coalescence stochastic fragmentation

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


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