Title: The Standard Additive Coalescent
Author: David Aldous and Jim Pitman
Date: September 1997
Abstract:
Regard an element of the set
$$\Delta := \{(x_1,x_2,\ldots): x_1 \geq x_2 \geq \ldots \geq 0, \sum_i x_i = 1\}$$
as a fragmentation of unit mass into clusters of masses $x_i$. The
additive coalescent of Evans and Pitman (1997) 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 $(\bX^\infty (t), - \infty < t < \infty)$
of this process arises as a $n \to \infty$ weak limit of the process
started at time $- \frac{1}{2} \log n$ with $n$ clusters of mass $1/n$.
We show this {\em standard additive coalescent} may be constructed
from the continuum random tree of Aldous (1991,1993) by Poisson
splitting along the skeleton of the tree. We describe the distribution
of $\bX^\infty (t)$ 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.