The standard additive coalescent
Aldous, David ; Pitman, Jim
Ann. Probab., Tome 26 (1998) no. 1, p. 1703-1726 / Harvested from Project Euclid
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.
Publié le : 1998-10-14
Classification:  Brownian excursion,  continuum random tree,  random forest,  splitting,  stable subordinator,  stochastic coalescence,  stochastic fragmentation,  60J25,  60C05,  60J65
@article{1022855879,
     author = {Aldous, David and Pitman, Jim},
     title = {The standard additive coalescent},
     journal = {Ann. Probab.},
     volume = {26},
     number = {1},
     year = {1998},
     pages = { 1703-1726},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1022855879}
}
Aldous, David; Pitman, Jim. The standard additive coalescent. Ann. Probab., Tome 26 (1998) no. 1, pp.  1703-1726. http://gdmltest.u-ga.fr/item/1022855879/