• DOI:10.1214/AOP/1022855879
• Corpus ID: 9056986

The standard additive coalescent

@article{Aldous1998TheSA,
  title={The standard additive coalescent},
  author={David J. Aldous and Jim Pitman},
  journal={Annals of Probability},
  year={1998},
  volume={26},
  pages={1703-1726}
}

• D. Aldous, J. Pitman
• Published 1 October 1998
• Mathematics
• Annals of Probability

Regard an element of the set Δ := {(x 1 , x 2 , . . .): x 1 ≥ x 2 ≥ ⋯ ≥ 0, ∑ 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 Δ-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 (X ∞ (t), -∞ < t < ∞) of this process arises as a n → ∞ 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 X∞(t) on Δ 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 → -∞, we establish a Gaussian limit for (centered and normalized) cluster sizes and study the size of the largest cluster.