The standard additive coalescent

Jim Pitman, David Aldous
1998 Annals of Probability  
Regard an element of the set = 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 -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
more » ... 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 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.
doi:10.1214/aop/1022855879 fatcat:bpcex5ubtbckjewgy2eagyauay