The standard additive coalescent

• 成果类型: Article
• 署名作者: Aldous, D; Pitman, J
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1998
• 页码: 1703-1726
• 关键词: continuum random tree; coagulation; subordinators; invariant; systems; models

摘要

Regard an element of the set Delta := {(x(1), x(2), ...): x(1) greater than or equal to x(2) greater than or equal to ... greater than or equal to 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 (X-infinity(t), -infinity < t < infinity) of this process arises as a n --> infinity 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-infinity(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 --> -infinity, we establish a Gaussian limit for (centered and normalized) cluster sizes and study the size of the largest cluster.