THE CONTINUUM RANDOM TREE-III

• 成果类型: Article
• 署名作者: ALDOUS, D
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989404
• 发表日期: 1993
• 页码: 248-289
• 关键词: brownian-motion

摘要

Let (R(k), k greater-than-or-equal-to 1) be random trees with k leaves, satisfying a consistency condition: Removing a random leaf from R(k) gives R(k - 1). Then under an extra condition, this family determines a random continuum tree l, which it is convenient to represent as a random subset of l1. This leads to an abstract notion of convergence in distribution, as n --> infinity, of (rescaled) random trees T(n) on n vertices to a limit continuum random tree l. The notion is based upon the assumption that, for fixed k, the subtrees of T(n) determined by k randomly chosen vertices converge to R(k). As our main example, under mild conditions on the offspring distribution, the family tree of a Galton-Watson branching process, conditioned on total population size equal to n, can be rescaled to converge to a limit continuum random tree which can be constructed from Brownian excursion.