REGENERATIVE TREE GROWTH: MARKOVIAN EMBEDDING OF FRAGMENTERS, BIFURCATORS, AND BEAD SPLITTING PROCESSES

• 成果类型: Article
• 署名作者: Pitman, Jim; Winkel, Matthias
• 署名单位: University of California System; University of California Berkeley; University of Oxford
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP945
• 发表日期: 2015
• 页码: 2611-2646
• 关键词: branching trees; Brownian bridge; fragmentations; asymptotics; partitions

摘要

Some, but not all processes of the form M-t = exp(-xi(t)) for a pure-jump subordinator xi with Laplace exponent Phi arise as residual mass processes of particle 1 (tagged particle) in Bertoin's partition-valued exchangeable fragmentation processes. We introduce the notion of a Markovian embedding of M = (M-t, t >= 0) in a fragmentation process, and we show that for each Phi, there is a unique (in distribution) binary fragmentation process in which M has a Markovian embedding. The identification of the Laplace exponent Phi* of its tagged particle process M* gives rise to a symmetrisation operation Phi bar right arrow Phi*, which we investigate in a general study of pairs (M, M*) that coincide up to a random time and then evolve independently. We call M a fragmenter and (M, M*) a bifurcator. For alpha > 0, we equip the interval R-1 = [0, integral(infinity)(0) M-t(alpha) dt] with a purely atomic probability measure itch which captures the jump sizes of M suitably placed on R-1. We study binary tree growth processes that in the nth step sample an atom (bead) from mu(n) and build (Rn+1, mu(n+1)) by replacing the atom by a rescaled independent copy of (R-1, mu(1)) that we tie to the position of the atom. We show that any such bead splitting process ((R-n,R- mu(n)), n >= 1) converges almost surely to an alpha-self-similar continuum random tree of Haas and Miermont, in the Gromov Hausdorff Prohorov sense. This generalises Aldous's line-breaking construction of the Brownian continuum random tree.