A line-breaking construction of the stable trees

Abstract

We give a new, simple construction of the $\alpha$-stable tree for $\alpha \in (1,2]$. We obtain it as the closure of an increasing sequence of $\mathbb{R}$-trees inductively built by gluing together line-segments one by one. The lengths of these line-segments are related to the the increments of an increasing ${\mathbb{R}}_{+}$-valued Markov chain. For $\alpha =2$, we recover Aldous' line-breaking construction of the Brownian continuum random tree based on an inhomogeneous Poisson process.

27 pages

References in corpus (3)

• Self-similar fragmentations derived from the stable tree II: splitting at nodes
• Spinal partitions and invariance under re-rooting of continuum random trees
• Growth of Levy trees