The stable trees revisited

TL;DR

This paper presents a new line-breaking construction for the alpha-stable tree, generalizing Aldous' Brownian CRT, and provides a new proof of an invariance principle linking branching processes to stable trees.

Contribution

It introduces a novel, simpler line-breaking construction for alpha-stable trees using a random rate process and offers a new proof of the invariance principle for branching process trees.

Findings

New line-breaking construction for alpha-stable trees.

Proof of convergence of branching process trees to alpha-stable trees.

Connection between discrete and continuous tree constructions.

Abstract

We introduce a new, relatively simple, line-breaking construction of the $\alpha$-stable tree which realises its random finite-dimensional distributions. This is a direct analogue of Aldous' line-breaking construction of the Brownian continuum random tree, which is based on an inhomogeneous Poisson process. Here, we replace the deterministic rate function from the Brownian setting by a random rate process, given by a certain measure-changed $(\alpha-1)$-stable subordinator. Rather than attaching uniformly, the line-segments now connect to locations chosen with probability proportional to the sizes of the jumps of the rate process. We also give a new proof of an invariance principle originally due to Duquesne, which states that the family tree of a Bienaym\'e branching process with critical offspring distribution in the domain of attraction of an $\alpha$-stable law (for $\alpha \in (1,2))$, conditioned to have $n$ vertices, converges on rescaling distances appropriately to the $\alpha$-stable tree. Our proof makes use of a discrete line-breaking construction of the branching process tree, which we show converges to our continuous line-breaking construction.