The Annals of Probability

Regenerative tree growth: Markovian embedding of fragmenters, bifurcators, and bead splitting processes

Jim Pitman and Matthias Winkel

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Abstract

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 $M=(M_{t},t\ge0)$ 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 $\Phi\mapsto\Phi^{*}$, 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 $R_{1}=[0,\int_{0}^{\infty}M_{t}^{\alpha}\,dt]$ with a purely atomic probability measure $\mu_{1}$, which captures the jump sizes of $M$ suitably placed on $R_{1}$. We study binary tree growth processes that in the $n$th step sample an atom (“bead”) from $\mu_{n}$ and build $(R_{n+1},\mu_{n+1})$ 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 $((R_{n},\mu_{n}),n\ge1)$ 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.

Article information

Source
Ann. Probab., Volume 43, Number 5 (2015), 2611-2646.

Dates
Received: April 2013
Revised: May 2014
First available in Project Euclid: 9 September 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1441792294

Digital Object Identifier
doi:10.1214/14-AOP945

Mathematical Reviews number (MathSciNet)
MR3395470

Zentralblatt MATH identifier
1330.60106

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Fragmentation self-similar tree continuum random tree $\mathbb{R}$-tree weighted $\mathbb{R}$-tree

Citation

Pitman, Jim; Winkel, Matthias. Regenerative tree growth: Markovian embedding of fragmenters, bifurcators, and bead splitting processes. Ann. Probab. 43 (2015), no. 5, 2611--2646. doi:10.1214/14-AOP945. https://projecteuclid.org/euclid.aop/1441792294


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