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September 2015 Regenerative tree growth: Markovian embedding of fragmenters, bifurcators, and bead splitting processes
Jim Pitman, Matthias Winkel
Ann. Probab. 43(5): 2611-2646 (September 2015). DOI: 10.1214/14-AOP945

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.

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Jim Pitman. Matthias Winkel. "Regenerative tree growth: Markovian embedding of fragmenters, bifurcators, and bead splitting processes." Ann. Probab. 43 (5) 2611 - 2646, September 2015. https://doi.org/10.1214/14-AOP945

Information

Received: 1 April 2013; Revised: 1 May 2014; Published: September 2015
First available in Project Euclid: 9 September 2015

zbMATH: 1330.60106
MathSciNet: MR3395470
Digital Object Identifier: 10.1214/14-AOP945

Subjects:
Primary: 60J80

Keywords: $\mathbb{R}$-tree , Continuum random tree , fragmentation , self-similar tree , weighted $\mathbb{R}$-tree

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 5 • September 2015
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