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

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 $M=(M_t,t\ge 0)$ 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 ${\Phi }^{\ast }$ of its tagged particle process $M^{\ast }$ gives rise to a symmetrisation operation $\Phi \mapsto {\Phi }^{\ast }$, which we investigate in a general study of pairs $(M,M^{\ast })$ that coincide up to a random time and then evolve independently. We call $M$ a fragmenter and $(M,M^{\ast })$ 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 $((R_n,{\mu }_n),n\ge 1)$ 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.

Published at http://dx.doi.org/10.1214/14-AOP945 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Asymptotic laws for compositions derived from transformed subordinators
• Spinal partitions and invariance under re-rooting of continuum random trees
• Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models
• Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions
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Cited by in corpus (5)

• Generalized Mittag Leffler distributions arising as limits in preferential attachment models
• Aldous diffusion I: a projective system of continuum $k$-tree evolutions
• Scaling limits for some random trees constructed inhomogeneously
• Regenerative tree growth: structural results and convergence
• A recursive distribution equation for the stable tree