174 research outputs found
Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions
We use a natural ordered extension of the Chinese Restaurant Process to grow
a two-parameter family of binary self-similar continuum fragmentation trees. We
provide an explicit embedding of Ford's sequence of alpha model trees in the
continuum tree which we identified in a previous article as a distributional
scaling limit of Ford's trees. In general, the Markov branching trees induced
by the two-parameter growth rule are not sampling consistent, so the existence
of compact limiting trees cannot be deduced from previous work on the sampling
consistent case. We develop here a new approach to establish such limits, based
on regenerative interval partitions and the urn-model description of sampling
from Dirichlet random distributions.Comment: Published in at http://dx.doi.org/10.1214/08-AOP445 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Invariance principles for pruning processes of Galton-Watson trees
Pruning processes have been studied
separately for Galton-Watson trees and for L\'evy trees/forests. We establish
here a limit theory that strongly connects the two studies. This solves an open
problem by Abraham and Delmas, also formulated as a conjecture by L\"ohr,
Voisin and Winter. Specifically, we show that for any sequence of Galton-Watson
forests , , in the domain of attraction of a L\'evy
forest , suitably scaled pruning processes
converge in the Skorohod topology on
cadlag functions with values in the space of (isometry classes of) locally
compact real trees to limiting pruning processes. We separately treat pruning
at branch points and pruning at edges. We apply our results to study ascension
times and Kesten trees and forests.Comment: 33 page
Restricted exchangeable partitions and embedding of associated hierarchies in continuum random trees
We introduce the notion of a restricted exchangeable partition of
. We obtain integral representations, consider associated
fragmentations, embeddings into continuum random trees and convergence to such
limit trees. In particular, we deduce from the general theory developed here a
limit result conjectured previously for Ford's alpha model and its extension,
the alpha-gamma model, where restricted exchangeability arises naturally.Comment: 35 pages, 5 figure
A new family of Markov branching trees: the alpha-gamma model
We introduce a simple tree growth process that gives rise to a new
two-parameter family of discrete fragmentation trees that extends Ford's alpha
model to multifurcating trees and includes the trees obtained by uniform
sampling from Duquesne and Le Gall's stable continuum random tree. We call
these new trees the alpha-gamma trees. In this paper, we obtain their splitting
rules, dislocation measures both in ranked order and in sized-biased order, and
we study their limiting behaviour.Comment: 23 pages, 1 figur
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