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dc.contributor.authorPagnard, Camille
dc.date.accessioned2018-02-16T11:39:21Z
dc.date.available2018-02-16T11:39:21Z
dc.date.issued2017
dc.identifier.issn1083-6489
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/17406
dc.language.isoenen
dc.subjectrandom treesen
dc.subjectMarkov branching treesen
dc.subjectlocal limitsen
dc.subjectscaling limitsen
dc.subjectself-similar fragmentation treesen
dc.subjectGHP topologyen
dc.subjectGalton-Watson treesen
dc.subject.ddc519en
dc.titleLocal limits of Markov Branching trees and their volume growthen
dc.typeArticle accepté pour publication ou publié
dc.description.abstractenWe are interested in the local limits of families of random trees that satisfy the Markov branching property, which is fulfilled by a wide range of models. Loosely, this property entails that given the sizes of the sub-trees above the root, these sub-trees are independent and their distributions only depend upon their respective sizes. The laws of the elements of a Markov branching family are characterised by a sequence of probability distributions on the sets of integer partitions which describes how the sizes of the sub-trees above the root are distributed. We prove that under some natural assumption on this sequence of probabilities, when their sizes go to infinity, the trees converge in distribution to an infinite tree which also satisfies the Markov branching property. Furthermore, when this infinite tree has a single path from the root to infinity, we give conditions to ensure its convergence in distribution under appropriate rescaling of its distance and counting measure to a self-similar fragmentation tree with immigration. In particular, this allows us to determine how, in this infinite tree, the "volume" of the ball of radius R centred at the root asymptotically grows with R. Our unified approach will allow us to develop various new applications, in particular to different models of growing trees and cut-trees, and to recover known results. An illustrative example lies in the study of Galton-Watson trees: the distribution of a critical Galton-Watson tree conditioned on its size converges to that of Kesten's tree when the size grows to infinity. If furthermore, the offspring distribution has finite variance, under adequate rescaling, Kesten's tree converges to Aldous' self-similar CRT and the total size of the R first generations asymptotically behaves like R².en
dc.relation.isversionofjnlnameElectronic Journal of Probability
dc.relation.isversionofjnlvol22en
dc.relation.isversionofjnldate2017
dc.relation.isversionofjnlpages53 p.en
dc.relation.isversionofdoi10.1214/17-EJP96en
dc.subject.ddclabelProbabilités et mathématiques appliquéesen
dc.relation.forthcomingnonen
dc.relation.forthcomingprintnonen
dc.description.ssrncandidatenonen
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.relation.Isversionofjnlpeerreviewedouien
dc.relation.Isversionofjnlpeerreviewedouien
dc.date.updated2018-02-16T11:37:28Z
hal.person.labIds60


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