Local limits of Galton–Watson trees conditioned on the number of protected nodes

We consider a marking procedure of the vertices of a tree where each vertex is marked independently from the others with a probability that depends only on its out-degree. We prove that a critical Galton–Watson tree conditioned on having a large number of marked vertices converges in distribution to...

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Bibliographic Details
Published inJournal of applied probability Vol. 54; no. 1; pp. 55 - 65
Main Authors Abraham, Romain, Bouaziz, Aymen, Delmas, Jean-François
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.03.2017
Applied Probability Trust
Cambridge University press
Subjects
Atomic Physics
Condensed Matter
Conditioning
Mathematics
Physics
Probability
Research Papers
Strongly Correlated Electrons
Trees (mathematics)
Galton > Watson tree
protected node
60B10
Primary 60J80
random tree
local limit
Secondary 05C05
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Summary:We consider a marking procedure of the vertices of a tree where each vertex is marked independently from the others with a probability that depends only on its out-degree. We prove that a critical Galton–Watson tree conditioned on having a large number of marked vertices converges in distribution to the associated size-biased tree. We then apply this result to give the limit in distribution of a critical Galton–Watson tree conditioned on having a large number of protected nodes.
ISSN:0021-9002
1475-6072
DOI:10.1017/jpr.2016.86