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Coding multitype forests: Application to the law of the total population of branching forests


Authors: Loïc Chaumont and Rongli Liu
Journal: Trans. Amer. Math. Soc. 368 (2016), 2723-2747
MSC (2010): Primary 60C05, 05C05
DOI: https://doi.org/10.1090/tran/6421
Published electronically: September 15, 2015
MathSciNet review: 3449255
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Abstract: By extending the breadth first search algorithm to any $ d$-type critical or subcritical irreducible branching forest, we show that such forests can be encoded through $ d$ independent, integer valued, $ d$-dimensional random walks. An application of this coding, together with a multivariate extension of the Ballot Theorem which is obtained here, allows us to give an explicit form of the law of the total population, jointly with the number of subtrees of each type, in terms of the offspring distribution of the branching process.


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Loïc Chaumont
Affiliation: LAREMA – UMR CNRS 6093, Université d’Angers, 2 bd Lavoisier, 49045 Angers cedex 01, France
Email: loic.chaumont@univ-angers.fr

Rongli Liu
Affiliation: Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China
Email: rongli.liu@gmail.com

DOI: https://doi.org/10.1090/tran/6421
Keywords: Multitype branching forest, coding, random walks, Ballot Theorem, total population, cyclic exchangeablity
Received by editor(s): September 19, 2013
Received by editor(s) in revised form: March 5, 2014
Published electronically: September 15, 2015
Additional Notes: This work was supported by MODEMAVE research project from the Région Pays de la Loire.
Ce travail a bénécié d’une aide de l’Agence Nationale de la Recherche portant la référence ANR-09-BLAN-0084-01.
This work was supported by NSFC, grant No. 11301261.
Article copyright: © Copyright 2015 American Mathematical Society