Journal of Theoretical Probability

, Volume 31, Issue 2, pp 757–788 | Cite as

Critical Multi-type Galton–Watson Trees Conditioned to be Large

  • Romain AbrahamEmail author
  • Jean-François Delmas
  • Hongsong Guo


Under minimal condition, we prove the local convergence of a critical multi-type Galton–Watson tree conditioned on having a large total progeny by types toward a multi-type Kesten’s tree. We obtain the result by generalizing Neveu’s strong ratio limit theorem for aperiodic random walks on \(\mathbb {Z}^d\).


Galton–Watson tree Random tree Local limit Strong ratio theorem Branching process 

Mathematics Subject Classification (2010)

60J80 60B10 



The authors would like to thank Jean-Philippe Chancelier for pointing out the references on convex analysis and his valuable advice as well as the two anonymous referees for their comments and suggestions. H. Guo would like to express her gratitude to J.-F. Delmas for his help during her stay at CERMICS. The research has also been supported by the ANR-14-CE25-0014 (ANR GRAAL).


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Romain Abraham
    • 1
    Email author
  • Jean-François Delmas
    • 2
  • Hongsong Guo
    • 2
    • 3
  1. 1.Laboratoire MAPMO, CNRS, UMR 7349, Fédération Denis Poisson, FR 2964Université d’OrléansOrléans Cedex 2France
  2. 2.Université Paris-Est, CERMICS (ENPC)Marne la ValléeFrance
  3. 3.Department of MathematicsChina University of Mining and TechnologyBeijingPeople’s Republic of China

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