Abstract
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 \(\mathbb {Z}^d\).
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Acknowledgements
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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Abraham, R., Delmas, JF. & Guo, H. Critical Multi-type Galton–Watson Trees Conditioned to be Large. J Theor Probab 31, 757–788 (2018). https://doi.org/10.1007/s10959-016-0739-8
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DOI: https://doi.org/10.1007/s10959-016-0739-8