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Conditioning Galton–Watson Trees on Large Maximal Outdegree

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Abstract

We propose a new way to condition random trees, that is, conditioning random trees to have large maximal outdegree. Under this conditioning, we show that conditioned critical Galton–Watson trees converge locally to size-biased trees with a unique infinite spine. For the subcritical case, we obtain the local convergence to size-biased trees with a unique infinite node. We also study the tail of the maximal outdegree of subcritical Galton–Watson trees, which is essential for the proof of the local convergence.

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Acknowledgments

The author would like to thank the anonymous referees for their comments and suggestions, which improved considerably the presentation of this paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Xin He

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  1. Xin He
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Correspondence to Xin He.

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Supported by NSFC (No. 11401012) and Ministry of Education (985 Project).

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He, X. Conditioning Galton–Watson Trees on Large Maximal Outdegree. J Theor Probab 30, 842–851 (2017). https://doi.org/10.1007/s10959-016-0664-x

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  • DOI: https://doi.org/10.1007/s10959-016-0664-x

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