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Local Convergence of Critical Random Trees and Continuous-State Branching Processes

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  • Xin He

    (University of Science and Technology of China)

Abstract

We study the local convergence of critical Galton–Watson trees and Lévy trees under various conditionings. Assuming a very general monotonicity property on the measurable functions of critical random trees, we show that random trees conditioned to have large function values always converge locally to immortal trees. We also derive a very general ratio limit property for measurable functions of critical random trees satisfying the monotonicity property. Finally we study the local convergence of critical continuous-state branching processes, and prove a similar result.

Suggested Citation

  • Xin He, 2022. "Local Convergence of Critical Random Trees and Continuous-State Branching Processes," Journal of Theoretical Probability, Springer, vol. 35(2), pages 685-713, June.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:2:d:10.1007_s10959-021-01074-9
    DOI: 10.1007/s10959-021-01074-9
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    References listed on IDEAS

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    1. Robin Stephenson, 2018. "Local Convergence of Large Critical Multi-type Galton–Watson Trees and Applications to Random Maps," Journal of Theoretical Probability, Springer, vol. 31(1), pages 159-205, March.
    2. Xin He, 2017. "Conditioning Galton–Watson Trees on Large Maximal Outdegree," Journal of Theoretical Probability, Springer, vol. 30(3), pages 842-851, September.
    3. Romain Abraham & Jean-François Delmas & Hongsong Guo, 2018. "Critical Multi-type Galton–Watson Trees Conditioned to be Large," Journal of Theoretical Probability, Springer, vol. 31(2), pages 757-788, June.
    Full references (including those not matched with items on IDEAS)

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