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

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.

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Acknowledgements

I am very grateful to an anonymous referee, especially for him or her pointing out a gap in the original proof of Theorem 4.4, but also for providing many useful suggestions which helped me to improve the presentation of this paper.

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Correspondence to Xin He.

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Supported by the Fundamental Research Funds for the Central Universities (WK0010000063).

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He, X. Local Convergence of Critical Random Trees and Continuous-State Branching Processes. J Theor Probab 35, 685–713 (2022). https://doi.org/10.1007/s10959-021-01074-9

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  • DOI: https://doi.org/10.1007/s10959-021-01074-9

Keywords

  • Galton–Watson tree
  • Lévy tree
  • Conditioning
  • Local limit
  • Immortal tree
  • Height
  • Width
  • Total mass
  • Maximal degree

Mathematics Subject Classifications 2010

  • 60J80
  • 60F17