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PreprintCC BY 4.0arXiv

Uniform integrability of the distance to the nearest leaf in random trees

  • Víctor J. Maciá
  • Benedikt Stufler
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Abstract

We study the distance from the root to the nearest leaf, the analogous quantity for a uniformly chosen vertex, and its protection number, in size-conditioned simply generated trees. We prove a uniform exponential tail bound for each of these quantities, valid for arbitrary offspring distributions. As a consequence, these random variables are uniformly integrable of every order. This yields convergence of all moments to those of the corresponding local limit. The argument is probabilistic and unified across the three quantities.

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Cite

BibTeXRIS
BibTeX
@misc{macia2026uniform,
  title = {{Uniform integrability of the distance to the nearest leaf in random trees}},
  author = {Víctor J. Maciá and Benedikt Stufler},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.15339},
  eprint = {2606.15339},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.15339},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Uniform integrability of the distance to the nearest leaf in random trees
AU  - Víctor J. Maciá
AU  - Benedikt Stufler
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2606.15339
UR  - https://arxiv.org/abs/2606.15339
AB  - We study the distance from the root to the nearest leaf, the analogous quantity for a uniformly chosen vertex, and its protection number, in size-conditioned simply generated trees. We prove a uniform exponential tail bound for each of these quantities, valid for arbitrary offspring distributions. As a consequence, these random variables are uniformly integrable of every order. This yields convergence of all moments to those of the corresponding local limit. The argument is probabilistic and unified across the three quantities.
ER  -