Fluctuations for the number of records on subtrees of the Continuum Random Tree
type: Dataset
description: We study the asymptotic behavior af the number of cuts $X(T_n)$ needed to isolate the root in a rooted binary random tree $T_n$ with $n$ leaves. We focus on the case of subtrees of the Continuum Random Tree generated by uniform sampling of leaves. We elaborate on a recent result by Abraham and Delmas, who showed that $X(T_n)/\sqrt{2n}$ converges a.s. towards a Rayleigh-distributed random variable $\Theta$, which gives a continuous analog to an earlier result by Janson on conditioned, finite-variance Galton-Watson trees. We prove a convergence in distribution of $n^{-1/4}(X(T_n)-\sqrt{2n}\Theta)$ towards a random mixture of Gaussian variables. The proofs use martingale limit theory for random processes defined on the CRT, related to the theory of records of Poisson point processes.
issued: 2026-05-29T16:47:21.402328
language: en
modified: 2026-05-29T16:47:21.402335
contact point: Organization
  fn: CCSD
keyword: 60f05--60g55--60j80
keyword: central-limit-theorem
keyword: continuum-random-tree
keyword: infoeu-reposemanticspreprint
keyword: mathmath-prmathematics-mathprobability-mathpr
keyword: point-processes
keyword: preprints-working-papers-
keyword: records
landing page: https://hal.science/hal-00768343

Fluctuations for the number of records on subtrees of the Continuum Random Tree
type: Distribution
format: HTML
issued: 2026-05-29T16:47:21.456050
modified: 2026-05-29T16:47:21.374061
access URL: https://hal.science/hal-00768343

test_moissonnage_selune
type: Agent