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Some central limit theorems for critical beta-splitting tree

Alexander Iksanov, Anatolii Nikitin, Roman Yakymiv

TL;DR

The paper studies critical beta-splitting trees and their connection to infinite occupancy schemes, proving joint central limit theorems for leaf-heights in both discrete and continuous versions. By coupling a balls-in-boxes scheme to a subordinator with Lévy measure $\nu$, the authors derive a unified Brownian-limit framework for $(L_{n,1},\ldots,L_{n,j},L_n,D_n)$ and establish explicit Gaussian mixtures for the coordinates $L_{n,r}$ with constants expressed through $\zeta(2)$ and $\zeta(3)$; these results hold jointly with their occupancy counterparts. The approach hinges on distributional equalities, Poissonization, and a mix of Skorokhod-space convergence and occupancy-limit theorems (notably Gnedin et al.), yielding a coherent bridge between discrete and continuous critical beta-splitting trees. The findings illuminate the asymptotic structure of tree heights and deepen the connection between random trees and infinite-occupancy models, with potential implications for related stochastic-tree models and limit theorems.

Abstract

We further explore a connection initially unveiled in Iksanov (2025) between critical beta-splitting trees and infinite `balls-in-boxes' schemes. Using the connection, we derive a new joint central limit theorem for components of the height of a leaf chosen uniformly at random in the discrete version of a critical beta-splitting tree. Also, we obtain a joint central limit theorem for the heights in the discrete and continuous versions of a critical beta-splitting tree.

Some central limit theorems for critical beta-splitting tree

TL;DR

The paper studies critical beta-splitting trees and their connection to infinite occupancy schemes, proving joint central limit theorems for leaf-heights in both discrete and continuous versions. By coupling a balls-in-boxes scheme to a subordinator with Lévy measure , the authors derive a unified Brownian-limit framework for and establish explicit Gaussian mixtures for the coordinates with constants expressed through and ; these results hold jointly with their occupancy counterparts. The approach hinges on distributional equalities, Poissonization, and a mix of Skorokhod-space convergence and occupancy-limit theorems (notably Gnedin et al.), yielding a coherent bridge between discrete and continuous critical beta-splitting trees. The findings illuminate the asymptotic structure of tree heights and deepen the connection between random trees and infinite-occupancy models, with potential implications for related stochastic-tree models and limit theorems.

Abstract

We further explore a connection initially unveiled in Iksanov (2025) between critical beta-splitting trees and infinite `balls-in-boxes' schemes. Using the connection, we derive a new joint central limit theorem for components of the height of a leaf chosen uniformly at random in the discrete version of a critical beta-splitting tree. Also, we obtain a joint central limit theorem for the heights in the discrete and continuous versions of a critical beta-splitting tree.
Paper Structure (4 sections, 7 theorems, 50 equations)

This paper contains 4 sections, 7 theorems, 50 equations.

Table of Contents

  1. Setting and result
  2. Our strategy
  3. Proofs
  4. Appendix

Key Result

Proposition 1.1

As $n\to\infty$, where $\zeta(s):=\sum_{n\geq 1}n^{-s}$ for $s>1$, and

Theorems & Definitions (12)

  • Proposition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • proof : Proof of Proposition \ref{['prop:eqdistr']}
  • proof : Proof of Theorem \ref{['thm:31a']}
  • proof : Proof of Proposition \ref{['prop:gnedin']}
  • Lemma 4.1
  • ...and 2 more