Critical beta-splitting, via contraction

TL;DR

This work provides an alternative contraction-method proof of the central limit theorem for the height of a uniformly chosen leaf in the critical beta-splitting tree ${\mathcal{T}}_n$, a model at the border between polynomial and poly-logarithmic height growth. The authors show that $H_n^*=(H_n-\mu_n)/\sigma_n$ converges to a standard normal and establish a rate bound in the Zolotarev metric $\zeta_3(H_n^*,Z)=O(\log^{-1/2+\varepsilon} n)$, with extensions to the continuous-time exponential-splitting variant for the time-height $\hat{H}_n$. The analysis carefully handles a borderline case where standard contraction with a trivial fixed point does not apply directly, by refining moment estimates and exploiting recent expansions for $\mu_n$ and $\sigma_n^2$ (due to AJ24b and earlier work by Aldous and Pittel). The work connects to coalescent processes and suggests potential routes for related CLTs in the $\beta(2,b)$-coalescent, and it demonstrates the robustness of contraction methods in driving asymptotic normality for complex recursive trees.

Abstract

The critical beta-splitting tree, introduced by Aldous, is a Markov branching phylogenetic tree. Aldous and Pittel recently proved, amongst other results, a central limit theorem for the height of a random leaf. We give an alternative proof, via contraction methods for random recursive structures. These methods were developed by Neininger and Rüschendorf, motivated by Pittel's article "Normal convergence problem? Two moments and a recurrence may be the clues." Aldous and Pittel estimated the leading order terms in the first two moments. More recently, Aldous and Janson obtained an asymptotic expansion for the average height. We show that a central limit theorem follows, and bound the distance to normality. Our results also apply to the continuous version of the model, in which branching times are exponential.

Figures (1)

• Figure 1: A critical beta-splitting tree with leaves labelled by $\{1,\ldots,23\}$. The location of an internal node indicates when and where the set of leaves above it was split into two subsets. Initially, $\{1,\ldots,23\}$ splits into $\{1,2\}$ and $\{3,\ldots,23\}$. Then, $\{1,2\}$ splits into $\{1\}$ and $\{2\}$; $\{3,\ldots,23\}$ splits into $\{3,4,5\}$ and $\{6,\ldots,23\}$; etc., until only the singleton sets $\{1\},\ldots,\{23\}$ remain.

Theorems & Definitions (17)

• Theorem 1: Aldous and Pittel AP23
• Lemma 2
• proof
• Theorem 3: Neininger and Rüschendorf NR04b
• Theorem 4
• Lemma 5
• proof
• Lemma 6
• proof
• proof : Proof of Theorem \ref{['T_CLT2']}