Some central limit theorems for critical beta-splitting tree

TL;DR

This paper establishes new joint central limit theorems for the heights of leaves in critical beta-splitting trees, connecting discrete and continuous models through an infinite balls-in-boxes scheme.

Contribution

It introduces novel joint CLTs for leaf heights in critical beta-splitting trees, linking discrete and continuous frameworks via a balls-in-boxes connection.

Findings

Joint CLT for leaf heights in discrete critical beta-splitting trees

Joint CLT for heights in discrete and continuous versions

Connection between beta-splitting trees and balls-in-boxes schemes

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.