Asymptotics for the harmonic descent chain and applications to critical beta-splitting trees

TL;DR

This paper analyzes the asymptotic behavior of the harmonic descent chain, a recursive sequence linked to phylogenetic trees, establishing convergence rates and implications for statistics of critical beta-splitting trees.

Contribution

It provides the first detailed asymptotic analysis of the harmonic descent chain and derives central limit theorems for related phylogenetic tree statistics.

Findings

Proved the convergence rate of the harmonic descent chain.

Established the asymptotic form of the sequence with an implicit exponent.

Derived central limit theorems for statistics of critical beta-splitting trees.

Abstract

Motivated by the connection to a probabilistic model of phylogenetic trees introduced by Aldous, we study the recursive sequence governed by the rule $x_n = \sum_{i=1}^{n-1} \frac{1}{h_{n-1}(n-i)} x_i$ where $h_{n-1} = \sum_{j=1}^{n-1} 1/j$, known as the harmonic descent chain. While it is known that this sequence converges to an explicit limit $x$, not much is known about the rate of convergence. We first show that a class of recursive sequences including the above are decreasing and use this to bound the rate of convergence. Moreover, for the harmonic descent chain we prove the asymptotic $x_n - x = n^{-\gamma_* + o(1)}$ for an implicit exponent $\gamma_*$. As a consequence, we deduce central limit theorems for various statistics of the critical beta-splitting random tree. This answers a number of questions of Aldous, Janson, and Pittel.