Weak convergence of the number of vertices at intermediate levels of random recursive trees

TL;DR

This paper investigates the asymptotic distribution of the number of vertices at intermediate levels in random recursive trees, establishing weak convergence to a Gaussian process using probabilistic methods.

Contribution

It proves weak convergence of the process of vertex counts at intermediate levels, connecting the model to Crump-Mode-Jagers branching processes, which is a novel approach.

Findings

Weak convergence to a Gaussian process with covariance (u+v)^{-1}

Extension of previous one-dimensional results to finite-dimensional distributions

Probabilistic proofs leveraging branching process connections

Abstract

Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive tree with $n+1$ vertices. We are interested in the asymptotic behavior of $X_n(k)$ for intermediate levels $k=k_n$ satisfying $k_n\to\infty$ and $k_n=o(\log n)$ as $n\to\infty$. In particular, we prove weak convergence of finite-dimensional distributions for the process $(X_n ([k_n u]))_{u>0}$, properly normalized and centered, as $n\to\infty$. The limit is a centered Gaussian process with covariance $(u,v)\mapsto (u+v)^{-1}$. One-dimensional distributional convergence of $X_n(k_n)$, properly normalized and centered, was obtained with the help of analytic tools by Fuchs, Hwang and Neininger in 2006. In contrast, our proofs which are probabilistic in nature exploit a connection of our model with certain Crump-Mode-Jagers branching processes.