A functional limit theorem for the profile of random recursive trees

TL;DR

This paper establishes a functional limit theorem showing that the scaled profile of a random recursive tree converges to a Gaussian process, specifically integrated Brownian motions, using branching process techniques.

Contribution

It introduces a new functional limit theorem for the profile of random recursive trees, linking it to Gaussian processes via branching process methods.

Findings

Convergence of the scaled profile to a Gaussian process.

Identification of the limit as integrated Brownian motions.

Application of branching process theory to tree profiles.

Abstract

Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive tree with $n+1$ vertices. We prove a functional limit theorem for the vector-valued process $(X_{[n^t]}(1),\ldots, X_{[n^t]}(k))_{t\geq 0}$, for each $k\in\mathbb N$. We show that after proper centering and normalization, this process converges weakly to a vector-valued Gaussian process whose components are integrated Brownian motions. This result is deduced from a functional limit theorem for Crump-Mode-Jagers branching processes generated by increasing random walks with increments that have finite second moment.