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

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 $k_n\to \mathrm{\infty }$ and $k_n=o(\log n)$ as $n\to \mathrm{\infty }$. 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 $n\to \mathrm{\infty }$. 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.