Functional limit theorems for the profile of random recursive trees

Let $X_n(k)$ denote the number of vertices at level $k$ in a random recursive tree with $n+1$ vertices. I am going to discuss a functional limit theorem for the vector-valued process $(X_{[n^t]}(1),\ldots, X_{[n^t]}(k))_{t\geq 0}$ , for each $k\in\mathbb{N}$. It will be explained that, after proper centering and normalization, this process converges weakly to a vector-valued Gaussian process whose components are integrated Brownian motions. The other topic of my interest is the asymptotic behavior of $X_n(k)$ for intermediate levels $k=k_n$ satisfying $k_n\to\infty$ and $k_n=o(logn)$ as $n\to\infty$. It turns out that the finite-dimensional distributions of the process $(X_n ([k_n u]))u>0$, properly normalized and centered, converge weakly as $n\to\infty$. The limit is a centered Gaussian process with explicitly known covariance. Both results are deduced from new functional limit theorems for Crump-Mode-Jagers branching processes generated by increasing random walks with increments that have finite second moment. The talk is based on two recent papers joint with Zakhar Kabluchko.