Open Access
2018 A functional limit theorem for the profile of random recursive trees
Alexander Iksanov, Zakhar Kabluchko
Electron. Commun. Probab. 23: 1-13 (2018). DOI: 10.1214/18-ECP188


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. Let $Y_k(t)$ be the number of the $k$th generation individuals born at times $\leq t$ in this process. Then, it is shown that the appropriately centered and normalized vector-valued process $(Y_{1}(st),\ldots , Y_k(st))_{t\geq 0}$ converges weakly, as $s\to \infty $, to the same limiting Gaussian process as above.


Download Citation

Alexander Iksanov. Zakhar Kabluchko. "A functional limit theorem for the profile of random recursive trees." Electron. Commun. Probab. 23 1 - 13, 2018.


Received: 14 January 2018; Accepted: 4 November 2018; Published: 2018
First available in Project Euclid: 23 November 2018

zbMATH: 07023473
MathSciNet: MR3882228
Digital Object Identifier: 10.1214/18-ECP188

Primary: 60F17 , 60J80
Secondary: 60C05 , 60F05 , 60G50

Keywords: Branching random walk , Crump-Mode-Jagers branching process , Functional limit theorem , Integrated Brownian motion , low levels , Profile , Random recursive tree

Back to Top