Electronic Communications in Probability

A functional limit theorem for the profile of random recursive trees

Alexander Iksanov and Zakhar Kabluchko

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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. 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.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 87, 13 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1542942176

Digital Object Identifier
doi:10.1214/18-ECP188

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G50: Sums of independent random variables; random walks 60C05: Combinatorial probability 60F05: Central limit and other weak theorems

Keywords
branching random walk Crump-Mode-Jagers branching process functional limit theorem integrated Brownian motion low levels profile random recursive tree

Rights
Creative Commons Attribution 4.0 International License.

Citation

Iksanov, Alexander; Kabluchko, Zakhar. A functional limit theorem for the profile of random recursive trees. Electron. Commun. Probab. 23 (2018), paper no. 87, 13 pp. doi:10.1214/18-ECP188. https://projecteuclid.org/euclid.ecp/1542942176


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