Electronic Journal of Probability

Asymptotics of heights in random trees constructed by aggregation

Bénédicte Haas

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Abstract

To each sequence $(a_n)$ of positive real numbers we associate a growing sequence $(T_n)$ of continuous trees built recursively by gluing at step $n$ a segment of length $a_n$ on a uniform point of the pre–existing tree, starting from a segment $T_1$ of length $a_1$. Previous works [5, 10] on that model focus on the influence of $(a_n)$ on the compactness and Hausdorff dimension of the limiting tree. Here we consider the cases where the sequence $(a_n)$ is regularly varying with a non–negative index, so that the sequence $(T_n)$ explodes. We determine the asymptotics of the height of $T_n$ and of the subtrees of $T_n$ spanned by the root and $\ell $ points picked uniformly at random and independently in $T_n$, for all $\ell \in \mathbb N$.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 21, 25 pp.

Dates
Received: 23 June 2016
Accepted: 26 January 2017
First available in Project Euclid: 21 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1487646307

Digital Object Identifier
doi:10.1214/17-EJP31

Mathematical Reviews number (MathSciNet)
MR3622891

Zentralblatt MATH identifier
1358.05054

Subjects
Primary: 05C05: Trees 60J05: Discrete-time Markov processes on general state spaces

Keywords
random trees line–breaking uniform recursive trees

Rights
Creative Commons Attribution 4.0 International License.

Citation

Haas, Bénédicte. Asymptotics of heights in random trees constructed by aggregation. Electron. J. Probab. 22 (2017), paper no. 21, 25 pp. doi:10.1214/17-EJP31. https://projecteuclid.org/euclid.ejp/1487646307


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