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

Part of: Limit theorems Markov processes Stochastic processes Combinatorial probability

Published online by Cambridge University Press:  16 January 2019

Alexander Iksanov  and
Zakhar Kabluchko
Alexander Iksanov*
Affiliation:
Taras Shevchenko National University of Kyiv
Zakhar Kabluchko*
Affiliation:
Westfälische Wilhelms-Universität Münster
*
* Postal address: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine. Email address: iksan@univ.kiev.ua
** Postal address: Institut für Mathematische Statistik, Westfälische Wilhelms-Universität Münster, 48149 Münster, Germany. Email address: zakhar.kabluchko@uni-muenster.de
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Abstract

Let Xn(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 Xn(k) for intermediate levels k=kn satisfying kn→∞ and kn=o(logn) as n→∞. In particular, we prove weak convergence of finite-dimensional distributions for the process (Xn ([knu]))u>0, properly normalized and centered, as n→∞. The limit is a centered Gaussian process with covariance (u,v)↦(u+v)−1. One-dimensional distributional convergence of Xn(kn), properly normalized and centered, was obtained with the help of analytic tools by Fuchs et al. (2006). In contrast, our proofs, which are probabilistic in nature, exploit a connection of our model with certain Crump–Mode–Jagers branching processes.

Keywords

Crump–Mode–Jagers branching processGaussian processintermediate levelrandom recursive treeweak convergence

MSC classification

Primary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G50: Sums of independent random variables; random walks 60C05: Combinatorial probability
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 4 , December 2018 , pp. 1131 - 1142
Copyright
Copyright © Applied Probability Trust 2018 

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