Open Access
December 2016 A functional central limit theorem for branching random walks, almost sure weak convergence and applications to random trees
Rudolf Grübel, Zakhar Kabluchko
Ann. Appl. Probab. 26(6): 3659-3698 (December 2016). DOI: 10.1214/16-AAP1188


Let $W_{\infty}(\beta)$ be the limit of the Biggins martingale $W_{n}(\beta)$ associated to a supercritical branching random walk with mean number of offspring $m$. We prove a functional central limit theorem stating that as $n\to\infty$ the process

\[D_{n}(u):=m^{\frac{1}{2}n}(W_{\infty}(\frac{u}{\sqrt{n}})-W_{n}(\frac{u}{\sqrt{n}}))\] converges weakly, on a suitable space of analytic functions, to a Gaussian random analytic function with random variance. Using this result, we prove central limit theorems for the total path length of random trees. In the setting of binary search trees, we recover a recent result of R. Neininger [Random Structures Algorithms 46 (2015) 346–361], but we also prove a similar theorem for uniform random recursive trees. Moreover, we replace weak convergence in Neininger’s theorem by the almost sure weak (a.s.w.) convergence of probability transition kernels. In the case of binary search trees, our result states that

\[\mathcal{L}\{\sqrt{\frac{n}{2\log n}}(\operatorname{EPL}_{\infty}-\frac{\operatorname{EPL}_{n}-2n\log n}{n})\Big |\mathcal{G}_{n}\}\overset{\mathrm{a.s.w.}}{\underset{n\to\infty}\longrightarrow}\{\omega \mapsto\mathcal{N}_{0,1}\},\] where $\operatorname{EPL}_{n}$ is the external path length of a binary search tree $X_{n}$ with $n$ vertices, $\operatorname{EPL}_{\infty}$ is the limit of the Régnier martingale and $\mathcal{L}\{\cdot |\mathcal{G}_{n}\}$ denotes the conditional distribution w.r.t. the $\sigma$-algebra $\mathcal{G}_{n}$ generated by $X_{1},\ldots,X_{n}$. Almost sure weak convergence is stronger than weak and even stable convergence. We prove several basic properties of the a.s.w. convergence and study a number of further examples in which the a.s.w. convergence appears naturally. These include the classical central limit theorem for Galton–Watson processes and the Pólya urn.


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Rudolf Grübel. Zakhar Kabluchko. "A functional central limit theorem for branching random walks, almost sure weak convergence and applications to random trees." Ann. Appl. Probab. 26 (6) 3659 - 3698, December 2016.


Received: 1 May 2015; Revised: 1 September 2015; Published: December 2016
First available in Project Euclid: 15 December 2016

zbMATH: 1367.60028
MathSciNet: MR3582814
Digital Object Identifier: 10.1214/16-AAP1188

Primary: 60J80
Secondary: 60B10 , 60F05 , 60F17 , 60G42 , 68P10

Keywords: almost sure weak convergence , Binary search trees , Branching random walk , functional central limit theorem , Galton–Watson processes , Gaussian analytic function , mixing convergence , Pólya urns , Quicksort distribution , Random recursive trees , stable convergence

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.26 • No. 6 • December 2016
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