The Annals of Applied Probability

A functional central limit theorem for branching random walks, almost sure weak convergence and applications to random trees

Rudolf Grübel and Zakhar Kabluchko

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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.

Article information

Ann. Appl. Probab., Volume 26, Number 6 (2016), 3659-3698.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles 60B10: Convergence of probability measures 68P10: Searching and sorting 60G42: Martingales with discrete parameter

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


Grübel, Rudolf; Kabluchko, Zakhar. A functional central limit theorem for branching random walks, almost sure weak convergence and applications to random trees. Ann. Appl. Probab. 26 (2016), no. 6, 3659--3698. doi:10.1214/16-AAP1188.

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