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

Rudolf Grübel, Zakhar Kabluchko|arXiv (Cornell University)|Oct 2, 2014

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper establishes a functional central limit theorem for branching random walks, proving almost sure weak convergence of conditional distributions to a Gaussian analytic function with random variance. It applies this result to derive central limit theorems for path lengths in random trees, recovering and strengthening Neininger's result on binary search trees by replacing weak convergence with almost sure weak convergence, and extending it to uniform random recursive trees.

ABSTRACT

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 $n\ o\\infty$ the process $$ D_n(u):= m^{\\frac 12 n} \\left(W_{\\infty}\\left(\\frac{u}{\\sqrt n}\ ight) - W_{n}\\left(\\frac{u}{\\sqrt n}\ ight) \ ight) $$ 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 [Refined Quicksort Asymptotics, Rand. Struct. and Alg., to appear], 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 $$ L\\left\\{\\sqrt{\\frac{n}{2\\log n}} \\left(EPL_{\\infty} - \\frac{EPL_n-2n\\log n}{n}\ ight)\\Bigg | G_{n}\ ight\\} \ o \\{\\omega\\mapsto N_{0,1}\\}, \\quad \ ext{a.s.w.},$$ where $EPL_n$ is the external path length of a binary search tree $X_n$ with $n$ vertices, $EPL_{\\infty}$ is the limit of the R\\'egnier martingale, and $L(\\,\\cdot\\, |G_n)$ denotes the conditional distribution w.r.t. the $\\sigma$-algebra $G_n$ generated by $X_1,\\ldots,X_n$. A.s.w. 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 P\\'olya urn.

Motivation & Objective

• To establish a functional central limit theorem for the Biggins martingale in supercritical branching random walks.
• To develop a framework for almost sure weak convergence of probability transition kernels, stronger than weak and stable convergence.
• To apply the functional CLT to derive second-order asymptotics for the total path length in random binary search trees and uniform random recursive trees.
• To refine Neininger's recent central limit theorem for binary search trees by replacing weak convergence with almost sure weak convergence.
• To prove that the limit distribution of normalized path length differences converges almost surely weakly to a standard normal distribution, conditionally on the tree history.

Proposed method

• Derive a functional central limit theorem for the process $ D_n(u) = m^{n/2} \left( W_\infty(u/\sqrt{n}) - W_n(u/\sqrt{n}) \right) $, showing weak convergence to a Gaussian analytic function with random variance.
• Use the convergence of $ D_n(u) $ to establish almost sure weak convergence of conditional distributions for path length functionals in random trees.
• Apply the law of the iterated logarithm to the Yule process and Poisson point process to obtain precise asymptotics for the time $ T_n $ when the $ n $-th particle is born.
• Decompose the normalized path length difference into three terms, showing two vanish almost surely and the third converges via the functional CLT.
• Leverage the product probability space and conditional independence to reduce the conditioning from $ \mathcal{F}_{T_n} $ to $ \mathcal{G}_n $, the history of the tree process.
• Use Proposition 4.10 and Proposition 4.12 to transfer convergence from the product space to the original probability space with almost sure weak convergence.

Experimental results

Research questions

• RQ1 Does the difference between the limit of the Biggins martingale and its finite-time version satisfy a functional central limit theorem?
• RQ2 Can almost sure weak convergence of conditional distributions be established for path length functionals in random trees?
• RQ3 Does the central limit theorem for the external path length in binary search trees hold under stronger convergence than weak convergence?
• RQ4 Is the limit distribution of normalized path length fluctuations almost surely weakly convergent to a standard normal distribution?
• RQ5 Can the functional CLT for branching random walks be extended to derive second-order asymptotics for random recursive trees?

Key findings

• The process $ D_n(u) = m^{n/2} \left( W_\infty(u/\sqrt{n}) - W_n(u/\sqrt{n}) \right) $ converges weakly to a Gaussian analytic function with random variance on a suitable space of analytic functions.
• For binary search trees, the conditional distribution $ \mathcal{L}\left\{ \sqrt{\frac{n}{2\log n}} \left( \text{EPL}_\infty - \frac{\text{EPL}_n - 2n\log n}{n} \right) \Big| \mathcal{G}_n \right\} $ converges almost surely weakly to the standard normal distribution.
• The convergence is stronger than weak and stable convergence, establishing almost sure weak convergence of transition kernels in the context of random trees.
• The result extends to uniform random recursive trees, proving a similar central limit theorem with almost sure weak convergence.
• The proof relies on precise asymptotics for $ T_n $, the time of the $ n $-th particle birth, using the law of the iterated logarithm for Poisson processes and Yule processes.
• The decomposition of the normalized path length difference into three terms shows that two terms vanish almost surely, while the third converges via the functional CLT, leading to the final convergence result.