[Paper Review] Convergence in law of the minimum of a branching random walk

Élie Aïdékon|Munich Personal RePEc Archive (Ludwig Maximilian University of Munich)|Jan 10, 2011

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper establishes the convergence in distribution of the minimum position in a super-critical branching random walk centered at $\frac{3}{2}\ln n$, proving that the limiting law is characterized by the derivative martingale $D_\infty$. Using the many-to-one lemma and spine decomposition, the authors derive the asymptotic tail behavior of the minimum, showing convergence to a Gumbel-type distribution shifted by the derivative martingale, extending Bramson's classical result for branching Brownian motion to discrete branching random walks under general moment and non-lattice conditions.

ABSTRACT

We consider the minimum of a super-critical branching random walk. Addario-Berry and Reed [Ann. Probab. 37 (2009) 1044-1079] proved the tightness of the minimum centered around its mean value. We show that a convergence in law holds, giving the analog of a well-known result of Bramson [Mem. Amer. Math. Soc. 44 (1983) iv+190] in the case of the branching Brownian motion.

Motivation & Objective

• To establish the convergence in distribution of the minimum position $M_n$ in a super-critical branching random walk around $\frac{3}{2}\ln n$.
• To extend Bramson’s classical result on branching Brownian motion to the discrete branching random walk setting.
• To characterize the limiting distribution of $M_n$ in terms of the derivative martingale $D_\infty$ under general moment and non-lattice conditions.
• To prove that the tail behavior of the minimum is governed by the derivative martingale, with explicit constants derived via spine decomposition and killed branching random walks.

Proposed method

• Uses the many-to-one lemma to relate additive martingales to expectations under a size-biased measure.
• Applies spine decomposition to analyze the path of the typical particle contributing to the minimum.
• Studies the minimum of a killed branching random walk (killed below zero) to derive tail estimates via the derivative martingale.
• Employs a two-stage approximation: first analyzing the killed process to obtain constant $C_1$, then relating it to the original process via a constant $c_0$.
• Uses the derivative martingale $D_n = \sum_{|x|=n} V(x) e^{-V(x)}$, which converges a.s. to $D_\infty > 0$ on non-extinction.
• Applies renewal theory and large deviation estimates for random walks to control the probability of rare paths that achieve the minimum.

Experimental results

Research questions

• RQ1 Does the minimum $M_n$ of a super-critical branching random walk converge in distribution when centered at $\frac{3}{2}\ln n$?
• RQ2 What is the limiting distribution of $M_n - \frac{3}{2}\ln n$ in terms of the derivative martingale $D_\infty$?
• RQ3 How does the tail behavior of $M_n$ relate to the derivative martingale and the killed branching random walk?
• RQ4 Can the convergence in law be established under general moment and non-lattice conditions, without assuming log-concavity or lattice structure?

Key findings

• The limit law of $M_n - \frac{3}{2}\ln n$ is characterized by $\mathbb{E}[\exp(-C^* e^x D_\infty)]$, where $C^* = C_1 c_0$ and $C_1, c_0$ are constants derived from the killed process and spine decomposition.
• The tail probability $\mathbb{P}(M_n < \frac{3}{2}\ln n - z)$ behaves asymptotically as $\frac{C_1 c_0}{z} e^{-z}$ as $n \to \infty$ and $z \to \infty$.
• The derivative martingale $D_n$ converges almost surely to a positive limit $D_\infty$ on non-extinction, which drives the randomness in the limiting Gumbel-type distribution.
• The non-lattice assumption is essential; convergence in law around $\frac{3}{2}\ln n$ fails in the lattice case.
• The result extends Bramson’s theorem for branching Brownian motion to discrete branching random walks under mild moment conditions.
• The proof relies on controlling the probability that particles stay above a moving boundary and reach the minimum, using spine techniques and large deviation estimates.