A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk

TL;DR

This paper establishes a central limit theorem and a law of the iterated logarithm for the Biggins martingale in supercritical branching random walks, under certain integrability conditions, advancing the understanding of their asymptotic behavior.

Contribution

It introduces new limit theorems for the Biggins martingale, specifically a functional CLT and a law of the iterated logarithm, under conditions of uniform integrability and finite variance.

Findings

Proves a functional central limit theorem for the tail process of the martingale.

Establishes a law of the iterated logarithm for the difference between the martingale limit and its partial sums.

Provides conditions under which these asymptotic results hold.

Abstract

Let $(W_n(\theta))_{n\in\mathbb N_0}$ be the Biggins martingale associated with a supercritical branching random walk and denote by $W_\infty(\theta)$ its limit. Assuming essentially that the martingale $(W_n(2\theta))_{n\in\mathbb N_0}$ is uniformly integrable and that $\text{Var} W_1(\theta)$ is finite, we prove a functional central limit theorem for the tail process $(W_\infty(\theta) - W_{n+r}(\theta))_{r\in\mathbb N_0}$ and a law of the iterated logarithm for $W_\infty(\theta)-W_n(\theta)$, as $n\to\infty$.