Stable-like fluctuations of Biggins' martingales

TL;DR

This paper studies the asymptotic fluctuations of Biggins' martingales in supercritical branching random walks, showing convergence to an $ ext{alpha}$-stable process under certain conditions.

Contribution

It establishes a new stable-like fluctuation limit for Biggins' martingales when the offspring distribution is in the domain of attraction of an $ ext{alpha}$-stable law.

Findings

Weak convergence of the tail process to an $ ext{alpha}$-stable autoregressive process.

Identification of the normalization needed for convergence.

Extension of classical results to stable-like fluctuation regimes.

Abstract

Let $(W_n(\theta))_{n \in \mathbb{N}_0}$ be Biggins' martingale associated with a supercritical branching random walk, and let $W(\theta)$ be its almost sure limit. Under a natural condition for the offspring point process in the branching random walk, we show that if the law of $W_1(\theta)$ belongs to the domain of normal attraction of an $\alpha$-stable distribution for some $\alpha \in (1,2)$, then, as $n\to\infty$, there is weak convergence of the tail process $(W(\theta) - W_{n-k}(\theta))_{k \in \mathbb{N}_0}$, properly normalized, to a random scale multiple of a stationary autoregressive process of order one with $\alpha$-stable marginals.