On large deviation probabilities for empirical distribution of branching random walks with heavy tails

TL;DR

This paper studies the probabilities of large deviations in the empirical distribution of branching random walks with heavy-tailed step sizes or offspring laws, extending previous results to heavy-tailed scenarios.

Contribution

It provides new asymptotic estimates for large deviation probabilities in heavy-tailed branching random walks, completing prior work in the field.

Findings

Derived asymptotic decay rates for large deviations

Extended results to heavy-tailed step sizes and offspring laws

Confirmed the universality of Gaussian limiting behavior under heavy tails

Abstract

Given a branching random walk $(Z_n)_{n\geq0}$ on $\mathbb{R}$, let $Z_n(A)$ be the number of particles located in interval $A$ at generation $n$. It is well known (e.g., \cite{biggins}) that under some mild conditions, $Z_n(\sqrt nA)/Z_n(\mathbb{R})$ converges a.s. to $\nu(A)$ as $n\rightarrow\infty$, where $\nu$ is the standard Gaussian measure. In this work, we investigate its large deviation probabilities under the condition that the step size or offspring law has heavy tail, i.e. the decay rate of $$\mathbb{P}(Z_n(\sqrt nA)/Z_n(\mathbb{R})>p)$$ as $n\rightarrow\infty$, where $p\in(\nu(A),1)$. Our results complete those in \cite{ChenHe} and \cite{Louidor}.