Lower deviation probabilities for level sets of the branching random walk

TL;DR

This paper studies the probabilities of lower deviations in the number of particles in the upper tail of a branching random walk, providing convergence rate results that extend previous research.

Contribution

It offers new insights into the lower deviation probabilities of level sets in branching random walks, completing and extending prior work in the field.

Findings

Derived convergence rates for lower deviation probabilities

Extended existing results to broader parameter ranges

Provided a comprehensive analysis of tail behavior in branching random walks

Abstract

Given a branching random walk$\{Z_n\}_{n\geq0}$ on $\mathbb{R}$, let $Z_n([y,\infty))$ be the number of particles located in $[y,\infty)$ at generation $n$. It is known from \cite{Biggins1977} that under some mild conditions, $n^{-1}\log Z_n([\theta x^* n,\infty))$ converges a.s. to $\log m-I(\theta x^*)$, where $\log m-I(\theta x^*)$ is a positive constant. In this work, we investigate its lower deviation, in other words, the convergence rates of $$\mathbb{P}\left(Z_n([\theta x^* n,\infty))<e^{an}\right),$$ where $a\in[0,\log m-I(\theta x^*))$. Our results complete those in \cite{Mehmet}, \cite{Helower} and \cite{GWlower}.