On large deviation probabilities for empirical distribution of branching random walks: Schr{\"o}der case and B{\"o}ttcher case

TL;DR

This paper studies the probabilities of large deviations in the empirical distribution of super-critical branching random walks, focusing on convergence rates in Schr{"o}der and B{"o}ttcher cases, extending known Gaussian convergence results.

Contribution

It provides new insights into the convergence rates of large deviation probabilities for empirical measures in branching random walks under different cases.

Findings

Derived bounds for large deviation probabilities in Schr{"o}der case.

Derived bounds for large deviation probabilities in B{"o}ttcher case.

Extended classical Gaussian convergence results to large deviation regimes.

Abstract

Given a super-critical branching random walk on $\mathbb{R}$ started from the origin, let $Z\_n(\cdot)$ be the counting measure which counts the number of individuals at the $n$-th generation located in a given set. Under some mild conditions, it is known in \cite{B90} that for any interval $A\subset \mathbb{R}$, $\frac{Z\_n(\sqrt{n}A)}{Z\_n(\mathbb{R})}$ converges a.s. to $\nu(A)$, where $\nu$ is the standard Gaussian measure. In this work, we investigate the convergence rates of $$\mathbb{P}\left(\frac{Z\_n(\sqrt{n}A)}{Z\_n(\mathbb{R})}-\nu(A)>\Delta\right),$$ for $\Delta\in (0, 1-\nu(A))$, in both Schr{\"o}der case and B{\"o}ttcher case.