A note on large deviation probabilities for empirical distribution of branching random walks

Abstract

We consider a branching random walk on R started from the origin. Let Zn(⋅) be the counting measure which counts the number of individuals at the nth generation located in a given set. For any interval A⊂R, it is well known that Zn(nA)Zn(R) converges a.s. to ν(A) under some mild conditions, where ν is the standard Gaussian measure. In this note, we study the convergence rate of PZ̄nnσ2A−ν(A)≥Δ,for a small constant Δ∈(0,1−ν(A)). Our work completes the results in Chen and He (2017) and Louidor and Perkins (2015), where the step size of the underlying walk is assumed to have Weibull tail, Gumbel tail or be bounded.