Moderate deviation probabilities for empirical distribution of the branching random walk

Given a super-critical branching random walk {Zn}n >= 0 on R, let Zn(A) be the number of particles located in some Borel set A subset of R at generation n. Under some mild conditions, it's well-known (e.g., [4]) that Zn(v/nA)/Zn(R) converges almost surely to nu(A) as n-+ infinity, where nu(center dot) is the standard Gaussian measure on R. Large deviation probabilities of Zn(v/nA)/Zn(R) have been well studied (see [10] and [27]). In this work, we investigate its moderate deviation probabilities, i.e. the convergence rate of P(Zn(v/nA)/Zn(R) >= nu(A) + Delta n) as n-+ infinity, where {Delta n}n >= 0 is some positive sequence tending to zero.(c) 2023 Elsevier Inc. All rights reserved.