Berry-Esseen bound and Cramér moderate deviation expansion for a supercritical branching random walk

Abstract

We consider a supercritical branching random walk where each particle gives birth to a random number of particles of the next generation, which move on the real line, according to a fixed law. Let $Z_n$ be the counting measure which counts the number of particles of n-th generation situated in a given region. Under suitable conditions, we establish a Berry-Esseen bound and a Cramér type moderate deviation expansion for $Z_n$ with suitable norming.