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

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.