On large deviation probabilities for empirical distribution of supercritical branching random walks with unbounded displacements

Abstract

Given a branching random walk on $${\mathbb {R}}$$ started from the origin, where the tail of the branching law decays at least exponentially fast and the offspring number is at least one, 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 (Biggins in Stoch. Process. Appl. 34:255–274, 1990) 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 $$\begin{aligned} {\mathbb {P}}\left( \frac{Z_n(\sqrt{n}A)}{Z_n({\mathbb {R}})}-\nu (A)>\Delta \right) , \end{aligned}$$ for $$\Delta \in (0, 1-\nu (A))$$ . We consider both the Schroder case, where the offspring number could be one, and the Bottcher case, where the offspring number is at least two.