On large-deviation probabilities for the empirical distribution of branching random walks with heavy tails

Given a branching random walk $(Z_n)_{n\geq0}$ on $\mathbb{R}$ , let $Z_n(A)$ be the number of particles located in interval A at generation n. It is well known that under some mild conditions, $Z_n(\sqrt nA)/Z_n(\mathbb{R})$ converges almost surely to $\nu(A)$ as $n\rightarrow\infty$ , where $\nu$ is the standard Gaussian measure. We investigate its large-deviation probabilities under the condition that the step size or offspring law has a heavy tail, i.e. a decay rate of $\mathbb{P}(Z_n(\sqrt nA)/Z_n(\mathbb{R})>p)$ as $n\rightarrow\infty$ , where $p\in(\nu(A),1)$ . Our results complete those in Chen and He (2019) and Louidor and Perkins (2015).