  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 (e.g., \cite{biggins}) that under some mild conditions, $Z_n(\sqrt
nA)/Z_n(\mathbb{R})$ converges a.s. to $\nu(A)$ as $n\rightarrow\infty$, where
$\nu$ is the standard Gaussian measure. In this work, we investigate its large
deviation probabilities under the condition that the step size or offspring law
has heavy tail, i.e. the 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 \cite{ChenHe} and \cite{Louidor}.
