Upper deviation probabilities for level sets of a supercritical branching random walk

Given a supercritical branching random walk $\{Z_n{\}}_{n\ge 0}$ on $\mathbb{R}$, let $Z_n([y,\mathrm{\infty }))$ be the number of particles located in $[y,\mathrm{\infty })\subset \mathbb{R}$ at generation $n$. Let $m$ be the mean of the offspring law of $\{Z_n{\}}_{n\ge 0}$ and $I(x)$ be the large deviation rate function of the underlying random walk of $\{Z_n{\}}_{n\ge 0}$. It is known from [6] that under some mild conditions, for $x\in (0,x^{\ast })$, $n^{-1}\log Z_n([nx,\mathrm{\infty }))$ converges almost surely to $\log m-I(x)$ on the event of nonextinction as $n\to \mathrm{\infty }$, where $x^{\ast }$ is the speed of maximal position of the branching random walk. In this work, we investigate its upper deviation probabilities, in other words, the convergence rates of $$\mathbb{P}(Z_n([xn,\mathrm{\infty }))\ge e^{an})$$ as $n\to \mathrm{\infty }$, where $x>0$ and $a>(\log m-I(x){)}^{+}$. This paper is a counterpart work of the lower deviation probabilities [28] and also completes those results in [1] for the branching Brownian motion.