Lower deviation and moderate deviation probabilities for maximum of a branching random walk

Given a super-critical branching random walk on $\mathbb R$ started from the origin, let $M_n$ be the maximal position of individuals at the $n$-th generation. Under some mild conditions, it is known from \cite{A13} that as $n\rightarrow\infty$, $M_n-x^*n+\frac{3}{2\theta^*}\log n$ converges in law for some suitable constants $x^*$ and $\theta^*$. In this work, we investigate its moderate deviation, in other words, the convergence rates of $$\mathbb{P}\left(M_n\leq x^*n-\frac{3}{2\theta^*}\log n-\ell_n\right),$$ for any positive sequence $(\ell_n)$ such that $\ell_n=O(n)$ and $\ell_n\uparrow\infty$. As a by-product, we also obtain lower deviation of $M_n$; i.e., the convergence rate of \[ \mathbb{P}(M_n\leq xn), \] for $x<x^*$ in B\"{o}ttcher case where the offspring number is at least two. Finally, we apply our techniques to study the small ball probability of limit of derivative martingale.