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Given a super-critical branching random walk on $\mathbb R$ started from theorigin, let $M_n$ be the maximal position of individuals at the $n$-thgeneration. 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 forsome suitable constants $x^*$ and $\theta^*$. In this work, we investigate itsmoderate deviation, in other words, the convergence rates of$$\mathbb{P}\left(M_n\leq x^*n-\frac{3}{2\theta^*}\log n-\ell_n\right),$$ forany 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 leasttwo. Finally, we apply our techniques to study the small ball probability oflimit of derivative martingale.

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