Consistent Minimal Displacement of Branching Random Walks

Let $\mathbb{T}$ denote a rooted $b$-ary tree and let ${S_v}_{v\in $\mathbb{T}$}$ denote a branching random walk indexed by the vertices of the tree, where the increments are i.i.d. and possess a logarithmic moment generating function $\mathrm{\Lambda }(\cdot )$. Let $m_n$ denote the minimum of the variables $S_v$ over all vertices at the $n$th generation, denoted by ${\mathbb{D}}_n$. Under mild conditions, $m_n\mathrm{/}n$ converges almost surely to a constant, which for convenience may be taken to be 0. With $\bar S_v=\max{S_w:{\rm $w$ is on the geodesic connecting the root to $v$}}$, define $L_n={\min }_{v\in {\mathbb{D}}_n}{\bar{S}}_v$. We prove that $L_n\mathrm{/}{n}^{1\mathrm{/}3}$ converges almost surely to an explicit constant $l_0$. This answers a question of Hu and Shi.