Consistent Minimal Displacement of Branching Random Walks (0912.1392v1)

Abstract: 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 $\Lambda(\cdot)$. 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, $m_n/n$ 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 $L_n/n^{1/3}$ converges almost surely to an explicit constant $l_0$. This answers a question of Hu and Shi.