Consistent Minimal Displacement of Branching Ran- Dom Walks

Let T denote a rooted b-ary tree and let {Sv}v∈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 Λ(·). Let mn denote the minimum of the variables Sv over all vertices at the nth generation, denoted by Dn. Under mild conditions, mn/n converges almost surely to a constant, which for convenience may be taken to be 0. With S̄v =max{Sw : w is on the geodesic connecting the root to v}, define Ln = minv∈Dn S̄v . We prove that Ln/n 1/3 converges almost surely to an explicit constant l0. This answers a question of Hu and Shi.