Tightness for the minimal displacement of branching random walk

Recursion equations have been used to establish weak laws of large numbers for the minimal displacement of branching random walk in one dimension. Here, we use these equations to establish the tightness of the corresponding sequences after appropriate centering. These equations are special cases of recursion equations that arise naturally in the study of random variables on tree-like structures. Such recursion equations are investigated in detail, in Bramson and Zeitouni (2006 Preprintmath.PR/0612382v1), in a general context. Here, we restrict ourselves to investigating the more concrete setting of branching random walk, and provide motivation for the rigorous arguments that are given in Bramson and Zeitouni. We also discuss briefly the cover time of symmetric simple random walk on regular binary trees, which is another application of the more general recursion equations.