Minimal Positions in a Branching Random Walk

We consider a branching random walk on the real line, with mean family size greater than 1. Let Bₙ denote the minimal position of a member of the nth generation. It is known that (under a weak condition) there is a finite constant γ, defined in terms of the distributions specifying the process, such that as n → ∞, we have Bₙ = γ n + o(n) a.s. on the event S of ultimate survival. Our results here show that (under appropriate conditions), on S the random variable Bₙ is strongly concentrated and the $o(n)$ error term may be replaced by O(log n).