Weak convergence for the minimal position in a branching random walk: A simple proof

Consider the boundary case in a one-dimensional super-critical branching random walk. It is known that upon the survival of the system, the minimal position after n steps behaves in probability like log n when n ¿ 8. We give a simple and self-contained proof of this result, based exclusively on elementary properties of sums of i.i.d. real-valued random variables