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

Abstract

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 $$ \frac{3} {2} $$ log n when n → ∞. 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.