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

TL;DR

This paper provides a simple, self-contained proof for the asymptotic behavior of the minimal position in a one-dimensional super-critical branching random walk, showing it behaves like (3/2) log n with high probability.

Contribution

It introduces an elementary proof relying solely on properties of sums of i.i.d. random variables, simplifying previous complex approaches.

Findings

Minimal position after n steps behaves like (3/2) log n

Proof is elementary and self-contained

Applicable to boundary case in super-critical branching random walk

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 ${3\over 2} \log n$ when $n\to \infty$. 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.