A short proof of the asymptotic of the minimum of the branching random walk after time $n$

TL;DR

This paper proves that the minimal position of a branching random walk, after proper normalization, converges in distribution to a shifted Gumbel distribution, providing insight into its asymptotic behavior.

Contribution

The paper offers a short proof of the asymptotic distribution of the minimum of a branching random walk in the boundary case.

Findings

$R_n - rac{1}{2} ext{log} n$ converges in law to a shifted Gumbel distribution

Provides a concise proof of the asymptotic behavior of the minimum

Clarifies the distributional limit for the minimal position after time $n$

Abstract

We write $R_n$ for the minimal position attained after time $n$ by a branching random walk in the boundary case. In this article, we prove that $R_n - \frac{1}{2} \log n$ converges in law toward a shifted Gumbel distribution.