How big is the minimum of a branching random walk?

TL;DR

This paper establishes a law of iterated logarithm for the maximum displacement in a branching random walk, revealing its asymptotic behavior and analyzing moderate deviations related to Mandelbrot cascades.

Contribution

It introduces a law of iterated logarithm for the upper limits of the minimal position in a branching random walk, a novel result in this context.

Findings

The limsup of the centered minimal position scaled by log log log n equals 1 almost surely.

Provides asymptotic estimates for moderate deviations of the minimal position.

Links the deviations to small deviations in Mandelbrot's cascades.

Abstract

Let $M_n$ be the minimal position at generation $n$, of a real-valued branching random walk in the boundary case. As $n \to \infty$, $M_n- {3 \over 2} \log n$ is tight (see [1][9][2]). We establish here a law of iterated logarithm for the upper limits of $M_n$: upon the system's non-extinction, $ \limsup\_{n\to \infty} {1\over \log \log \log n} ( M_n - {3\over2} \log n) = 1$ almost surely. We also study the problem of moderate deviations of $M_n$: $p(M_n- {3 \over 2} \log n > \lambda)$ for $\lambda\to \infty$ and $\lambda=o(\log n)$. This problem is closely related to the small deviations of a class of Mandelbrot's cascades.