How big is the minimum of a branching random walk?

Abstract

Let $M_n$ be the minimal position at generation $n$, of a real-valued branching random walk in the boundary case. As $n\to \infty$, $M_n-\frac{3}{2}\log n$ 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, $\mathrm{limsup}\_n\to \infty \frac{1}{\log \log \log n}(M_n-\frac{3}{2}\log n)=1$ almost surely. We also study the problem of moderate deviations of $M_n$: $p(M_n-\frac{3}{2}\log n>\lambda )$ 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.