Lower deviation and moderate deviation probabilities for maximum of a branching random walk

Abstract

Given a super-critical branching random walk on $\mathbb{R}$ started from the origin, let $M_n$ be the maximal position of individuals at the $n$-th generation. Under some mild conditions, it is known from \cite{A13} that as $n\to \infty$, $M_n-x^{\ast }n+\frac{3}{2{\theta }^{\ast }}\log n$ converges in law for some suitable constants $x^{\ast }$ and ${\theta }^{\ast }$. In this work, we investigate its moderate deviation, in other words, the convergence rates of $$\mathbb{P}\left(M_n\le x^{\ast }n-\frac{3}{2{\theta }^{\ast }}\log n-{\ell }_n\right),$$ for any positive sequence $({\ell }_n)$ such that ${\ell }_n=O(n)$ and ${\ell }_n\uparrow \infty$. As a by-product, we also obtain lower deviation of $M_n$; i.e., the convergence rate of \[ \mathbb{P}(M_n\leq xn), \] for $x<x^{\ast }$ in Böttcher case where the offspring number is at least two. Finally, we apply our techniques to study the small ball probability of limit of derivative martingale.

33 pages, 2 figures