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

TL;DR

This paper studies the probabilities of deviations for the maximum position in a super-critical branching random walk, focusing on moderate and lower deviations, and also explores small ball probabilities of the derivative martingale's limit.

Contribution

It provides new results on the convergence rates of moderate and lower deviation probabilities for the maximum in branching random walks, extending understanding beyond known law of large numbers.

Findings

Established the convergence rates for moderate deviations of M_n.

Derived the lower deviation probabilities for M_n in the Böttcher case.

Analyzed small ball probabilities of the derivative martingale's limit.

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 $n\rightarrow\infty$, $M_n-x^*n+\frac{3}{2\theta^*}\log n$ converges in law for some suitable constants $x^*$ and $\theta^*$. In this work, we investigate its moderate deviation, in other words, the convergence rates of $$\mathbb{P}\left(M_n\leq x^*n-\frac{3}{2\theta^*}\log n-\ell_n\right),$$ 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 $x<x^*$ in B\"{o}ttcher 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.